Algebra 2 With Trigonometry Honors Classroom Policies
Mrs. Gapinski
Materials: Everyday you will need to be prepared and organized. I will expect that you have the following
supplies:
1. Required Foerster textbook
2. Three-ring class notes binder (purchase in bookstore)
3. Homework (can be done in a spiral notebook that is inserted in your binder)
*Your notebook/binder will be your ACT & SAT study guide and review materials!!
4. TI-Nspire CX CAS calculator is required for this course
Learning Targets:
In this course, students will learn to…
 Make sense of problems and persevere in solving them.
 Reason abstractly and quantitatively.
 Construct viable arguments and critique the reasoning of
others.
 Model with mathematics.
 Use appropriate tools strategically.
 Attend to precision.
 Look for and make use of structure.
 Look for and express regularity in repeated reasoning.
Expectations: Respect and responsibility are the cardinal rules!!
1. Be to class on time!! It is disruptive to be late.
2. Avoid missing class!!! Your attendance is essential to your success in this class. For every
unexcused absence, 3% will be subtracted from your quarter grade. It is your responsibility
to check your attendance record with me and to follow up with clearing any absences with
the attendance office.
3. New Attendance Policy:
3rd absence: notification by teacher to parent
5th absence: notification by teacher to parent
8th absence: re-entry conference required before allowed to return to class;
grade reduction of 5% will result in this class
4. Extra Help: Don’t hesitate to go to the math lab for extra help, to check the answers on your
homework, or to work in a small group. You are also welcome to set-up a time to meet with
me during one of my free periods: 2, 4, 6 or 9.
5. http://class.dist113/HPHS/math/Gapinski; Check my portal site weekly for tutorial help,
links to various websites, announcements, and answer keys to daily homework assignments!
1
Assessments:
I will assess and evaluate your understanding of the coursework with random homework checks,
homework quizzes, chapter quizzes, tests, TI- Nspire CX calculator activities.
1. You will be graded on a point system. There will be no curves on any assessments including
the final exams.
2. Few exceptions will be made if any for turning in assignments late. If you are absent, it
must be handed in for credit within 2 days when you return to school. It is your
responsibility to check with me as to what assignment was collected/missed and to turn it in
completed for credit. I will not ask you for it! If you forget to turn in an assignment, you
will receive a zero on the assignment!
3. If you are absent, contact your partner/friend in class or e-mail me at:
rgapinski@dist113.org. I will leave extra worksheets in the classroom on the side table. It is
your responsibility to ask for any worksheets that you may have missed receiving when they
were passed out in class. Secondly, it is your responsibility to complete the assignment by
the scheduled date.
4. Quizzes are given to encourage you to apply the concepts that you have learned from the
homework and classroom participation. If you are absent the day of a quiz (not chapter
test!), your grade will be averaged without the quiz points. You may schedule or arrange to
take your quiz beforehand if you are aware of the absence in advance. By grading in this
manner, I am able to give immediate feedback to your classmates by returning the quizzes
the next school day.
5. Each student should e aware of the academic policy stated in the student handbook. If a
student is “caught” cheating or “gives” the impression of cheating, they will receive a zero
on the assignment, test, or quiz. You have the responsibility to never give the impression of
cheating at any time!
6. Generally, I follow a next day rule with grading on assessments. I will return any quiz
graded that day or on the day after you have taken it. Graded exams will be returned
within two school days as well.
Feedback:
You are encouraged to go to Infinite Campus via our district website to access your grades at any time.
I will update your grade at least twice a week on Infinite Campus and always by the day after you have taken a
chapter test. It will be your responsibility to monitor your performance on a daily basis and communicate
regularly with your parents regarding your current performance day-to-day as well as your quarter/semester
goals for this course. Don’t hesitate to meet with me at any time to discuss your grade and performance in
this class. I am here to support you with your math studies however I can so that you are most productive and
successful in this course.
2
Ice Breaker Activity
Beings with the
Letter
Something You
Find in School
Actor’s Last
Name
Fruit or
Vegetable
Math Term
C
H
P
S
3
Ice Breaker: People Scavenger Hunt
Find one person in the class who has done what is described below. Have them initial your paper. Happy
“hunting”!
has gone swimming
in Lake Michigan
this summer.
has more than
three brothers and
sisters.
has been to another is scared of heights.
country.
is an only child.
has an unusual pet.
has broken a bone.
has lived in another
state.
has performed in a
school play or
performance.
has an afterschool
or weekend job.
knows what Focus
on the Arts is.
knows what Charity
Drive is.
plays a musical
instrument.
plays a sport.
has an unusual
talent.
has won an award.
read a book for fun
over the summer.
has never broken a
bone.
Can speak a second
language fluently.
Went on vacation
out-of-state this
summer.
4
THE HISTORY OF ALGEBRA
The word "algebra" comes from Arabic: al-jebr because the subject was studied and written about in
something like the modern sense, by scholars who spoke Arabic in what is now the Middle East, in the 9th
century CE. Although classical Greeks and various of their predecessors and contemporaries had investigated
problems we now call "algebraic", these investigations became known to speakers of European languages not
from the classical sources but from Arabic writers. So that is why we use a term derived from Arabic.
Muhammad ben Musa al-Khwarizmi seems to have been the first person whose writing uses the term al-jebr.
As he used it, the term referred to a technique for solving equations by performing operations such as
addition or multiplication to both sides of the equation – just as is taught in first-year high school algebra. alKhwarizmi, of course, didn't use our modern notation with Roman letters for unknowns and symbols like "+",
"×", and "=". Instead, he expressed everything in ordinary words, but in a way equivalent to our modern
symbolism.
The word al-jebr itself is a metaphor, as the usual meaning of the word referred to the setting or straightening
out of broken bones. The same metaphor exists in Latin and related languages, as in the English words
"reduce" and "reduction". Although they now usually refer to making something smaller, the older meaning
refers to making somethng simpler or straighter. The Latin root is the verb ducere, to lead – hence to re-duce
is to lead something back to a simpler from a more convoluted state. In elementary algebra still one talks of
"reducing" fractions to lowest terms and simplifying equations.
The essence of the study of algebra, then, is solving or "reducing" equations to the simplest possible form. The
emphasis is on finding and describing explicit methods for performing this simplification. Such methods are
known as algorithms – in honor of al-Khwarizmi. Different types of methods can be used. Guessing at
solutions, for instance, is a method. One can often, by trying long enough, guess the exact solution of a simple
equation. And if one has a guess that is close but not exact, by changing this guess a little one can get a better
solution by an iterative process of successive approximation. This is a perfectly acceptable method of "solving"
equations for many practical purposes – so much so that it is the method generally used by computers (where
irrational numbers can be specified only approximately anyhow). Some approximation methods are fairly
sophisticated, such as "Newton's method" for finding the roots of polynomial equations – but they're still
based essentially on guessing an initial rough answer.
www.scienceandreason.blogspot.com
5
Section 1-1: Classification of Numbers
Classification of Numbers
Natural or “Counting”
Whole
Integer
Rational
Irrational
Transcendental
Real
Imaginary
Complex
Symbol
Definition
Examples
6
Section 1-1: Classifying Numbers
Transcendental
Numbers
Counting
Numbers
Natural
Numbers
Real
Numbers
Imaginary
Numbers
Irrational
Numbers
Rational
Numbers
Negative
Numbers
Positive
Numbers
Even
Numbers
Digits
Integers
5
2/3
-7
3
16
16
15
44

1.765
-1000
-1.5
 6
0
1
1/9
Convert the following terminating decimals into fractions of integers:
1. 0.6
3. 0.85
4. 0.123
Convert the following repeating decimals into fractions of integers:
1. 4.3
2. 2.54
3. 12.34
7
Section 1-2: The Field Axioms
Axiom: ______________________________________________________________________________
Name of Property
Commutative
Addition (+)
Multiplication (x)
Associative
Identity
Inverse
Closure
Distributive
A Field: ___________________________________________________________________________________
Identify the property shown.
1. 5  2  2  5
1. ________________________________
2. 6   6  0
2. ________________________________
 1 
3. 24    1
 24 
3. ________________________________
4. x  y  z    y  z  x
4. ________________________________
5. 6 8z   8  6 z 
5. ________________________________
6. 1   2  5  1  2   5
6. ________________________________
7.  a  b   0   a  b 
7. ________________________________
8. 2  y  c   2 y  2c
8. ________________________________
8
Section 1-2: The Field Axioms
Additional Practice
Name the property: (Assume no variable equals zero.)
 a b   1 a 1 b 
1.
2.
 xy  
1 1
 1 1
  x y 
a b
 a b
1. __________________________________
2. __________________________________
 1 1
1  1
 x
y
3. x  y 
 a b
a  b
3. __________________________________
 1  1  1  1 
4. x  y    x  y 
 a  b  a  b 
4. __________________________________
5. c  b  2   b  2 c
5. __________________________________
6. c  b  2  c  2  b 
6. __________________________________
7. 2b  2c  2  b  c 
7. __________________________________
8.  b  c  1  b  c 
8. __________________________________
9.  ac 
1
1
ac
9. __________________________________
1 1
 1 1
10. a  b    a b  

 x y
x y
10. _________________________________
a
a
 0  where x  0
x
x
11. _________________________________
11.
12. a  bc  0  2  a  bc   2
12. _________________________________
13. 2 x  ac  de   2xac  2xde
13. _________________________________
9
What is so special about variable “x” in algebra?
Question
For variables, why is the “known” variable, x? What is so special about x? What is the origin of the variable,
x? Like, most exams for algebra, teachers will use the letter x in lieu of other letters.
Answer
There's nothing special about "x". You could use any variable. Somehow, "x" has become associated with the
unknown or mysterious (X-Files, for example). The axes on a graph are usually referred to as the x and y axes
where x is the independent variable.
www.en.allexperts.com
10
Section 1-3: Variables and Expressions
What is a variable? ______________________________________________________________________
What is an expression? ___________________________________________________________________
What is the difference between an expression and an equation? __________________________________
_______________________________________________________________________________________
PEMDAS: _______________________________________________________________________________
Part I: Carry out the appropriate operations using PEMDAS.
2. 152  10 2  4  33
1. 6  10  5  8
Part II: Evaluate the following expression for x = -2, y = 3, and z = ½ .

2 x  3.5 y  4 x 

2 yz  6 x
1
5z 
Part III: Simplify the given expression.


5x  2 3  2  x  2  x  3
11
Section 1-4: Polynomials
What is a polynomial? _____________________________________________________________
What is a term? ________________________ What is a factor? __________________________
We name a polynomial by ________________ and number of ___________________________!!
Number of Terms
1
2
3
4 or more
Degree
0
1
2
3
4
5
6 or more
Name of degree
Examples:
What IS a polynomial
What IS NOT a polynomial
12
Tell whether or not the given expression is a polynomial. If it is, name it by degree and term. If it is not
a polynomial, tell why not.
1. 4 x3  2 x 2  5 x
2. 6abc  10
3. 7 x  5
4.
5. 4x  x 4
6.
7. 64
8.
x
 x2
2
ab  3a3
5
 54 x 3
x
9. 2 x5 y3 z
10. 2 x  7
11. 0
12. 3 yz  4.3
13. 8bcde
14. 5 x 2  10 x 5
15. 2b  4ac
16. 3x 2  6 x  8
3
13
Section 1-5: Equations & Absolute Values
Definition of Absolute Value:
a, if a  0

a, if a  0

a  
What are the differences between the following: 32  _______ 32  _________  3  __________
2
Part I. Solve in the indicated domain.
1. 2 x  5  3 {reals}
3.
 x  1 2x  5  0 {integers}
2. x 2  9 {whole numbers}
4. 6x  11  4x  21 {reals}
Part II: Simplify: 3  2  2 x  3  x  4   3  4  x  
Part III: Evaluate: 4  5  4  x   2 3  2  x  5   , if x = 5
14
Section 1-6: Inequalities
Solve the following inequalities, write the solution set, and graph in the specified domain.
1. 4x  6  14 {reals}
1. S = __________________________________
----------------------------------------------
2. 5  3x  9  11 {reals}
2. S = _________________________________
----------------------------------------------
3. 2 x  9  4 {reals}
3. S = ________________________________
----------------------------------------------
4. 5x  8  3 {reals}
4. S = _______________________________
---------------------------------------------
5. 3x  6  22 {integers}
5. S = _______________________________
---------------------------------------------
15
Section 1-5 & 1-6
Extra Practice
“Think-Pair-Share” Activity
1. Solve each in the indicated domain.
Initials
a. x 2  7
{rationals}
_________________
b. x 2  7
{irrationals}
_________________
a.
 x  33x  2  0
b. x 2 
1
9
{positive numbers}
_________________
{integers}
_________________
a. x  9
{real numbers}
_________________
b. 5x  3  2
{real numbers}
_________________
2. Graph the solutions on a number line.
a. 3x  5  4
{integers}
_________________
b. 2 x  3  10
{positive numbers}
_________________
16
Section 1-6: More Challenging Inequalities
Use the “dot n’ dash” method to graph the solution set for each inequality/equation. Then, write the solution
set in interval notation.
1.
3.
5.
 x  1 x  3  0
 x  2  x  4 
x 1
1
1

x  2 x 1
0
2.  5x  2 x  1  0
4.
 3x  1 x  5
0
 x  6
6.
1
1

x4 x2
17
Chapter 1: Inequalities WK
Do all work on a separate piece of paper. Graph the solution set for each inequality/equation and write the
solution set in interval notation.
1. 5 x  3  17
14.
4 x
0
2 x
18.
2. 3x  5x  2
15.
2x
0
5 x
19. 3x  12
3. 4x  7  x  5
16.
5
4

0
2 x  8 3x  12
20.
y2
5
3
4. 4  x  5  2  3x 1
17.
5
8
3
x7
x7
21.
3x  2
1
4
5.
6 x x
  x2
2
3
22.
1
3
x
6.
1 3
  12
x x
23.
1
6
x
7.
 x 1 x  2  0
24.
5x
2 7
6
8.  2 x  1 3x 1  0
9.
 x  3 x  2  0
1
2

x3 x2
25. 2 x  3  5
3x
4
5
3
27. 1 
x4
26. 1 
10.
 x  2 4  x   0
11.
 x  3 x  3  7
12.
 x  2  x  3  0
29.
x
5

0
x2 x4
13.
 2 x  1 3  x   0
30.
 x  3 x 1  0
x
x2
 1
 1

 1
 1  0
28. 
 x  1  x  1 
18
Chapter 1 Review WK (“Hot Seat QUIZ”)
1. Classify each number below according to the ALL sets of numbers.
___________________________
b.
7
10
__________________________
7
___________________________
d. 7
__________________________
e.  7
___________________________
f.
__________________________
g. 0
___________________________
g. 7
a. -7
c.
49
__________________________
2. Evaluate each expression for x = -1 and x = 5.
a. 13  3x  4
b. 2 x 2  10
for x = -1: _____ for x = 5: _________
for x = -1: ________ for x = 5: _________
3. Find the solution set for each of the following and graph their solutions.
a. 4 x  13  25
b.
1
2

x3 x2
c. 10  x  x  7   0
d.
 3x  4  x  1  0
 x  5
19
Section 2-2: Graphs of Functions
Domain: ____________________________________________________________________________
Range: _____________________________________________________________________________
Plot the graph of the function in the indicated domain. Identify the range.
1. y 
2
x 1 { x  0}
3
Range: ________________________
2. y 
4
{1  x  4 }
x
Range: ________________________
20
3. y  x 2  1 {-1, 0, 1, 2, 3}
Range: _____________________________
Analyze the graph to identify the domain and range. Assume the axes are marked in increments of 1 unit.
4.
5.
Using the domain and range given, sketch a graph that supports the data.
6. Domain:
6  x  4 ; Range: 4  y  3
7. Domain:
2  x  3 ; Range: 1  y  4
21
Section 2-3: Functions in the Real World
Vocabulary:
ordered pairs: ________________________________________________________________________
relation: _____________________________________________________________________________
function: _____________________________________________________________________________
independent variable: __________________________________________________________________
dependent variable: ____________________________________________________________________
******************************************************************************************
Sketch a reasonable graph showing how the dependent variable is related to the independent variable.
Label your axes and write one complete sentence explaining your graph!
1. The height of your head above the ground as you ride a Ferris wheel is a function of the time since you
got on.
2. The amount of water in a pan on a burner that is turned on “high” is a function of the time since the burner
was turned on.
3. The height of a ball that is dropped from a height of 10 feet is a function of the time since it was dropped.
22
Section 2-3: Practice Problems
1. Horace’s Bicycle Trip: Write a description for the graph below that describes Horace’s bicycle trip last
Saturday with his friends.
Distance (miles)
48
28
18
0
2
3
7
9
t (hours)
2. Bathtub Problem: The graph below depicts the water level of a bathtub with respect to time. Write a
paragraph that might describe what occurred.
Water level
Time
23
Section 2-3: Extra Practice
Drawing Function Graphs
1. Sara walks from her home to the store. Halfway to the store, she realizes that she forgot to
bring money, so she turns around, returns home, gets her money and then walks all the way to
the store. Graph time on the horizontal axis and distance from home on the vertical axis.
2. Rashid is jumping on a trampoline. Graph time on the horizontal axis and his distance off
the ground on the vertical axis.
3. Kendra is speeding along the highway and is stopped by a police officer. The officer gives
her a ticket and she continues on her way. Graph time on the horizontal axis and her speed on
the vertical axis.
24
Section 2-3: Absolute Value Graphs
Sketch a graph of each function below, labeling the vertex and at least two other key points.
1. y = |x| + 1
2. y = |x| - 3
3. y = -|x| + 2
4.. y = |2x| + 1
1
5. y = | x| - 2
2
6. y = -|3x| + 4
7. y = |x - 3| + 4
8. y = |x + 2| - 1
9. y = -|x – 1| + 3
25
Section 2-4: Graphs of Functions and Relations
When is a relation a function? ______________________________________________________________
x=a
x=a
(a,c)
(a,b)
(a,b)
0
a
0
Graph of a function
a
Not a graph of a function
Tell whether or not the relation graphed is a function.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
26
Absolute Value Graphs: Extra Practice WK
Graph the following functions on the grids at the right. Label on your sketch of each graph, the vertex and at
least two other points.
1. y  x  2
2. y  x  2
3. y   x  2
4. y  2 x  3
5. y  x  4  1
6. y  2 x  6  4
7. y  2 x  2  1
*8. y  x  x
27
The Greatest Integer Function WK
Graph: y   x below. Mark clearly the points on your graph!
Use a graphing calculator to assist you in sketching each equation onto the axes provided. Mark your points
clearly!!
1. y   x  1
4. y   x  3
2. y   x  4
3. y   x  2
5. y   x  2
6. y    x
28
Chapter 2 Review WK
Solve and graph the solution set for each inequality below. Write the solution set in interval notation.
1. 17  3x  2x 13
2.
4x
2 2
7
3. 3x 15  2 x   0
4.
x  x  4
0
2  x
29
5.
2
4

0
x  3 x 1
 4
 4

7. 
 2 
 2  0
 x2
 x  2

2
1

9.
x  2 x 1
6.
x
2

x 1 x  3
8.
3
4

x2 x2
 x  2  x  4  0
2
10.
x3
30
Sketch a reasonable graph showing how the dependent variable is related to the independent variable of
the following situations.
9. Your vertical position on a carousel horse depends on the time since the carousel began.
10. The amount of fuel in your boat’s outboard motor is related to the amount of time you have been
pulling skiers.
11. The temperature of your home in the summer is related to the amount of money spent on air
conditioning.
12. The speed of a ceiling fan blade and the amount of air moved by it are related.
31
13. Plot the graph of the given equations in the indicated domain. Identify the functions. For each relation,
define its domain and range.
a. 2 x  3 y  12;
3  x  6
Function:
Yes
No
Range: ______________________
b. x2  y  3;
x  R
Function?
Yes
No
Range: ______________________
32
c. y 2  x;
{ 0  x  9}
Function?
Yes
No
Range: _________________________
d. y  x 1  2;
3  x  5
Function?
Yes
No
Range: __________________________
e. y   x  2;
1  x  4
Function?
Yes
No
Range: __________________________
33
14. Tell whether or not the relation graphed is a function.
a.
b.
34
Section 3-2: Slope
Definition(s) of Slope: _______________________________________________________________
Types of Slopes:
35
36
Section 3-2: Properties of Linear Function Graphs
Plot the graphs of these linear functions. If possible, use the slope and y-intercept to do so.
2
x4
5
1. y  2 x  3
2. y   x  3
3. y 
3
4. y   x  5
2
5. x  y  2
6. 4 x  3 y  12
7. 2 x  y  3
8. x  y  0
9. x  3 y  9
37
SLOPE FORMULA:
3-3: Other Forms of the Linear Function Equation
Slope-Intercept Form
General Form:
Point-Slope Form
General Form:
Standard Form
General Form:
Uses:
Uses:
Uses:
Plot the graph, showing clearing the point and slope that appear in the equation. Then, transform the
equation to slope-intercept form and then to standard form.
1. y  3 
1
 x  4
2
2. y  4 
2
 x  1
5
38
Section 3-4: Equations of Linear Functions from Their Graphs
“Recipe” for writing the equation of a line:
Step #1: ________________________________________________
Step #2: ________________________________________________
Case #1:
Case #2:
Case #3:
Case #4:
Case #5:
Case #6:
Case #7:
Case #8:
39
Section 3-4: Equations of Linear Functions from Their Graphs
For problems #1-10, do the following:
a. Write the particular equation of the line described (i.e. point-slope form when appropriate).
b. Transform the equation into slope-intercept form.
c. Transform the equation to Ax + By = C, where A, B, and C are integers.
1. Has a y-intercept of -3 and a slope of -2
2. Passes through (1, 6) and (3, 9)
3. Passes through (2, -3) and is parallel to y = 3x – 5
3
4. Passes through (-2, 3) and is perpendicular to y   x  4
4
5. Has a y-intercept of 5 and a slope of
2
3
6. Has an x-intercept of 4 and a y-intercept of 3
7. Passes through (2, -1) and is parallel to x + 2y = 5
40
8. Passes through (-2, 1) and is perpendicular to 3x – 2y = 8
9. Passes through the origin with a slope of 1.3
10. Passes through (2, -6) and (4, -6)
41
Car Choice Problem:
Ms. Elain Eous has decided to buy a car from the local Bee Plus Car Company. After shopping for several days
she chose to test drive a “racy red” 1962 Rambler and a “mellow melon” 1976 Mustang. The price of the
Rambler is $12,500, and the cost of the Mustang is $29,600. She likes both cars, but she began to think about
the cost of owning an “older” car; so she decided to ask her mechanic about problems she might face with the
maintenance of the auto she chose. Mel Mechanic told her that the Rambler would average $1,500 per
month in expenses, but the Mustang would operate on $600 per month in expenses.
a. If she buys the Rambler, what is her total cost in 10 months?
a. _______________
b. If she buys the Mustang?
b. ______________
c. Find the equation of these functions. Let t= time in months.
Rambler equation: _____________________________
Mustang equation: _____________________________
d. Find R(25).
d. ______________
e. Find M(25).
e. ______________
f. Find the break even point. Explain its meaning.
break even point: _________________
42
Section 3-5: Linear Functions as Mathematical Models
1. Polka Dot Cab Problem: Bob takes a Polka Dot Cab from the airport to his home which is exactly six
miles away. The cab costs him $7.70. Wally takes a cab from his hotel to a golf course 10 miles away.
His fare is $11.50.
a. Write the particular equation of this function expressing cost in terms of distance.
b. Maria uses a cab as a delivery service vehicle. The distance driven one morning was 60 miles.
What was the cost of the cab?
c. The fare from the international airport to the local airport was $14.54. What is the distance
between the two places?
d. Sketch a graph of the function to the right.
e. Identify the domain and the range.
f. Explain the real-world meaning of the
cost-intercept.
43
2. Teeth Problem: The number of teeth of an average child and the age of the child in months are related by
a linear function. According to the published reports, an average child has no teeth at six months. After that
age, the child’s “milk teeth” begin a slow steady growth. The average 30 month old child has 20 teeth and has
completed the growing cycle of his first set of teeth.
a. Which variable is dependent and which is independent?
b. What is the domain of the independent variable?
What is the range of the dependent variable?
c. Using the given ordered pairs,
plot the graph of this function.
d. Write the particular equation of this function,
expressing number of teeth in terms of age.
e. How many teeth will a child have at 8 months?
Explain the need for a fraction!!
f. How old is a child that has 5 teeth?
g. What reasons can you think of that would make
this model incorrect?
44
Chapter 3: Practice ACT QUIZ
1. Plot each point on the coordinate plane at the right.
Label each point accordingly.
A (-2, 5)
B (4, 3)
C (2, -3)
D (-3, -1)
2. Graph the line with equation 3x + 6y = 12 by plotting points quickly.
3. What is the slope of the line in problem #2?
3. m = ________
4. What are the coordinates of the point on the y-axis and the coordinates of
the point on the x-axis of the graph of 2x + 4y = 4?
4. x-int:
y-int:
45
5. Find the slope and the y-intercept of the line with equation 10x + 5y = 20.
5. m = ___ b = ____
6. State whether the slope of each line is positive, negative, zero, or undefined.
a. m = _____________________
b. m = _________________
c. m = _____________________
d. m = __________________
7. What is the sum of the y-intercept and the slope of the linear equation 6x – 9y = 3?
a. -6
b. -3
c.
1
3
d. 
1
3
e. 9
8. What is the slope of a line parallel to the line with equation 12x – 3y = 17?
a. 
17
3
b. 4
c.
1
4
d. -4
7. _______
e.
8. _______
3
17
46
9. Which equation creates an infinite number of solutions when solved in a system with
y = 5x – 7?
a. 2y + 10x = -14
b. y = 7x – 5
d. 4y – 20x = -28
c. 3y – 15x = -28
e. 4y + 15x = -21
10. What is the equation of a line with a y-intercept of -3 and is perpendicular to the line
with equation 3y – 4x = 21?
a. y 
4
x 3
3
b. y 
d. y  3 x 
3
4
9. _______
3
x3
4
c. y  3 x 
10. ______
4
3
3
e. y   x  3
4
47
TI – Nspire CX calculator: Linear Regression or “Line of Best Fit”
Enter the Data
1) From the Home screen, select New Document and then select Add Lists
and Spreadsheets. If you are already working within a document, you can
add a page (ctrl i)and then select Add Lists and Spreadsheets.
2) At the “A” and “B” headings type in a header name. For example,
Minute and Speed.
3) Enter the x-values in row 1,and the y-values in row 2.
Plot Data On Your Viewing Window
1) Add another page to your document (ctrl i) and select Add Data &
Statistics.
2) On the x-axis move cursor to “Click to add variable.” Choose the
appropriate title of your x column.
3) On the y-axis move cursor to “Click to add variable.” Choose the
appropriate title of your y column.
Calculate the Line of Best Fit
1) Press menu. Choose #4: Analyze. Choose #6: Regression. Choose
#1: Linear (mx + b).
48
Linear Regression
1. The table below shows the number of bicycles produced in the United States from 1989 to
1993. Enter the given data on your calculator and plot the points on a scatterplot.
Year Number of Bicycles Produced (millions)
1989
5.3
1990
6.0
1991
7.3
1992
7.4
1993
8.0
a. What is the equation of the line of best fit?
a. _________________________
b. Use this line to predict the number of bicycles produced in 2009.
b. _____________
c. When will there be 50 million bicycles produced?
c. _____________
by hand:
or
use SOLVE feature on your TI-Nspire CX calculator
(Set equation = 0 to use solver to find x.)
d. Give the meaning of the slope in context.
d. ______________________________________
49
Linear Regression
2. The table below shows the heights and numbers of stories of some of the tallest buildings in the United States. Use
this table to answer the questions that follow.
Name of building
Number of Stories
Height (in feet)
Sears Tower, Chicago
110
1454
Empire State Building, N.Y.C
102
1250
Amoco, Chicago
80
1136
Chrysler Building, N.Y.C.
77
1046
Allied Bank Plaza, Houston
71
972
Columbia Cener, Seattle
76
954
InterFirst Plaza Tower, Dallas
71
921
Society Center, Cleveland
57
888
First Interstate Bank, L.A.
62
858
First National Bank, Chicago
60
850
a.
a. Make a scatter plot of the data on your calculator.
b. Use your calculator to find an equation for the line of best fit.
b. ______________________________
c. Donald Trump is building a new skyscraper in Chicago. If he wants his building to be 94 stories, approximately
how high can he expect his building to be?
c. ____________________
d. Does the data show a positive or negative correlation?
d. ____________________
50
Linear Regression
3. The table below shows the average daily energy requirements for male children and adolescents.
Age (years)
1
2
5
8
11
14
17
Energy
Needed
(Calories)
1100
1300
1800
2200
2500
2800
3000
a. Make a scatter plot of the data on your calculator.
b. Use your calculator to find the line of best fit.
b. ____________________________________
c. What is the vertical (y-intercept)? What does this number indicate for this particular problem?
c. y-intercept: _______ meaning: _____________________
d. Use your equation to estimate the daily energy requirements for a male 16 years old.
d. ____________________
e. How old is a male who requires 2450 calories per day?
e. _____________________
f. Do you think your model also applies to adult males? Explain.
51
Linear Regression
4. The table below shows the average speed of an airplane during the first 7 minutes of a flight, with x in
minutes and y in miles per hour.
Minutes since
take-off
1
2
3
4
5
6
7
Average Speed
180
250
290
310
400
420
410
a. Draw a scatter plot for the data. Approximate the best-fitting line.
b. Predict the average speed of the airplane 10 minutes after take-off.
52
A2TH: Linear Regression
Practice Worksheet
Name __________________________________________
Problem #1: The Cody Company ran a study on its sales force and learned that the average number of years
experience for each sales team was the direct relation to annual sales volume. Use the data below to answer
the following:
Average sales in thousands 46 35 51 42 33 50 30
Average years of experience 6 4 8 5.5 3 7 2.5
1. What is your independent variable (L1)?
1. _____________
2. What is your dependent variable (L2)?
2. _____________
3. Enter the data in to the lists and find a prediction equation.
3. ___________________________
4. What is the average years of experience if you sell 100,000 dollars worth of product? 4. ______________
(Remember the data is in 1000’s)
Problem #2: The table below shows the population and the number of representatives for Congress in the
states listed.
State
CA
NY
NC FL
NC IN AL
Population (millions) 29.8 18.0 17.0 12.9 6.6 5.5 4.0
Representatives
52
31
30
23
12 10 7
1. Make a scatter plot [0, 30] scl 5, [0, 60] scl 10
2. What is your independent variable?
2. _____________
3. What is your dependent variable? Why?
3. _____________
4. Find the line of regression for the data.
4. _________________________
5. If there are 21 million people in Illinois, how many representatives should IL get?
5. _____________
53
Problem #3: Manatees are large, gentle sea creatures that live along the Florida coast. Many manatees are
killed or injured by powerboats. Here are data on powerboat registrations (in thousands) and the number of
manatees killed by boats Florida in the years from 1977 to 1993.
Year # of Powerboat Registrations # of Manatees Killed
1977
447
13
1979
481
24
1981
513
24
1983
526
15
1985
585
33
1987
645
39
1989
711
50
1991
716
53
1993
716
35
1. We want to examine the relationship between the number of powerboats and the number of manatees
killed by boats.
a. What is the independent variable?
a. ____________
b. What is the dependent variable?
b. ____________
2. What is the prediction equation for this set of data?
2. ________________________
3. How many manatees would be predicted to be killed in Florida if only 700,000
boats were allowed to register?
3. ____________
4. How many powerboats do you think would be registered if 87 manatees were
killed a year?
4. ____________
54
Section 4-2: Solution of Systems of Linear Equations
“Types” of Systems of Linear Equations
Example System:
What happens when you
solve the system
A ‘”visual” of what the
system looks like
Descriptors of the
system
Methods for solving a linear system:
1. ____________________________________________________
2. ____________________________________________________
3. ____________________________________________________
4. ____________________________________________________
5. ____________________________________________________
6. ____________________________________________________
Part I: Solve by substitution. (When should you use this technique? _________________________________)
a.
x  12 y  68
x  8 y  12
b.
2 x  y   1
6 x  2 y  2
55
Part II: Solve by elimination or linear combination.
a.
c.
4x  2 y  4
b.
6x  2 y  8
3x  5 y  7
6 x  10 y  14
5 4
  17
x y
e.
1 10
  2
x y
d.
f.
3 x  y  6
6 x  2 y  25
4 x  5 y  13
6 x  3 y  12
0.02a  1.5b  4
0.5b  0.02a  1.8
g. Multiple Choice: The equation 3x  4 y  2 and which equation below form a system with no solutions?
a. 2 y  1.5 x  2
b. 2 y  1.5 x  1
c. 3 x  4 y  2
d. 4 y  3 x  2
56
Section 4-2: Systems of Linear Equations Mathematical Models
1. Mrs. Gapinski is writing a test for her algebra 2 with trig classes. The test will have true/false questions
worth 2 points each and multiple-choice questions worth 4 points each for a total of 100 points. She wants to
have twice as many multiple-choice questions as true/false. Let x be true or false questions and y be the
multiple-choice questions.
a. Write a system of equations that represents the number of each type of question.
b. How many true/false questions and multiple-choice questions will be on the test?
c. If most of her students can answer true/false questions within 1 minute and multiple-choice
questions within 1.5 minutes, will they have enough time to finish the test in 45 minutes?
2. Mrs. Gapinski exercises every morning for 40 minutes. She does a combination of step aerobics, which
burns about 11 calories per minute, and stretching, which burns about 4 calories per minute. Her goal is to
burn 335 calories during her routine.
a. Write a system of equations that represents Mrs. G's morning workout.
b. How long should she participate in each activity in order to burn 335 calories?
3. Airports: According to the Airports Council International, the busiest airport in the world is Atlanta's
Hartsfield International Airport, and the second busiest is Chicago's O'Hare Airport. Together, they handled
150.5 million passengers in the first six months of 1999. If Hartsfield handled 5.5 million more passengers
than O'Hare, how many were handled by each airport?
57
Section 4-3: Second-Order Determinants
Given the following system, solve by using second-order determinants.
x  2 y  10
x y 6
Nx 
D
Ny 
 N Ny 
Solution to the system is found by:  x ,
= __________________
 D D 


Solve the following systems by using Cramer’s Rule.
1.
3.
3x  4 y  9
3 x  2 y  3
2m  4n  4
3m  5n  3
2.
4.
2 x  5 y  24
3 x  5 y  14
2x  y  3
4x  y  9
58
Section 4-4: F(x) Terminology, and Systems as Models
Given:
f  x  2x  5
g  x   3  x2
h  x  x 1
Evaluate the following:
1. f  5
2. g  1
3. h  a 
4. g  a  b 
5. f  g  4  
6. h  f  1 

7. g f  h  2  
9.


8. f g  h  x  
f  4
g  4

11. g f  h  1 
10.


f  3
g  6

12. h g  f  x  

59
Section 4-5: Linear Equations With Three or More Variables
Describe the location of each point in coordinate space.
1.  2,1, 5
2.  3,  3, 4
3. (4,  7,  1)
4. Which point is NOT on the graph of 2 x  3 y  z  12?
a. (6, 0, 0)
b. (3, 3, 3)
c. (0, 4, 0)
d. (1, 1, 7)
5. What are the intercepts of 3x  5 y  2 z  60 ?
a. x = -180, y = 300, z = -120
b. x = -20, y = 12, z = -30
c. x = -3, y = 5, z = -2
d. x = -60, y = 60, z = -60
6. What is the xy-trace of 2 x  4 y  z  8 ?
a. -4y + z = 8
b . x – 2y =4
c. 2x + z = 8
d. z = 8
60
Section 4-5: Graphing equations with three or more variables
1. 2x + 3y + 4z = 12
x y z
2. 3x – 9y – 3z = 18
x y z
3. Write an equation of the plane given its x-, y-, and z-intercepts, respectively.
a. 8, -3, 6
a. _____________________________
b. 10, 4, -5
b. _____________________________
61
Section 4-5: Isometric Paper for homework problems
62
Section 4.6: Systems of Linear Equations with Three or More Variables
Solving systems with three or more variables
3x  2 y  4 z  1

Solve: 5 x  3 y  7 z  28
2 x  4 y  3z  17

(eq. 1) 2( 3 x  2 y  4 z  1) 
(eq. 3) 2 x  4 y  3z  17
________________
(call this equation 4)
(eq. 1) 3( 3 x  2 y  4 z  1 )

(eq. 2) 2( 5 x  3 y  7 z  28 ) 
_________________
(call this equation 5)
63
Easier method: Use MATRICES!!!
2 x  y  z  2

3x  2 y  z  5
 x y z  0

To solve a system of equations using matrices, first write a matrix equation. Then enter the coefficient and
constant matrix into your calculator. Next, multiply the inverse of the coefficient matrix by the constant
matrix. We must use inverse matrices to solve this system because we cannot divide matrices.

 2 1 1  x   2
3 2 1   y   5

   
1 1 1   z  0 
Easiest method: By “Reduced Row Eschelon Form” or rref
 2 1 1 2 


3 2 1 5 
1 1 1 0 


64
Additional Practice:
Solve each system using matrices. Be sure to write the matrix equation first.
a) 2x + y – z = 2
3x + 2y + z = 5
x+y+z=0
b) 2x + 6y – z = 4
4x + 5z = 7
3y – 2z = 0
65
Section 4-6: Matrices to solve systems of linear equations
Part I: Dimensions of a Matrix (___________ x ______________)
Give the dimensions of each matrix below.
2
A   0 
 1
 7 0
B

 3 1 
x
Part II: Scalar Multiplication a 
w
3 1 
C   0 4 
 2 5 
D  5 1 2 1
y   ax ay 

z   aw az 
Find the following using the matrices from Part I:
a. 3B
b. -2A
c.
1
D
2
Part III: Addition/Subtraction of Matrices
 2 1  8 4
a. 


 4 0   3 2
b. 12 8   5 2
 6 2
 1 0 
 2
c. 4 


 3 1 
 1 4
 4   2
d. 7  1    6 
   
 5  3 
Part IV: Matrix Multiplication: Is A  B defined? Give the dimensions of each possible product.
1. Dimensions
a. A3 x 4  B4 x 2  C
b. A2 x1  B1x5  C
c. A7 x 2  B7 x 2  C
d. A  B4 x3  C2 x3
66
2. Find the matrix product.
 2 3  1 1
a. 


1 1  4 2 
4 1  0 1
b.  3 2  1 2



1 3  5 1 
 1 8 
c.  3 2 3 0 5 1


 2 4
10 
 4 0 8  
d. 
  8
 1 5 1  5 
 
3. Square Matrix
What is a square matrix?
When do you use them?
4. Identity Matrix
I2x2
I3x3
I4x4
a b 
1  d b 
Part V: Inverse Matrix: Given A  
, then A1 
if ad  bc  0

ad  bc  c a 
c d 
a. Find the inverse matrix for each of the following if it exists.
3 4
B

6 7 
 2 5
C

4 6 
b. Solve each matrix equation for X.
5 1 
3 0 5 
X

7 2
1 4 2




5 3 
1 3  1 2
X

3 2 
 2 1   1 3



 

Solve the linear systems using matrices.
12 x  4 y  0
1.
17 x  5 y  16
x  2 y  3z  3
2. 2 x  y  5 z  8
3x  y  3z  22
67
Practice Exercises
I. Solve for the variables.
 3x 4 y   27 16
a. 


 48 49   3w 7 z 
x
b. 4 
6
y  2 10
25 


z   2 z 30 x  5 y 
II. Perform the indicated operations.
a.
3  8 12  2  27 9 

4  16 20 3 54 18
 2 4   3 2 7 
b. 


 3 1 6 0 5
 3 2   3 2 
d. 7 6  7 6 



 0 5  0 5
 2 4   3 0 
 3 0 
c. 
 2




 3 1  2 5
 2 5
III. Determine the dimensions of each matrix M.
a. A7 x 4  B4 x3  M
b. A7 x 4  B4 x3  M
c. M
 A1x 6  B2 x 6
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IV. Find the inverse of each matrix, if it exists.
 3 1
a. 

 4 2 
4 5
b. 

 4 3
4 6
c. 

6 9
V. Solve each system using a matrix equation. Set-up a matrix equation to justify your work.
a. 2 x  y  3 z  4
 x  4 y  z  15
3 x  4 y  2 z  28
b. 2 x  3 y  5 z  6
4 x  2 y  3 z  21
5 x  4 y  3z  2
c. 2 x  3 y  2 z  3
4x  y  6z  1
2x  6 y  z  4
d. 4 x  5 z  7
3y  2z  0
5 x  y  z  12
I.
Solve for the variables.
a2
a. 
2
9 15 36 9 5b 


16 18  2 2c 18 
 2 x  y   23
b. 
 
 x  3 y  15 
69
Section 4-10: Systems of Linear Inequalities
Solve each system of inequalities by graphing.
1.
3.
y  2x  2
y  x  1
y  3x  1
6 x  2 y  5
2.
x  y  2
y  2x  1
4.
y4
y  x 1
70
x y 8
5. x  0
y0
7.
y  x 1
y   x  2 1
y  2 x  4
6. x  3
y 1
8.
2x  y  3
y  x3 2
How to graph an inequality on a TI-Nspire CX:
1. Go to New Document. Add a graph page.
2. Type in equations to Y1 & Y2. Why did the f(x) notation change to y???
71
Section 4-11: Linear Programming
1. Publishing Problem: A local company reads for publication two types of books, romances and
westerns. The number of books reviewed in one month must fit the following parameters.
Let x = number of romances per month
y = number of westerns per month
i. x < 20 and y < 16
ii. 5x + 9y > 90
iii. y < 2x – 8
iv. x + 5y > 25
v. x + y < 28
a. Graph the parameters and shade the feasible region.
b. Can the company read only westerns?
c. Can the company read only romances?
d. It costs $150.00 to read a romance and $180.00 to read a western. Write an
equation for the total cost to read the books.
e. Shade the area in which the cost per month is at most $2700.
f. In the feasible region of part a, what point represents the least number of books to
be read in one month?
g. In the feasible region of part a, what point represents the most number of books to
be read in one month?
h. What is the least cost to the company?
i.
What is the greatest cost to the company?
72
2. Publishing Problem: A local company reads for publication two types of books, romances and
westerns. The number of books reviewed in one month must fit the following parameters.
Let x = number of romances per month
y = number of westerns per month
i.
x < 20 and y < 16
iii. y < 2x – 8
v.
ii. 5x + 9y > 90
iv. x + 5y > 25
x + y < 28
a. Graph the parameters and shade the feasible region.
b. Can the company read only westerns?
c. Can the company read only romances?
d. It costs $150.00 to read a romance and $180.00 to read a western. Write an
equation for the total cost to read the books.
e. Shade the area in which the cost per month is at most $2700.
f. In the feasible region of part a, what point represents the least number of books to
be read in one month?
g. In the feasible region of part a, what point represents the most number of books to
be read in one month?
h. What is the least cost to the company?
i.
What is the greatest cost to the company?
73
3. Miner Problem: Jennifer Wood is in charge of the handling of gold and silver each day. She has restrictions
on how many ounces of gold and silver she can mine, ship, sell, and refine per day. Let x = the amount of gold
and y = the amount of silver she works with each day. The following equations establish the parameters of her
job.
i.
x < 15, y < 14
iii.
1
y  4 x
4
v.
ii. x + y > 10
iv. y > 12 – 3x
y < 16 – x
a. Graph the inequalities, shade the feasible region.
b. It costs $10 to refine an ounce of silver and $5 for an ounce of gold. Write an equation that
states this fact.
c. Show the region of the graph in which the cost to refine gold and silver is less than $100.
d. The optimum point is at the minimum cost. What is that point and what is the minimum
cost?
4. Corn Chip Problem: Joe P. is president of Joe’s Corn Chips, Inc. His company is divided into two
departments which put out two types of corn chips, Extra Larges and Really Smalls. Each department has
separate regulations concerning the number of bags produced per day.
a. Write your “let” statements. (i.e. Define your variables.)
b. Write inequalities/equations for the following regulations.
i.
ii.
iii.
iv.
No more than 20 kilobags of Extra Larges and no more than 30 kilobags of Really Smalls
can be put out per day. (A kilobag is 1000 bags.)
No more than 45 kilobags, total, can be manufactured each day.
3
The number of Extra Larges must be no less than the number of Really Smalls
4
produced per day.
More than 300 hours of labor must be used each day to meet union requirements. It
takes 10 hours to make a kilobag of Extra Larges and 15 hours to make a kilobag of
Really Smalls.
74
c. Draw a graph of the feasible region. Label each constraint on your graph!
d. If Joe’s Corn Chips, Inc. makes a profit of $200 per kilobag of Extra Larges and $150 per kilobag
of Really Smalls, show the region on your graph in which the daily profit would be at least
$6000.
e. How many bags of each kind should be produced each day to give the greatest feasible profit?
What is this profit?
5. Video King has two different rental plans for videos. The basic plan is $2.00 per video tape. The
membership plan includes a membership fee of $21.00 per year, and $1.40 per video rental.
a. Write the cost of each rental plan as a function of the number of tapes.
b. Find f(100) for each rental plan and explain what this represents in the real-world.
c. Choose an appropriate WINDOW to investigate costs for a year for both rental plans. Then,
graph both functions on your TI-calculator. Use AUTOSCALE.
Xmin = _____
Xmax = _____
Xscl = _____
Ymin = _____
Ymax = _____
Yscl = ______
TI Window Screen
d. Use the CALC: Intersection function on your TI-graphing calculator. Write down the number of
tapes and the cost. What does this mean in the real-world?
e. Solve for the point algebraically below. Compare this answer to your graphic solution.
75
f. Which plan is a better bargain? How would you decide which plan to use? Under what
conditions would the membership plan cost you less money than the basic plan?
g. Find the number of tapes that must be rented to recover the original $21.00 fee. Compare this
result to the result found graphically and algebraically.
h. Suppose you plan to rent about 1 video a week. Find the amount of money saved per tape with
the membership plan.
i.
Suppose Video King manager Lisa Flick offers a Royalty Membership. A Royalty member pays
$80 and rents any video for just 25 cents! Compare the Royalty Membership to the $2.00 plan.
Find the number of tapes you would need to rent in one year in order to save money with the
Royalty plan. Show your solution both algebraically and graphically below.
Xmin = _____
Xmax = _____
Xscl = _____
Ymin = _____
Ymax = _____
TI Window Screen
j.
Compare the Royalty membership to the $1.40 plan. Find the number of tapes you would need
to rent in one year in order to save money with the Royalty plan. Show your solution both
algebraically and graphically.
Xmin = _____
Xmax = _____
Xscl = _____
Ymin = _____
Ymax = _____
TI Window Screen
k. If you were Lisa Flick, would you offer the Royalty membership? Why or why not?
76
6. A cabinet company makes two types of cabinet drawers, one plain and one fancy. Each plain drawer takes
2 hours of work to assemble and 1 hour of sanding. Each fancy drawer takes 1 hour of work to assemble and 4
hours of sanding. The four assembly workers and six sanding workers will each work up to 12 hours per day.
a. Define your variables.
b. Write the constraints.
c. Graph the system of constraints. Find the vertices of the feasible region. Be sure to
label the axes!
77
7. A manufacturer of tennis rackets produces two different types, the Set Point racket and the Double Fault
racket. To meet dealer demand, daily production of Double Faults should be between 30 and 80, whereas the
number of Set Points produced should be between 30 and 80. To maintain high quality, the total number of
rackets produced in a day should not exceed 80.
a. Define your variables.
b. Write the constraints.
c. Graph the system of constraints. Find the vertices of the feasible region. Be sure to label your
axes!!
78
8. Graph the solution set of the system:
x  y  11
3x  y  5
x0
y0
a. What does your solution look like?
b. Find the vertices of the region formed.
9. Graph the solution set of the system:
y  3x  2
2
y  2    x  3
5
2 x  3 y  15
y  1
79
Chapter 4 Review WK
Sketch the graph by drawing its three traces.
1. 2 x  5 y  2 z  20
2. 3x  8 y  4 z  24
x y z
3. Solve by linear combination:
x y z
3x  5 y  21
4 x  9 y  19
80
4. Solve by substitution:
6 x  y  29
2x  3y  7
2x  3y  z  9
5. Write a matrix equation and solve. x  3 y  z  6
3 x  y  4 z  31
6. Let f  x   2 x  5 , g  x   7  3x , and h  x   4  3x  x2 . Evaluate the following.
a. f(4)

e. g f  h  2  

4
b. g  
3
c. g  h 1 
d. h  0 
f. g  f  a  
g. g  2  a 
h. f g  h  c  


81
7. A Tip Problem: At Walker Brother’s Pancake House in downtown Highland Park, IL, a server earns $64
and a cook earns $96 in a normal shift. In addition, a server gets 80% of the tip money received,
and the cook gets 20%.
Let x = total dollars in tips received in a shift
S(x) = total dollars a server gets in a shift
C(x) = total dollars a cook gets in a shift
a. Write particular equations expressing S(x) in terms of x and C(x) in terms of x.
b. Find T(50) and H(50). Who is ahead with how much they have received in tips after $50 in tip
money has been collected?
c. How much would have to be received in tips for a server and a cook to break even?
82
TI-Calculator Writing Assignment
Due Date: ___________________________
As a way of thanking your parent(s), guardian, sibling, etc. for buying you that expensive, but very
helpful calculator, I want you to show them one interesting “thing” that shows the power of the calculator.
Show one of your parents a lesson on the TI-83/84 and then type 1 page (double spaced) on what you taught,
how well they learned, and how they enjoyed it. After you have written the page, have your parents read it
over and then sign it.
(I will assume that you will write complete sentences with proper punctuation and spelling!)
83
Section 5-1: Plotting Quadratic Functions Quickly
Partner Activity
With your partner, use your TI graphing calculator to graph the following functions. Plot at least five points for each
graph. Look for patterns! Can you and your partner come up with a “shortcut” or quick method that allows you to
graph the points of any quadratic function quickly without using a calculator or making a t-chart??! Good luck!
1. y  x 2
2. y  2 x 2
3. y  3x 2  8
4. y  2  x  1
2
84
Section 5-2: Graphs of Quadratic Functions
Part I: Squaring a binomial
 3x  5
2
2x  7
= _____________________
2
= _________________________
Part II: Completing the square to make a perfect square trinomial (“going backwards”)
x 2  12 x  _________  

3x 2  30 x  __________  3 
2

2
Part III: Solve each by completing the square.
x 2  8 x  20  0
For the following equations, complete the square to transform from standard form into vertex form. Then,
graph it and find the vertex and axis of symmetry.
Standard Form: y = ax2 + bx + c
Vertex Form: y = a(x – h)2 + k
where the Vertex is (h,k) and the
Axis of Symmetry (a.o.s.) is x = h
4. y = x2 – 6x + 5
Vertex form: ________________________________
y-intercept: _____________________
x-intercept(s): _____________________
Vertex: ________________ a.o.s. ______________
symmetric point: _____________________
85
5. y  4 x 2  16 x  15
Vertex form: ________________________________
y-intercept: _________________________
x-intercept(s): _______________________
Vertex: ________________ a.o.s. ______________
symmetric point: _____________________
6. y  x 2  5x  24
Vertex form: ________________________________
y-intercept: ____________________
x-intercept(s): __________________
Vertex: ________________ a.o.s. ______________
symmetric point: ________________
86
Section 5-3: X-intercepts & the Quadratic Formula
Review: Sketch the graph of the quadratic function with the given vertex and intercept.
Vertex (4, -3) and y-intercept: 5
Part I: Simplifying complex numbers
1. What does i equal?
2. Simplify the following:
a.
4
b.
20
c.
32
d.
72
Part II: Derivation of the Quadratic Formula by Completing the Square!!
87
Part III: Solve the equation by either factoring or using the Quadratic Formula.
a. x 2  8 x  15  0
b. x 2  6 x  4  0
Part IV: The Discriminant
a. What is the discriminant?
b. What does it tell us regarding the kind of solutions?
# of solutions: ________
# of solutions: _____
Description: ____________ __________________
# of solutions: ______
____________________
# of solutions: ______
___________________
88
c. Find the discriminant of each of the following. Then, tell what kind of solutions they will be.
i. x 2  x  1  0
ii. x 2  6 x  3  0
Discriminant value: _____________
Discriminant value: _____________
Type of solutions: __________________________ Type of solutions: __________________________
“Shortcut” for finding the Vertex Quickly!!
h
b
2a
Part V: Find the vertex quickly of the following:
y  x2  4x  3
y  2 x 2  8x  7
y  3x 2  7 x  2
89
Section 5-4: Imaginary and Complex Numbers
Review:
The imaginary number “i” is just equal to
1 . For example, 5i can also be written as 5 1 .
Simplify the following expressions using imaginary numbers. The first 2 have been done for you.
1.
9
2.
5.
512
6.
What’s a complex number?
98
3.
4  12
2
7.
4
9  18
12
4.
8.
125
1  22
2
__________________________________________________________________
Plot on a complex number plane:
a. 2  3i
b.
4  5i
90
9. Solve the following quadratic equation and express your answer as a complex number(s) in the form a + bi:
4 x 2  3x  1  0
What does 1 1 or i i or i 2 equal?
_____________________________________________________
10. Simplify each expression:
a. 3(i  4)  3(4i  5)
b. (5  6i )i
c. (1  7i )(2  3i)
91
11. “Powers of i’s”: Complete the table.
Power
i1
i2
Value
i
-1
i3
i4
i5
i6
i7
i8
i
i4 
i2 
i3 
Is it true that
ab  a b for any negative real numbers a and b ?
12. Simplify the following:
a.
i14
d. i1234
b. i 27
c. i100
*e. i 7
*f. i 10
92
Section 5-5: Evaluating Quadratic Functions
1. Suppose f(x) = 2 x 2  6 x  3 .
a. Find f(-2).
b. Find f(5).
c. Find x algebraically and graphically on your TI-calc, if f(x) = 4.
d. Find the x-intercepts algebraically and graphically on your TI-calc.
2. Given y  3x2  2 x  4 , find x if y = -2.
93
3. Use the discriminant to determine if the function has a real value for the given value of y.
In simple terms, what are you being asked to do and/or find??? ____________________________
________________________________________________________________________________
a. y  7 x2  x  8; y  3
b. y  8x 2  2 x  15; y  2
94
Section 5-6: Equations of Quadratic Functions from Their Graphs
Find the particular equation of the quadratic function containing the given ordered pairs:
(-2, 15), (5, 36), (3, 10)
Write your answer in standard form, y  ax 2  bx  c .
Method #1: By linear system
Method #2: By matrices
Method #3: By calculator techniques
95
Piecewise Functions
A piecewise function has different rules for different parts of its domain.
Graph each piecewise function.
 x  6, if x  2
1. y  
 x  2, if x  2
2 x  1, if x  3
2. f  x   
 x  8, if x  3
 x  7, if x  2

3. y   3, if  2  x  2
9  x, if x  2

 5  x, if x  2
4. y   2
 x  4, if x  2
96
3x  2, if 0  x  4
5. g  x   
10  x, if 4  x  8

2, if x  0

6. f  x    x  1, if 0  x  10
 1
 x  8, if x  10
 2
Write the piecewise function represented by each graph.
7.
97
Section 5-7: Quadratic Functions as Mathematical Models
Test Cramming Problem: Jeri begins cramming for her algebra test at 10:00 pm Wednesday evening. Her
grade depends on the number of hours she studies. She figures that with no studying she would make only
40. With one hour of studying she could make 75, and with 2 hours she might make 90. Assume that her
grade is a quadratic function of the number of hours.
a. Find the particular equation for this function.
a. ________________
b. Predict Jeri’s grade if she studies 4 hours.
b. ________________
c. How long must Jeri study to make 100?
c. ________________
d. How long must Jeri study to make the highest grade?
d. ________________
e. What is the highest grade?
e. ________________
98
Chapter 5 Review
1. Solve by using the quadratic formula and give your answers in exact form: 3x 2  10 x  10  0 .
2. If f  x   5x2  8x  13 ,
a. find f(7)
b. Find x when f(x) = 10.
3. Sketch the following parabolas below:
a. vertex at (5, -7) and y-intercept = 20
b. Exactly one x-intercept at x = 1 and
y-intercept of -2
4. Transform into vertex form: y  7 x 2  42 x  13 . Then, find the vertex and axis of symmetry.
Vertex: _________________
a.o.s.: ___________________
99
5. Evaluate:
a. (4 + 6i)(3 – i)
b. 2i(5 + i)(2 + 3i)
c. i25
d. i903
e. 3 121
f.
16  4
7. Find the particular equation of the quadratic function in vertex form that passes through a vertex of
(5, -1) and the point (8, -28). Show all work!
100
8. Find the particular equation of a quadratic function that passes through the points (1, 4), (-3, 32), and
(4, 25).
9. Cattle Problem: A rancher wants to yield the maximum amount of edible beef off of his square mile of
land. At first he finds that as he adds more cattle, his yield goes up. However, if he overgrazes, the harvest
goes down. An agricultural agent tells him that his yield will vary quadratically with the number of animals
that he grazes. For five head of cattle, his production is 8750 pounds of beef, and for 10 head, he reaps 15,000
pounds of beef. He had no production when he was grazing no cattle.
a. Define the variables. Write the three ordered pairs for this function. Find the particular equation of this
function expressing pounds of beef produced in terms of the number of cattle.
b. What number of cattle will give him his maximum profit in number of pounds?
c. What is this weight?
d. What is the domain and range of this function?
101
Quadratic Function Activity
Quadratic functions have important applications in science and engineering. For example, the parabolic path
of a bouncing ball is described by a quadratic function. In fact, the motion of all falling objects can be
described by quadratic functions.
A basketball game is about to begin. The referee holds the ball at a height of 3 feet above the floor. The
referee tosses the ball upward with a velocity of 24 feet per second. A video camera follows the motion of the
ball as it rises to its maximum height and then begins to fall. The ball is subject to the acceleration due to
gravity at the Earth’s surface, which is approximately 32 feet per second squared.
The vertical motion of the basketball can be described by the quadratic model below:
1
h  t    gt 2  v0 t  h0
2
In this activity, you will compare the vertical motion of the basketball on Earth with its vertical motion on each
of the other 8 planets. On each planet, assume that the initial vertical height of the ball is 3 feet and that the
initial velocity of the ball is 24 feet per second. However, the acceleration due to gravity near the surface of
each planet is different from that on Earth. The table below contains the acceleration due to gravity near the
surface of each planet as a fraction of that on Earth.
Complete the table.
Planet
Gravity at surface (as a
fraction of Earth’s)
Mercury
0.37
Venus
0.88
Earth
1.00
Mars
0.38
Jupiter
2.64
Saturn
1.15
Uranus
1.15
Neptune
1.12
Pluto
0.04
Vertical height model:
2
Gravity at surface, g(ft/s )
32
1
h  t    gt 2  v0 t  h0
2
h  t   16t 2  24t  3
102
2. Complete the table below by using the quadratic functions obtained in #1. Use your TI-calculator to obtain the
approximate values.
Planet
Maximum height of
basketball
Time required to
reach maximum
height
Time required to
return to planet’s
surface
Time required to
reach a height of
10 feet
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
3. Are there any planets on which the ball would never reach a height of 10 feet? If so, name them.
4. On which planet would the basketball achieve the highest maximum height?
5. On which planet would the basketball achieve the lowest maximum height?
6. Make a generalization about the relationship between the acceleration due to gravity, g, and the maximum
height, h, that is reached.
103
Section 6-1: Introduction to Exponential Functions
An exponential function is in the form:
y  a  bx
where “a” is the ___________________________ and “b” is the ___________________________
Example: Given y  3x , do the following:
a. Complete the table and graph the function.
x f(x)
5
4
3
2
1
0
-1
-2
-3
-4
b. What do you notice about the pattern of x-values?
c. What do you notice about the pattern of y-values?
d. How does the negative x-axis seem to be related to the graph of y?
e. How is the equation of this exponential function different than that of a quadratic function,
y  a  b3 ?
104
Section 6-2: Exponentiation for Positive Integer Exponents
“POWER”
Xn  exponent
MEMORIZE the following powers!!
base
Perfect Squares Perfect Cubes Powers of “2” Powers of “3”
8
22=
4 23=
21=
2 31=
3
32=
42=
52=
62=
72=
82=
92=
102=
112=
122=
132=
142=
152=
162=
172=
182=
192=
202=
212=
242=
252=
33=
43=
53=
63=
73=
83=
93=
103=
22=
23=
24=
25=
26=
27=
28=
29=
210=
32=
33=
34=
35=
36=
105
Section 6-3: Properties of Exponentiation
Product Property
Quotient Property
Power Property
Simplify each expression.
1.  3a
2
 4a 
6
 3x 2 
4. 

 2 
12 x y 
2
7.
2
6 2
8x4 y7
2.  4x y
3
8a 5
3.
2a 2

5 2
5.  4 p 2 q  p 2 q 3 
3
 xy 2   6 x 
8. 
  2 
 2  y 
6. x 5  2 x 
3
11 
11 
34 17
2
9.
33 17
106
Section 6-4: Exponentiation for Rational Exponents
What is the “reciprocal” key on my calculator?
Fill in the table at the right:
23=
8
22=
21=
20=
2-1=
2-2=
2-3=
What do you notice about an exponent “raised to the 0 power”? __________________________
What do you notice about negative exponents in comparison to positive exponents? __________
_______________________________________________________________________________
Exponentiation for 0 exponent:
Exponentiation for negative exponents:
x0=
x-n=
Simplify each expression as products of powers with no variables in the denominator.
1.  2x y
5

4 3
2.
 2r

1 2 0 2
st
2rs
3.  84h 4 m6 
0
Simplify each expression. Use only positive exponents.
x 2 y 3 z 1
4.
x 5 yz 3
3
6
5. 3  3
a
a
6. 4 x

2
5
 2x
1
4
107
Section 6-5: Powers of Radicals without Calculators
Definition of n
Why is
4
x:
4 not considered simplified?
*******************************************************************************************
Evaluate without your calculator. Write your answer in exact form.
3
3
1. 16 4
2. 16 4
4. 343

2
3
5. 100
3
2
3
3.
 16 4
2
 8 3
6. 

 27 
Simplify by transforming the radicals into exponential form. Leave your answer in exact form.
7.
10.
22  11
3
32
8.
11.
32  3 16
9.
8
12.
71.4 74.2 70.8
3
61.4  60.2
5 4 3
81
108
Section 6-6: Scientific Notation
Why do we use scientific notation? __________________________________________________________
_______________________________________________________________________________________
2.35 x 108
On the number in scientific notation form above, underline the mantissa and circle the characteristic. The
mantissa can only be a number between 1 inclusively and 10 exclusively. What does “EE” stand for in terms
of scientific notation on your TI-Nnspire? Is it okay to express your answer as 2.35E8?
Write in scientific notation form.
1. 7634
2. .09462
3. 165.2 million
4. 5 ten thousandths
Simplify without your calculator.
5.  3 x 104  8 x105 
2.4 x1014
6.
1.2 x103
Write the answer in scientific notation with the correct number of significant digits.
7.  8.73 x 106 1.84 x 1013 
8.
4.25 x 1012
3.76 x 108
9. Find the reciprocal of 3.45 x 106 .
109
Section 6-7: Exponential Equations Solved by Brute Force
Using your calculator, solve correctly to three significant digits.
1. 10 x  70
2. 3x  8
Now, use the properties of logarithms to solve the above equations.
110
Section 6-8: Exponential Equations Solved by Logarithms
Solve and check your answer.
1. 6  4 x  31
2. 8.4  105 x  76
3. 9.21 107.1x  28.4
4. 3.21  546.7 x  89.01
111
Section 6-9: Logarithms with Other Bases
Exponential Form
Logarithmic Form
Write each equation in exponential form.
1. log 4 256  4
1
 2
144
3. log10  1
5. 104  0.0001
6. 252  625
2. log12
Write each equation in logarithmic form.
4. 83  512
What are the restrictions of the definition of log a y ? ________________________________________
Find x. (Check for possible restrictions!)
7. x  log 2 16
10. log x
1
 2
25
13. log 1 x  2
8
8. x  log 3
1
27
9. x  log81 9
3
5
12. log x 1  0
11. log x 8 
14. log7 x  2
15. log 8 x  
27
2
3
112
Section 6-10: Properties of Logarithms
Using your calculator, evaluate log 15.
log 15 = ___________________
Using your calculator, evaluate (log 5)(log 3).
(log 5)(log 3) = ______________
Using your calculator, evaluate log 5 + log 3.
log 5 + log 3 = ______________
Using your calculator, evaluate (log 30)/(log 2).
(log 30)/(log 2) = ____________
Using your calculator, evaluate log 30 – log 2.
log 30 – log 2 = _____________
Using your calculator, evaluate log 8.
log 8 = ____________________
Using your calculator, evaluate log(2)3.
log(2)3= __________________
Using your calculator, evaluate log 2 + log 2 + log 2
log 2 + log 2 + log 2 = ________
Using your calculator, evaluate 3log2
3log2 = ___________________
Property
Exponentiation
Logarithms
Product
Quotient
Power
Part I: Let a  log 3 , b  log 4 , and c  log 5 . Use the properties of logarithms to express each value in terms
of constants or the above logarithms (a, b, or c). Do not use a calculator!
3
5
1. log 12
2. log 16
3. log
4. log 75
5. log 50
6. log 0.8
113
Part II: Write each logarithmic expression as a single logarithm.
1. log5 4  log5 3
2. log 2 4  log 2 2  log2 8
3. 2 log x  3log y
4.
1
2
log 3 x  log 3 x
3
3
5. 5log x  3log x 2
6.
1
 log 2 x  log 2 y 
3
7. x log 4 m 
1
log 4 n  log 4 p
y
8. log5 y  4  log5 q  2log5 x 
Part III: Expand each logarithm.
9. log xyz 4
11. log
2rst
5w
10. log
5x
4y
12. log 4
x9
1
s2t 3
114
Part IV: Solve each equation. Check your answer.
13. log8  x  1  log8  2 x  2
14. log7  6 x  4  log7  3x  5
15. log2 x  log2  x  4  5
16. log8  3x  1  log8  x  1  2
17. 2logb x  logb 2  logb  2 x  2 
18. 2log3 x  log3 5  log3 14 x  3
115
Algebra 2 With Trigonometry Honors
Logarithmic Applications
Name ______________________________________
Application #1: The Richter scale: The Richter scale is used to measure the strength of an
earthquake. It is a logarithmic scale based on the powers of ten. The table below gives the effects
of earthquakes of various intensities
Richter
Number
Intensity
Effect
1
2
3
101
Only
detectable
by
seismograph
102
Hanging
lamps
sway
103
Can
be
Felt
4
5
6
7
8
104
105
106
107
108
Glass
Furniture Wooden Buildings Catastrophic
breaks, collapses houses
collapse
damage
buildings
damaged
shake
The 1906 San Francisco earthquake measured 8.3 on the Richter scale. The Loma Prieta earthquake
that interrupted the 1989 World Series in San Francisco measured 7.1.
How many times more severe was the 1906 San Francisco quake than the 1989 quake?
How much more intense was the San Francisco earthquake of 1989 than its strongest aftershock, a
4.3 on the Richter scale?
Application #2: Chemistry: The pH of a solution is a measure of its acidity and is written as a
logarithm to the base 10. A low pH indicates an acidic solution, and a high pH indicates a basic
solution. Neutral water has a pH of 7. Acid rain has a pH of 4.2. How many more times acidic is the
acid rain than neutral water?
116
pH Levels of Products Common Products
pH Level
Lemon juice
2.1
Tomatoes
4.2
Black coffee
5.0
Milk
6.4
Pure water
7.0
Eggs
7.8
Baking soda
8.5
Now try these:
a. How much more acidic is black coffee than pure water?
b. How much more acidic is milk than eggs?
Application #3. Medicine: The pH of a person’s blood can be found by using the HendersonB
Hasselbach formula. The formula is pH  6.1  log10 , where B represents the concentration of
C
bicarbonate, which is a base, and C represents the concentration of carbonic acid, which is an acid.
Most people have a blood pH of about 7.4.
a. Use properties of logarithms to write the equation without a fraction.
b. A pH of 7 is neutral, and pH numbers less than 7 represents acidic solutions. pH levels greater
than 7 represent basic solutions. Is blood normally an acid, a base, or a neutral?
c. Use your TI-Calc to find the pH of a person’s blood if the concentration of bicarbonate is 25 and
concentration of carbonic acid is 2.
117
Application #4: Acoustics and Sound decibels: Common logarithms are used in the measure
I
of sound. The loudness, L, in decibels, of a particular sound is defined as L  10 log , where I is the
I0
intensity of the sound and Io is the minimum intensity of sound detectable by the human ear.
Soft recorded music is about 4000 times the minimum intensity of sound detectable by the human
ear. Use the definitions of logarithms to find the loudness in decibels.
Other common sounds and their approximate decibel levels are listed in the chart below:
Decibels
Sounds
120
Jet engine/threshold of pain
110
Pneumatic drill
100
Food blender
90
Moderate discotheque
80
Noisy city street
70
Accounting office
60
Normal conversation
50
Average residence area
40
City night noises
30
Broadcast studio—no program in progress
20
Average whisper (4 feet)
10
Rustle of leaves
0
Threshold of hearing
Now try these:
a. The threshold of the music at a rock concert registered 66.6 decibels several miles away. How
many times the minimum intensity of sound detectable by the human ear was this sound, if Io is
defined as 1?
b. Mrs. Gapinski had a new muffler installed on her car. As a result, the noise level of the engine of
her car dropped from 85 decibels to 73 decibels. How many times the minimum intensity of sound
detectable by the human ear was the car with the old muffler if Io is defined to be 1? How many
times the minimum intensity of sound detectable by the human ear is the car with the new muffler?
Find the percent of decrease of the intensity of the sound with the new muffler?
118
Section 6-11: Proofs of Properties of Logarithms
How do you evaluate log 7 9 ?
Change of Base Property:
loga b =
Evaluate each logarithmic expression to the nearest hundredth.
1. log 4 92
2. log 6 18
How do you evaluate 3log3 7 ?
blogb x 
Evaluate each logarithmic equation.
3. x  23log2 4
4. x  3log9 100
119
The Natural Base, e
(John Napier—1618) He is credited with discovering the constant, “e”. He is also known as the inventor of
logarithms! However, Leonhard Euler popularized the use of “e”.
Using your TI-calculator:
x
 1
1. Enter the function y1  1   in your calculator. Use the following window:
x

xmin= 0
ymin= -2
xmax= 10
ymax= 5
xscl= 1
yscl= 1
2. What happens to the y-values as x-values get larger? (Change xmax = 500 and use the trace feature
and the table feature to get a visual and numerical understanding. Change table minimum to 200) The
definition lends itself to problems involving continuous compounding of interest (i.e. Think of adding
1/n of the current total to the current total “n times”.)
3. Estimate the limit of the function to four decimal places.
This number is _____________ and is written as a transcendental number, ________, as its decimal
expansion continues forever without repeating patterns.
4. Find the inverse of y  e x .
5. Graph the two functions on the same window. What do you notice?
Logarithms of base e are called ________________________ and are written in the form ____________.
120
“e” and “ln e”
Evaluate each expression to the nearest thousandth. If the expression is undefined, write undefined.
2
1. e6
2. 2e-0.5
3. e
4. ln 10,002
5. ln (-2)
6. ln
8. e3 ln 2
9. 2 ln e 4
1
5
Simplify each expression.
7. e ln 2
Solve each equation for x.
10. ln  2 x  3  21
11. ln x  ln  x  1  ln 2
12. 2ln x  2  1
13. e2 x  20
14. e4 x  22  56
2 x 1
15. e    2
121
Meet “e” in St. Louis Project
How does a 630-foot high arch stand up to the forces of nature? Will it last for its projected life of 1000 years?
How does it withstand winds up to 150 miles per hour?
The secret is in the shape of the arch, which transfer forces downward through its legs into huge underground
foundations. You can learn more about this remarkable shape, called a catenary curve, by looking at some of its simpler
x
x
 
a a
a
forms. The general equation for a catenary curve is y   e  e  where a is a real nonzero constant.
2

Complete the following steps to explore the catenary curve.
1. Graph f  x   e x .
2. Graph g  x   e x  e x
3. What are the similarities that you noticed between the two graphs above?
4. What are the differences that you noticed between the two graphs above?
122
5. Begin exploring the catenary by letting a = 2.
a. Write the equation for the catenary curve with a = 2. ________________________
b. Complete the table at the right.
x 0 1 2 3 4 5
y
c. What is y when x = -1? ________ Compare this value for y with the value of y when x = 1.
d. Why will the y-values for –x and x always be equal for this equation?
6. Now explore the graph of the equation for the curve with a = 2.
a. Graph the equation for the catenary curve
with a = 2. Describe the graph.
b. Your graph should look like an upside-down arch. What can you do to the equation to invert
the graph? (Hint: Think about how to invert, or reflect, a parabola.) Check your new
equation by graphing it.
123
7. Your catenary curve may look similar to a parabola. To see how it is different, follow the steps
below.
a. Write an equation and draw the graph for a parabola that resembles your graph from part a
of Step 6. (Hint: How do the values of a, h, and k in a quadratic equation of the form
y  a  x  h   k affect the location and shape of the parabola?
2
b. Compare the graph of your parabola with that of the catenary curve.
8. Do some research about the St. Louis Gateway Arch and its construction. State some “cool” facts
that you learned about it below!
124
Section 6-12: Inverses of Functions
1
x – 2, find the following:
2
 Given f(x) = 2x + 4 and g(x) =
1a. f(-2) = __________ (
,
)
1b. g(0) = __________ (
,
)
2a. f(-1) = __________ (
,
)
2b. g(2) = __________ (
,
)
3a. f(0) = __________ (
,
)
3b. g(4) = __________ (
,
)
4a. f(1) = __________ (
,
)
4b. g(6) = __________ (
,
)
What can you conclude about the points of f and g?______________________________________
 f and g are inverses of each other.
 Graph the line y = x and plot the points for f and g below.
The points of f are __________________ over the line y = x. The points of g are ______________ over
the line y = x. This will be true for all inverses.
125
 To find an inverse relation, you should do the following two steps:
Step #1: _____________________________________________________
Step #2: _____________________________________________________
2. Find the inverse relation of the following functions. Graph the functions on your calculator and do the
horizontal line test to determine whether or not the inverse relation is also a function.
a)
f  x   3x  6
b) g  x  
1
x 8
5
c) h  x   x 2  6
2. Sketch the inverse of each function on the coordinate system.
a)
b)
c)
126
3. Given f(x) =
1
x – 4 and g(x) = 2x + 8
2
a) Find f(g(x))
b) Find g(f(x))
c) Graph f and g
d) Do you think f and g are inverses of each
other?
4. Graph f  x   2x . Then, find f 1  x  .
127
Section 6.13: The Add-Multiply Property of Exponential Functions
We have in the past used lines and quadratic functions to model data; we consider some actual data points
and fit a function to the data to use as a predictor for other values. How can we decide when it is appropriate
to use an exponential function to model data?
Tell whether or not the function could be an exponential function.
1.
2.
x
f(x)
x
f(x)
1
2
3
100
3
6
0
20
5
18
-3
6
7
54
-6
0.8
3.
4.
x
f(x)
x
f(x)
3
1.1
6
24
10
1.21
-1
6
21
1.331
-8
1.5
28
1.4641
-15
.375
Assume that f is an exponential function. Use the property of exponential functions to calculate values of f(x)
for values of x, two larger and two smaller than the two given.
5. f 1  8 and f  3  12
6. f  5  12 and f  9  14.4
128
“Time Value of Money” (TVM) Notes:
1. Every time value of money problem has either 4 or 5 variables (corresponding to the 5 basic financial keys).
Of these, you will always be given 3 or 4 and asked to solve for the other. To solve these problems, you simply
enter the variables that you know on the appropriate lines and then scroll to the line for the variable you wish
to solve for. To get the answer, press ALPHA ENTER. Be sure that any variables NOT in the problem are set to
0, otherwise they will be included in the calculation.
2. The order in which the numbers are entered does not matter.
3. Always make sure that the P/Y (payments per year) and C/Y (coupons per year) are set to 1 unless specified
otherwise. (i.e. 1 for annual compounding, 2 for bi-annual compounding, 4 for quarterly compounding, 12 for
monthly compounding, 365 for daily compounding)
4. When entering an interest rate, input the value as a percentage not as a decimal. This is because the
calculator automatically divides any number entered on the I/Y line by 100.
5. The TVM solver was designed on purpose to follow the “Cash Flow Sign Convention.” This is simply a way
of keeping the direction of the cash flow straight. Cash inflows are entered as positive numbers and cash
outflows are entered as negative numbers. For example, if you are investing $100 into a savings account (i.e.
a cash outflow) versus being given a loan of $100 (i.e. a cash inflow).
TI-Calculator Application with Exponential Functions: TVM Solver
129
Introduction: You deposit $2000 into an account that pays 2% annual interest compounded quarterly. How
long will it take for the balance to reach $2500?
TVM Solver Directions
F1  r IJ
TVM stands for the Time Value of Money and is based on the compound interest formula: A  PG
H nK
nt

 This finance program will allow you to do the kind of everyday financial calculations that you will
encounter often as an adult.
 To access the finance program on the TI-83, press 2nd, x-1. If you have a TI-83 or TI-84 Plus, press the blue
APPS key, select option 1: Finance and press Enter. On either calculator, the finance calculation screen
appears with option 1: TVM Solver. Press Enter.
N:
I%:
PV:
PMT:
FV:
P/Y:
C/Y:
130
To use the TVM Solver:
1. Plug in the values you know
2. Put your cursor in the location for what you are trying to solve for
3. Without moving your cursor from that position, press ALPHA ENTER to solve.
Example #2: You deposit $2000 into an account that pays 2% annual interest compounded quarterly. How
long will it take for the balance to reach $2500?
N=
I% =
PV =
PMT =
FV =
P/Y =
C/Y =
Example #3: Buying a Car – You have your eye on a brand new Ford Mustang convertible. Unfortunately, it
costs $25,365. The bank has given you a 4 year loan at 6.9% annual interest, and you have $2000 to make a
down payment.
a. What will your monthly payment be?
N=
I% =
PV =
PMT =
Monthly payments = $_____________
FV =
P/Y =
C/Y =
b. Find the total amount you will end up repaying the bank over the 4 years:
c. Find the total amount spent in interest over the course of your 4 year loan:
131
Example #4: You buy a car for $28,620 and give a 20% down payment on a loan with a 4% interest rate. How
long will it take to pay off your loan if your monthly payments are $421.67?
N=
I% =
PV =
PMT =
FV =
P/Y =
C/Y =
Example #5: Buying a Home – Jim and Lisa are buying a house for $250,000. They have 5% of the total cost for
the down payment. The remainder of the cost will be financed at 6.75% annual interest for 30 years. Find
their total monthly payment, the total amount the bank will be paid and total amount spent in interest.
N=
I% =
PV =
Monthly payments = $_____________
PMT =
FV =
Total amount paid back = $_____________
P/Y =
C/Y =
Total amount spent in interest = $_____________
132
Example #6: Shopping Spree – Amy went on a shopping spree and charged $812.25 to her mother’s VISA. She
plans to pay the minimum amount of $30 each month at an annual interest rate of 19.5%.
N=
I% =
PV =
PMT =
FV =
P/Y =
C/Y =
a. How many months will it take to pay the bill entirely?
b. By using the charge card and only paying the minimum, how much did Amy actually pay for the shopping
spree?
Example #7: You deposit $600 in a savings account that is compounded semiannually. If the amount in your
account triples in 6 years, what interest rate did the savings account have?
N=
I% =
PV =
PMT =
FV =
P/Y =
C/Y =
133
TVM Solver Extra Practice Problems:
1. Suppose that Mrs. Gapinski is planning to send her three year old daughter to college in 15 years. Assume
that Mrs. Gapinski has determined that she will need $200,000 at that time in order to pay for tuition, room
and board, books, etc. If you believe that Mrs. Gapinski can earn an average annual rate of rate of 8% per
year, how much money would she need to invest today as a lump sum to achieve her goal?
2. Suppose that you have $1,250 today and you would like to know how long it will take you to double your
money to $2,500. Assume that you can earn 9% per year on your investment.
3. Suppose Mrs. Gapinski wants to send her son to college in 13 years. She has determined that she will need
$150,000 at that time in order to pay for his tuition, room and board, etc. If she has $20,000 to invest today,
what interest rate of return does she need to earn in order to reach her goal?
4. Mrs. Gapinski is saving for a pair of diamond earrings. She puts away $10 every week into an account that
pays 8% compounded weekly. How long until she has $1200?
134
5. You would like to have $10,000 available in 5 years. You will deposit a lump sum now into an account
paying 10% compounded quarterly. How much needs to be deposited now?
6. Let’s say you are an HPHS senior who just finished paying off your car. Your monthly payments were $325
and your car is worth $4000 if you sold it now to buy a new car. You are thinking about selling the car and
buying a new one that costs $20,000. You have several choices:
a) You can save your $325 per month until you have enough money to buy a new car for cash. You
make the deposits each month into an account paying 6% monthly.
b) You can sell your car and buy a new car and pay it off in 3 years. You can get a deal for 8% with
monthly payments on the unpaid balance.
c) You can sell your car and buy a new car and pay it off in 5 years. You can get a deal for 8% with
monthly payments on the unpaid balance.
d) How much is each option going to cost you? Which one is the best deal?
135
Section 6-9 to 6-11 Review of Logarithms
Find the value of the variable in each expression.
1. log 2 x  3
2. log2 5 4  r
4. log x 5  1
5. log y
7. log3 1  y
8. log32 t  .6
9. r  log 27
11. 0  log b 1
12. a  log 2   3 4 


10. log5 5 5  t
1
3

8
2
3. log x 256  .375
6. 0  log 6 t
1
9
1
2


136
Find the value of x in each statement.
13.
alogx t  t
16. x  4log 2 8
19.
x  51log5 3
14.
xlog2 7  7
15.
18. 0 
17. x  log 3 34
20.
5log5 x  8
2
log 5 2 x
3
x  72log 3
7
III. Solve for x.
21. 23 x 1  22 x
22. log.5 5  log 2 x 2  log 2 5 x
23. log(2x - 4) – log(x + 2) = 1
24. log(x + 1) + log (2x + 1) = log(6)
137
Simplify.
25. log3 27  log3 3 81
Solutions:
64
1) 8
2) 2/5
3) 2 3
4) 5
8) t = 8
9) -2/3
10) 3/2
11) b  1, b > 0
12) -1/3
13) a
14) 2
15) 8
16) 12
18) 0
19) 15
20) 9
21) x = 1
22) x = 1
23) x = -3, solution = 
24) 1
25) 9/8
5) 4
17) 4
6) 1
7) 0
138
Chapter 6 Review: How well do you know the logarithmic properties?
1. Express as a single logarithm:
1
4
log10 x 2  log10 y  log10 5
3
3
2. If log10 2 =0.3010 find log 2 100
3. Simplify:
a. 16
2
1
2
 1  3
b. 

 8 
1 5
4. Solve for x: x  log 3 
9
5
c. 9 2
 
d. 5
3
3

27 

139
5. Solve for x:
4x  5
=0
x
log6
6. Solve for x:
3
log x  3
2 3
x
256
 3
7. Solve for x:   
81
 4
9. Solve for x: 3
x 4 

1
27
8. Solve for x: 3x  92 x1
10. Solve for x: log6  6x  30  2  log6 x
140
Chapter 6 Calculator QUEST Review Worksheet
For each of the following, solve for x.
1. 2.57 x  1008
2. 2e x 3  19
3. eln 5 x  43
4. 4 ln e8 x  18
5. 6ln 8x 1  13
6. 326  74  2.18x
7. If f(x) varies exponentially with x, and f(0) =12 and f(3)= 4322, find the particular equation in terms of x.
141
8. If g  x   82  3.8x , find x when g(x) = 180.
9. If f is an exponential function and f(7) = 19, and f(11) = 5, use the add-multiply property to calculate:
f(3) = ________
f(-1) = __________
f(15) = ___________ f(19) = ______________
10. Rabbit Problem: When rabbits were first brought to Australia last century, they had no natural
enemies so their numbers increased rapidly. Assume that there were 60,000 rabbits in 1865, and that by
1867 the number had increased to 2,400,000. Assume that the number of rabbits increased exponentially
with the number of years that elapsed since 1865.
a. Write a particular equation for this function.
b. How many rabbits would you predict in 1870?
c. According to your model, when was the first pair of rabbits introduced into Australia?
11. Bacteria Problem: Suppose that the number of bacteria per square millimeter in a culture in your
biology lab is increasing exponentially with time. On Tuesday, you find that there are 2000 bacteria per
square millimeter. On Thursday, the number has increased to 4500 per square millimeter.
a. Find the particular equation for this function.
b. Predict the number of bacteria per square millimeter on Saturday of the same week.
c. Predict the time when the number of bacteria per square millimeter reaches 22,000.
142
Section 7-2: Factoring a Difference of Two Perfect Squares
Rule:
a 2  b2 
Factor completely.
1. x 2  1
2. y 2  16
3. m 2  64
4. x 2  100
5. 4 x 2  25
6. 16 y 2  49
7. 9 y 2  16
8. 4  25x 2
9. 4  m2
10. 25 y 2  64
11. 1  9x 2
12. 25  m 2
13. 4 y 6  x 2
14. 4 y 2  9
15. 9 x 4  4
16. 16 x 2  25 y 2
17. x 2  16 y 2
18. 36m 2  121
19. 49m2  100
20. x 2 y 2  1
21. 25  4x 2
22. x8  81
23. 4x 2 y 4  m2
24. 100y 2  x 2
25. 9 x 2  121
26. x 2  16
27. 16x 4  y 2
28. x 4 y 2  25
143
Section 7-2: Factoring A Perfect Square Trinomial
Rule:
a 2  2ab  b 2 
Factor completely.
1. x 2  4 x  4
2. 9 y 2  24 y  16
3. 25m2  30m  9
4. x 2  6 x  9
5. x 2  12 x  36
6. 16n 2  72n  81
7. 16 y 2  8 y  1
8. 36m2  60m  25
9. 9 x 2  42 x  49
10. y 2  10 y  25
11. 81n 2  90n  25
12. 16  40 x  25 x 2
13. 9 y 2  6 y  1
14. 64m2  16m  1
15. 64  80 x  25 x 2
16. 4 y 2  20 y  25
17. n 2  14n  49
18. x 2  8 x  16
19. 16 y 2  24 y  9
20. 9m2  30m  25
21. 25 x 2  60 x  36
22. 49 x 2  28 x  4
23. 4  20m  25m 2
24. n 2  14n  49
25. 16 y 2  56 y  49
26. 9 y 2  12 y  4
27. 36 x 2  12 x  1
28. 25 x 2  10 x  1
144
2
Section 7-2: Factoring ax  bx  c (where a = 1)
Rules:
Factor completely.
1. x 2  5 x  6
2. n 2  12n  27
3. x 2  3 x  4
4. m2  4m  45
5. y 2  4 y  21
6. x 2  11x  30
7. n 2  3n  18
8. y 2  6 y  16
9. x 2  6 y  7
10. 24  10m  m 2
11. y 2  5 y  36
12. x 2  12 x  36
13. m 2  12m  32
14. n 2  8n  9
15. x 2  x  30
16. n 2  n  6
17. x 2  3 x  4
18. y 2  10 y  21
19. n 2  3n  18
20. 7  8x  x 2
21. m 2  13m  36
22. y 2  4 y  32
145
Section 7-2: Factoring ax 2  bx  c (where a > 1)
Factor completely: 10 x 2  21x  10
Factor completely.
1. 2 x 2  15 x  7
2. 3 y 2  13 y  10
3. 6 x 2  31x  5
4. 8m 2  2m  15
5. 6 y 2  5 y  6
6. 6n 2  25n  14
7. 15 y 2  23 y  28
8. 6 x 2  x  40
9. 15n 2  8n  16
10. 10 x 2  13 x  30
11. 14 y 2  33 y  18
12. 15n 2  43n  30
13. 2 x 2  5 x  12
14. 8 x 2  27 x  20
146
Section 7-3: Special Products and Factoring
Factor:
1.
12 x 2  9 x  3
2.
9 x 2  25
3.
5 x 2  14 x  3
4.
4 x 2  20 x  25
Multiply:
5.
(5 x  4)( x  3)
6.
( x  1)( x 2  2 x  1)
7.
( x  2)3
8.
(2 x  3 y)2
147
100% Factoring QUIZ (Practice!!)
10 minutes maximum!
Solve the quadratic equations by factoring. You must show the factored form and the solutions.
1. 4 x 2  25  0
1. ________________________________
2. x 2  10 x  24  0
2. ________________________________
3. x 2  11x  28  0
3. ________________________________
4. x 2  64  0
4. ________________________________
5. 25 x 2  9  0
5. ________________________________
6. 3 x 2  20 x  7  0
6. ________________________________
7. 3x 2  10 x  8  0
7. ________________________________
8. x 2  2 x  15
8. ________________________________
9. 6 x  5  x 2
9. ________________________________
10. 4 x 2  19 x  12  0
10. _______________________________
148
Section 7-4: More With Factoring and Graphing
Multiply:
 x  2  x2  2x  4
a3  b3 
a3  b3 
Factor the sum or difference of cubes:
1.
x 3  64
2.
216m3  512
3.
125 x 3  343
Factor by splitting the middle term.
4.
6 x 2  29 x  35
6.
8 x 2  26 x  15
5.
24 x 2  2 x  15
Calculate the discriminant and use it to decide if the polynomial factors. Factor it if possible.
7.
20 x 2  61x  45
8.
24 x 2  30 x  88
149
Section 7-5: Long Division of Polynomials
How do you divide
4309
?
6
Divide by using long division:
1.
x
3
 5 x 2  18 x  48    x  6 
2.
x
3
 3x 2  18 x  40    x  4 
3.
x
3
 7 x  6    x  1
5.
x
3
 4 x 2  4 x  3   x 2  x  1
150
Section 7.6: Factoring Higher-Degree Polynomials—The Factor Theorem
Example #1: Given P( x)  x3  9 x 2  23x  15 ,
a.
Evaluate P(5) by direct substitution.
b.
Divide x 3  9 x 2  23x  15 by x – 5 using long division.
THE FACTOR THEOREM: ___________________________________________________________
Example #2: Factor the following polynomial: x3  9 x 2  23x  15  (
)(
)
This linear factor must have a constant that is a factor of -15.
151
Example #3: Factor 3x3  2 x 2  7 x  2 .
3x3  2 x 2  7 x  2  (ax - b) (
)
a must be a factor of 3; b must be a factor of –2.
If (ax – b) is a factor of P(x), then P(x) = 0 when (ax - b) = 0  x 
Possible values for
b
.
a
b
factors of  2
are
a
factors of 3
THE RATIONAL ROOT THEOREM: ___________________________________________________
Example #4: Factor x3  5 x 2  2 x  24
Example #5: Factor 3 x3  4 x 2  17 x  6
Example #6: 12 x3  20 x 2  x  3
152
7-2 to 7-6 Factoring Review
Part I. Factor completely.
1. x 2  64
2. 4 x 2  20 x  25
3. x 2  8 x  16
4. 9 x 2  25
5. x 2  8 x  15
6. x 2  11x  24
7. x 2  7 x  10
8. x 2  19 x  42
9. 2 x 2  5 x  3
10. 4 x 2  4 x  3
11. x 2  5 x  36
12. 5 x 2  14 x  8
13. 3 x 2  7 x  20
14. 6 x 2  16 x  8
15. 5 x 2  2 x  16
16. 14 x 2  x  3
17. 6 x 2  x  1
19. 5 x3  5 x 2  6 x  6 20. 4 x3  x 2 y  4 xy 2  y 3
22. x3  x 2 y  xy 2  y 3
23. x3  216
18. xy  xz  wy  wz
21. 10 xy  10 xz  7 wy  7 wz
24. 1000  27a 3
153
25. x 6  1
28. x3  5 x 2  8 x  12
26. x 3  x 2  10 x  8
27. x3  10 x 2  31x  30
29. x3  13x  12
Part II. Do the long division and write the expression in mixed number form, or in polynomial form if the
remainder is zero.
30. x  2 x3  x 2  11x  10
31. x  4 x3  2 x 2  7 x  4
154
Section 7-3 to 7-6: More Challenging Factoring Problems
Factor Completely.
1. 8a 3  27b3
2. 24 x2  6 xy 18 y 2
3. 64c 2  d 2  4  4d
4. a2  a  1  5a  5
5. ac  6bd  2ad  3bc
6. x 4  y 2  9
7. 2 x 1  x   3  x 1
8. x2   y  1


2
2
155
9. 3a  x  y   2b( y 2  x2 )
10. 56  x  x 2
11. x3  y 2  2 xy  y3  x 2
12. x  x  1 4x  5  6  x 1

 
13. 2 c3  1  7 1  c 2

14. a 4  2a 2b 2  b 4
15. x 2  6 y  9  y 2

17. 4a 2b2  a 2  b2  c2
16. 6 x 2  11x  10

2
18. a 6  b 6
156
19. 20 x 2  x  12
20. 4 x2  25 y 2  2 x  5 y
21. x 2  2 x  48
22. 1  6mn  9n 2  m 2
23. 45  12x  x 2
24. b3  b 2  c3  c 2
25. x3  x 2  x  1
26. x4  y 4  2 x2 y 2  x2  y 2


2
157
a
1   a 1
27. x4  x3 y  xy3  y 4
28.
29. c3  8h3
30. 5a 2  30a  45
31. 36  12  a  5   a  5
33.
 x  2y   x  2y
3
35. x 4  x3  x 2
5
2
2
2
2
32. x 2 a  2 x a  1
34. a 3  a 2b  a  b
36. a 6 c  5a 3c  6
158
37.  3x 1  3  3x 1 10
38. 12 x 2  8 x  15
39. 6 x3  7 x 2  7 x  6
40. r 3  3r 2  9r  5
41. d 3  18  9d  2d 2
42. 3 x 2  10 x  8
43. 24 x 2  2 x  15
44. 10 x 2  43x  35
2
159
45. x 4  x 2 y 2  y 4
46. 125 x 3  1
47. 9 x 4  8 x 2  4
48. 9 x 2  42 x  49
49. x 4  33x 2  16
50. 216x3  a 3  b3
160
Section 7-7: Products and Quotients of Rational Expressions
Multiply or divide. Write the answer in simplest form. State any restrictions on the variables.
2x4
1.
10 y 2
x2  5x  6 x2  7 x  6
2. 2
x  5x  6 x2  x  6
5 y3
4 x3
3.
x2  4
1  x2
x 1
2
x  2x
4.
x2  5x  6 x2  4 x  3
 2
x3  8
x  3x  2
5.
2 x2  5x  2 x2  x  2
 2
4x2  1
2x  x  1
6.
x3  x 2  14 x  24 x 2  4 x  5

x2  6 x  5
8  2 x  x2
Perform the indicated operation by simplifying each complex fraction.
x
3
7.
x
x
3
x
1
x2
8.
1
4
x2
4
5
x
9.
5
10  2
x
10 
161
Section 7-8: Sums and Differences of Rational Expressions
How do you add or subtract fractions together?
1. ___________________________________________________________________
2. ___________________________________________________________________
3. ___________________________________________________________________
Add or subtract. Simplify when possible!
1.
5 y  2 2x  4

4 xy
xy 2
2.
1
1

x  2 x 1
3.
3y
8

y  25 5  y
4.
2x
4x
 2
x  x  2 x  3x  2
5.
5x
4
 2
x  x  6 x  4x  4
6.
5y
4
9


y  7 y 2 y  14 y
2
2
2
2
162
Section 7-9: Graphs of Rational Algebraic Functions
Graph: f  x  
x 1
 x  1 x  3
x-intercept(s): ___________
y-intercept: ________
HA: ___________
VA: ___________
RD: ___________
Definition of a rational function: ________________________________
What is a point of discontinuity? _______________________________________________
Two types of discontinuities that can occur on a rational function graph:
1. ________________________________________________________________
2. ________________________________________________________________
Describe the vertical asymptotes and holes for the graph of each rational function.
1. y 
3
x2
2. y 
x5
x5
3. y 
x3
 2 x  3 x  1
VA = ___________
VA = ____________
VA = _______________
RD = ___________
RD = ____________
RD = _______________
intercepts: __________
intercepts: _____________ intercepts: ______________
4. y 
 x  3 x  2
 x  2 x  1
5. y 
x2  4
x2
6. y 
x2  x  2
x2  5x  6
VA = ___________
VA = ____________
VA = ______________
RD = ___________
RD = ____________
RD = ______________
intercepts: ___________
intercepts: _____________ intercepts: ______________
163
Section 7-9: Graphs of Rational Algebraic Functions
What does x   mean? __________________________________________________________
What does x   mean? ________________________________________________________
Use your calculator to sketch each rational function on the axes below.
1. f  x  
1
x
2. g  x  
2x
x 1
3. h  x  
x2
x 1
as x   , f  x   ________
as x   , g  x   ________
as x   , h  x   ________
as x   , f  x   ________
as x   , g  x   ________
as x   , h  x   ________
What did you notice between the three graphs as the x-values got increasingly large or increasingly small?
__________________________________________________________________________________________
A horizontal asymptote: _____________________________________________________________________
The three cases for horizontal asymptotes:
“bottom heavy”
“equal”
“top heavy”
y = _________
y = _________
y = _________
164
Find the domain of each rational function. Identify all asymptotes and “holes” in the graph of each rational
function. Then sketch the function.
1. h  x  
2x  2
2x  2
2. f  x  
2x
2 x  18
2
domain: ___________________________
domain: _______________________________
VA: _____________ HA: _____________
VA: _____________ HA: _________________
RD: _____________ int: ______________
RD: _____________ int: _________________
3. g  x  
x3  4 x
x2  x  6
4. g  x  
2 x2  8x
x 2  7 x  12
domain: ___________________________
domain: _____________________________
VA: _____________ HA: _____________
VA: _____________ HA: _______________
RD: _____________ int: _____________
RD: _____________ int: _________________
165
Section 7-10: Fractional Equations and Extraneous Solutions
A fractional (or rational) equation is an equation that contains at least one rational expression.
Solve. Check your solution.
x
1

x6 x4
Solve. Check your solution.
x
2x
18

 2
x3 x3 x 9
Practice problems: Solve and check your solution(s).
1. 5 
3.
4
6
x 1
b
b
10

 2
b3 b2 b b6
2.
5
x
41  2 x

 2
x  3 x  4 x  x  12
4.
x2
x2
21

 2
2x  3 2x  3 4x  9
166
Section 7-11: Variation Notes
Variation Functions: Where y is equal to a constant multiplied or divided by a power of x.
*****************************************************************************************
DIRECTLY: If the constant is multiplied by the variable, then y varies directly with the power of x.
INVERSELY: If the constant is divided by the variable, then y varies inversely with the power of x.
CONSTANT: k, the proportionality constant
DIRECTLY
y  kxn
INVERSELY
y
k
xn
Write the general equations for the following examples:
1. y varies directly with x.
1. _________________________
2. y varies inversely with x.
2. _________________________
3. y varies inversely with the square of x.
3. _________________________
4. y varies directly with the cube of x.
4. _________________________
5. y varies inversely with the 0.3 power of x.
5. _________________________
6. y varies directly with the square root of x.
_________________________
167
Chapter 7: “Hot Seat” QUIZ
1. Factor the following:
a. 9 x 2  64
b. 2 x 2  5 x  3
c. 6 x3  4 x 2  6 x  4
2. Do the long division and write the expression in mixed number form.
a. x  2 x3  x 2  9 x  6
b. 2 x 1 2 x3  3x2  x  1
3. Do the following by synthetic division.
a.
x3  2 x 2  3x  8
x 3
b.
x4  2 x  8
x2
168
4. Factor completely!
a. x3  x 2  10 x  8
b. x3  10 x 2  31x  30
5. Factor completely!
a. c3  216
b. 125  8a3
6. Perform the indicated operation.
x2  2 x  8 x2  4x  4
 2
a.
x2  2 x
x  6x  8
x2 1 x2

b.
x
x 1
169
7. Perform the indicated operation.
a.
x2 y
xy 2
 2
x  9 x  81
b.
x 2  16 x  4
 3
x2
x  64
7. Perform the indicated operation.
2
x
a.
1
2
x
x
3
x 1
b.
3
3
x 1
3
170
Section 8-1: Graphs of Irrational Functions
Graph: f  x   x
General Form:
a:
b:
h:
k:
y  a b x h   k
vertical stretch or compression by a factor of |a|; for a < 0, the graph is a reflection across the x-axis.
horizontal stretch or compression by a factor of |1/b|; for b < 0, the graph is a reflection across the y-axis.
horizontal translation h units to the right for h > 0 and |h| units to the left for h < 0
vertical translation k units up for k > 0 and |k| units down for k < 0
For each function, describe the transformations applied to f  x   x . Graph each transformed function and
give its domain.
1. g  x   x  4
2. h  x   x  4
3. h  x   2 x  3  1
domain: _______________
domain: ________________
domain: _______________
range: _________________
range: _________________
range: ________________
171
Section 8-2: Simple Radical Form
Example: Simplify
24
Two Methods for simplifying radicals:
Simplify each radical expression.
3
54
1.
50
2.
128
3.
4.
32x3
5.
98x8 y 3 z
6. 3 5 96
Simplify each product or quotient. Assume that the value of each variable is positive.
7. 3 2 x3  5 4 x3
1
8.
3
3a 2 b4   a3b5  3
9.
24 x 5
6x 
3
1
2
172
Section 8-3: Radicals and Simple Radical Form
Transform to simple radical form.
1.
3.

3 5
12
3
7.
8 3
12
5.
5 1
5
3
9

3 5



2. 7 6  3 5 4 2  6 3
4 6
4.
6
 54
10
6.
8.
2

7 2
3
4
49
173
Section 8-4: Radical Equations
Steps for solving a radical equation:
1. _______________________________________________________________
2. _______________________________________________________________
3. _______________________________________________________________
4. _______________________________________________________________
Solve each radical equation. Check your solution.
1. 2 x  5  8
3.
5.
3
x2  3 x3
4 x  1  3x  2  5
2.
3
2x  9  2
4. x  24  4 x  3
6.
3x  5  x  1
174
Section 8-5: Variation Functions with Non-Integer Exponents
2
power of x
5
a. Find the value of the constant if the function contains the order pair (32, 8).
1. Given that y varies directly with the
b. Find y, if x = 11.
c. Find x, if y = 20.
2. Given that y varies indirectly with the 1.9 power of x
a. Find the value of the constant if the function contains the ordered pair (8, 5).
b. Find y, if x = 15.6.
c. Find x, if y = 18.4.
175
Section 8-6: Functions of More Than Once Independent Variable
1. Given: y varies directly with the square of x, directly with z, and inversely with the square of w.
a. Write the general equation of this function.
b. If x = 2, w = 6, z = 3, and y = 14, find the constant.
c. If x = 3, z = 4, and w = 8, what is the value of y?
d. If y = 18.5, w = 4, and z = 9, what is the value of x?
e. Solve the general equation for z.
176
1st Semester Review Packet
Algebra 2 With Trigonometry Honors
ARE YOU READY???
Be able to…
1.
2.
3.
4.
Classify a number
Know your field axioms (i.e. closure, commutative, associative, distributive, identity, and inverse)
Know the Order of Operations (PEMDAS)
Know the Definition of an Absolute Value
 x, if x  0
x 
 x, if x  0
5. Know the classification of a polynomial by
--number of terms (monomial, binomial, trinomial, or polynomial)
--by degree (constant, linear, quadratic, etc.)
6. Know what an extraneous solution is
7. Know that when you divide or multiply an inequality by a negative value, that you REVERSE the
inequality sign.
8. Know how to find the solution set and graph of
 x  2  x  3  0
1  x 
9. Know the domain and the range of a function
10. Know how to use the vertical line test
11. Know EVERYTHING about a linear function & how to write the equation in slope-intercept form,
point-slope form, or standard form
12. Know how to graph a line
13. Know how to find both the x and y intercepts
14. Know how to find the slope of a line
15. Know how to graph horizontal and vertical lines
16. Be able to solve a linear system by either substitution, linear combination or matrix equation
(calculator!)
17. Be able to graph a system and the find the solution (intersection point).
18. Be able to evaluate with the f(x) terminology (i.e. find f(g(-2)))
19. Be able to sketch and find the intercepts of 2x + 3y + z = 6
20. Be able to graph and shade the solution set to a linear inequality system
21. Know EVERYTHING about a quadratic function (i.e. vertex and standard forms of the equation).
22. Be able to find the vertex, axis of symmetry, and a symmetric point of a quad. Eqn.
23. Know how to complete the square.
24. Be able to find the x-intercepts of a quadratic eqn by factoring or by the quadratic formula.
25. Know the quadratic formula!
177
b 2  4ac  0; 2 real solutions
26. Know the discriminant test! ( b 2  4ac  0; 1 real solution
b 2  4ac  0; 2 imaginary solutions
b
27. Know the shortcut to finding the vertex quickly ( h 
)
2a
28. Be able to evaluate imaginary solutions (i.e. i 43 )
29. Know complex numbers and their conjugates
30. Be able to set up a matrix equation to find a particular equation given 3 points for a quadratic
function.
31. Know EVERYTHING about an exponential function and its properties (i.e. product, quotient,
power of a power, power of a product, and power of a quotient)
32. Know negative and rational exponents
33. Be able to simplify radicals and rational exponents
34. Be able to write in scientific notation (the mantissa must be a number s.t. 1  x  10 )
35. Know how to add/subtract/multiply/divide numbers in scientific notation

36. Know the definition of a logarithm and its properties b y  x  y  logb x

37. Know the “special” properties and “Change of Base” property of logarithms:
logb b  1
x  blogb x
logb 1  0
log a b 
log b
log a
38. Know how to find the inverse of a function
--Switch x & y
--Solve for y and rewrite in inverse notation
39.
40.
41.
42.
43.
44.
45.
46.
Be able to find the horizontal, vertical, and points of discontinuities with rational functions
Know what an asymptote is!
Know how to factor all cases
Know how to factor by splitting the middle term if necessary
Know how to do long division and synthetic division
Know the Factor Theorem
Know how to add/subtract/multiply/divide rational expressions
Be able to solve a rational equation and check for extraneous solutions
47. Be able to solve variation functions (direct: y  kx n ) and (inverse: y 
48.
49.
50.
51.
52.
k
)
xn
Be able to graph an irrational function and find its domain
Be able to simplify a radical and a radical expression
Be able to use a conjugate to simplify a radical expression in the denominator
Be able to solve a radical equation and check for extraneous solutions
Be able to solve a variation function with non-integer exponents
Mathematical Models
1. Linear functions (chapter 3: pg. 93)
Variation Functions (chapter 7: pg. 383)
2. Quadratic functions (chapter 5: pg. 204) Var. Fcns w/ Non-integer(ch 8: pg. 435)
3. Exponential functions (chapter 6: pg. 297)
178
Chapter 1 Review:
1. Name the property being illustrated:
(ab)c = a(bc) __________________
mx(
1
) = 1 _________________
m
2. Check all that apply:
Integer
Digits
Even
(a + b) + c = c + (a + b)
a(b + c) = ab + ac
Rational
Irrational
__________________
__________________
Imaginary
Real
Transcendental
4  2i
3
5
5
4
7

.01492
0
-49
2.718…
3. Tell whether or not the given expression is a polynomial. If it is, then name it according to its degree
and number of terms. If it is not a polynomial, then tell why not.
a. 42 x 2 y 3  x 4
b. 8 xyz  5
c.
3
 ab 2
a
d. x 2  
179
Chapter 2 Review:
Tell whether the following graphs are functions and give their range and domain.
Function: ___________
Function: ______________
Function: ____________
Domain: ____________
Domain: _______________
Domain: ______________
Range: _____________
Range: ________________
Range: _______________
Chapter 3 Review:
1. What is the slope of a horizontal line? _______
2. What is the slope of a line parallel to the y-axis? ____
3. Which quadrant does each of the following ordered pairs belong?
a. (-2, 3)
b. (8, 12)
c. (-6, -7)
d. (5, -4)
4. Graph the following equations:
a. 3x + 2y = 6
b. y = 
5
x+2
4
c. 6x – y = 6
180
5. Write the equation of a line in STANDARD FORM that is parallel to the given equation and passes
through the ordered pair.
a. y = -3x (1, 4)
b. x + 2y = 6 (3, 7)
c. 5x – 7y = 8 (2, 1)
6. Write the equation of a line in POINT-SLOPE FORM that is perpendicular to the given equation
and the ordered pair.
a. x – 2y = 9 (-3, 1)
b. 3x + 4y = 5 (-3, -2)
c. 2x + y = 1 (-3, 1)
7. Find the slope and the y-intercept, given the following ordered pairs: (1, 3) and (3, -5)
m = ___________ b = _________
181
Chapter 5 Review:
1. Given y  x 2  6 x  11 , transform into vertex form by completing the square. Then, find the
vertex, axis of symmetry, and a point of symmetry. Graph it.
Vertex: __________ a.o.s. = _______
Point of symmetry: _____________
2.
Given 3x 2  2 x  5  0 , solve by using the quadratic formula.
3. Given y  3x 2  24 x  31, transform into vertex form. Find the vertex.
Vertex: __________
182
4. Identify the quadratic term, linear term, and constant term of g  x   5x  7 x2  2 .
Quadratic term: _______ linear term: ________ constant term: _________
5. Solve for x in exact form: x 2  4 x  15  0
6. Simplify the following:
a. (2i)(5i)
d.
9
25
g. i 72
b.
9  25
c.
27  3
e. i10
f. i 33
h. i 33
i. (3  2i)(5  4i)
7. Find the coordinates of the vertex of the parabola y  x 2  4 x  3 quickly.
Vertex: ____________
183
Chapter 6 Review:
1. Evaluate the following expressions:
64
4
23
a.
b.
4  82  2  33  3(4)2
2. Simplify each expression with no variables written in the denominator.
a.
 2x  3x
2
3
1 2
y

4
b.
 7a
b    21a 4b2 
5 6
 18 x 4 y 5 
d. 
3 6 
 27 x y 
8 x 1 y 5 z 3
c.
4 x 2 y 2 z 5
2
3. Simplify.
a. 32
d.
4 3
6
5
16
3
 9 2
b. 

 49 
e. 3x
1
2
 4x
c.
2
3
f.
3
512  32
3
7
x y
2
7
4
5
x y
3
5
184
4.
Write in scientific notation.
a. 0.00352
b. 986.4
5. Find x.
a. log5 x  2
d. log x 64 
3
4
g. x  10
log10 2
1log5 3
i. x  5
b. log 4 x 
1
2
c. log 3
e. x  3log 2 4
1
x
9
f. x  log 7 73
h. x  log 2 48  log 2 12  log 2 4
j. 43 x 1  83 x
k. log32 x  0.6
l. log10  2 x  4  log10  x  2  1
185
Chapter 7 Review
Factor completely.
1. 36 x 2  81y 2
2. 8a 3  125
3. 2 x2  21xy  36 y 2
4. 2 x3  3ax 2  18 x  27a
5. p 2 14 p  49  9k 2
6. x 4  x 2  16
7. x3  3x 2  18 x  40
8. a 2b5  a 5b 2
Simplify.
1.
x2  9 3  x

x2  x x2 1
2.
x 2  3x  10 x 2  2 x  3

x2  7 x  6 x2  x  6
186
3.
x
y

x y x y
4.
3
5
 2
x  x2 x  x6
2
Graph each of the following rational functions. Find any points of discontinuity and/or asymptotes.
5. f  x  
Points of
2
2
x  3x  10
Eqns. of
Discontinuities: ______ Asympt: _______
6. f  x  
Points of
x2  x  6
x 2  3x  2
Eqns. of
Discontinuities:_______ Asympt: _________
187
7.
f  x 
2x
x 3
Points of
Eqns. Of
Discontinuities: ____Asympt: _____________
8. f  x  
x 1
x  3x  4
2
Points of
Eqns. of
Discontinuities______ Asympt: _______
x3  2 x 2  11x  12
9. Given f  x  
, tell the ordered pair(s) at which there is (are) removable
x2  x  6
discontinuities.
188
1st Semester Cumulative Review (AAT-1)
Are you ready yet???
1. Explain how you can simplify an expression that has the imaginary number that is raised to
any exponent.
2. Write the equation of the line in slope-intercept form that passes through (6, 0) and is
perpendicular to 7x – 3y = 7.
3. For what value of k is the set of points (7, -2), (0, 5), and (3, k) collinear?
4. Simplify:
5.
6
.
3 5
Simplify: a. i10
6. Simplify:
5
b. i 203

c. i 576

3  10 5 3  10 .
189
7. Simplify the expressions.
x 2  xy x 2  y 2

xy  y 2 xy 2  y 3
a 2  6a  8
a2 1

a 2  a  2 a 2  8a  16
8. Solve the system: 9x – 8y = 1
3x = 2y
9. Graph the following parabolas by labeling the vertex and at least two other points and their
symmetric points.
a. y  2  2( x  3)2
V: (
,
)
b. y  2 x 2  4 x 1
V: (
,
)
190
10. Graph, on the same grid, the two functions. Then, use your calculator to find all points of
intersection. Round to three decimal places.
b. y   x  2   1
a. –3x + 2y = 12
2
2
y   x5
3
y = 4x
Pts of intersection: ________________
11.
Pts of intersection: _________________
Shade the intersection: y  2 x  1 and 3x  4 y  9
12. Solve and graph the solutions: 3x  2  4 .
S = _______________________
----------------------------------
191
13a. Graph: y  x  7  3
13b. Graph: y   x   2 for 1  x  3
Label the vertex & at least 3 other points.
14. Graph f  x  
discontinuities.
x2
. Identify vertical and horizontal asymptotes and any removable
x  x6
2
VA: __________________
HA: _________________
Remov. Discont: _____________
15. Solve and graph the solutions:
 x  2  2  3x   0 .
x 1
---------------------------------------------------
S = ________________________
192
16. If c = log 5 and d = log 3, then find log 4 0.6 in terms of c and d only.
17. Solve for x if
18. Evaluate:

3
log x  3 .
2 3
2 2 8
2

2
19. Find x if log x = log 3 + 2 log 6
20. Solve using a matrix equation:
3x – y + z = 1
2x + y + z = 2
x–y–z=3
193
21. Describe the nature of the roots in the following quadratic equations. (i.e. 2 rational, 2
irrational, 1 rational, or 2 imaginary).
a. 3x 2  2 x  8  0
b. x 2  6 x  2  0
c. x 2  6 x  9  0
22. Write the equation of a line that passes through (2, -3) and (4, 1) in standard form.
23. Write the equation of a line in standard form that passes through (-1, 4) and is parallel to
–7x + y = -3.
24. Solve the systems.
a.
3x – 8y = 5
-5x + 2y = 3
b. 7x + y – 2z = 1
-5x + y + z = 3
3x + 2y – z = 5
194
25. Solve for x. x 
26. Simplify:
2
x 3

x 1 1  x
2  2 x  1
2
x


2
x  x6 x 3 2 x
27. Find x if 7 x  3 . Round to 3 decimal places.
28. Simplify:
2
7
a.
b.
2
7 3
c.
3 5
4 7
29. Simplify the following:
a.
6
36
b. 7 45 
10
5
c. log9 27
195
d. 2log 4 20  2log 4 5
30.
 5
e. 
 x2
 5
 1 
  x3

 1

Factor completely.
a. x(a  y)  b(a  y)
b. 6 x 2  13x  7
c. 25 x 4  4
d. 4 x 2  6 y  9  y 2
e. y 3  11y 2  31y  21
f. 16 x3  54
31. Solve for x: 8
4 x3
 7  61
196
32. Use your calculator to find log5 8 to three decimal places.
33. Find x if x  4
log2 7
34. Solve for x: 5
log5 x
35. Simplify:
9
3log3 4  25log5 7
36. Solve for p if 27 2 p 1  33 p  2
37. Simplify:
33  92
33
197
38. Simplify:
12
5
27
2
3
x

1
39. Simplify:
13
3
x4
40. Solve the equation: 1  7  x  x
41. Graph the piecewise function:
 x 2  5, x  1

f ( x)   x  1,  1  x  1
4  x, x  1

198
41. Skin Problem—Phoebe Small goes to the dermatologist to have a skin disease cured. She has 70
disease spores per square millimeter of skin before the treatment starts. While she is under the
treatment lamp, the number of spores decreases exponentially with time, dropping to 56.7 per square
millimeter after 2 minutes.
a. Write the particular equation expressing spores per square millimeter in terms of time.
b. How many spores per square millimeter are left after 10 minutes?
c. How long a treatment is needed to reduce the disease to 0.2 spores per square millimeter?
42. S. Bones is the doctor in Deathly, IL, a suburb of Chicago. One day, John Garfinkle comes in
with a high fever. Dr. Bones takes a blood sample and finds that it contains 1300 flu viruses per
cubic millimeter and is increasing. John immediately gets a shot of penicillin. The virus count
should continue to increase for awhile, then (hopefully!) level off and go back down. After 5
minutes the virus count is up to 1875, and after 5 more minutes, it is 2400. Assume that the
virus count varies quadratically with the number of minutes since the shot.
a. Write an equation expressing the number of viruses per cubic millimeter in terms of the number of
minutes since the shot.
b. Dr. Bones realizes that if the virus count ever reaches 4500, John must go to the hospital. Must he
go?
199
43. Assume that your height and your age are related by a linear function. Consulting your health
records, you find that at age 5 years old, your height was 39 inches and when you were 9 years
old, you were 55 inches tall.
a. Write an equation expressing the dependent variable in terms of the independent variable.
b. Predict your height at age 16.
c. What does the h-intercept equal, and what does it represent in the real world?
d. Since you are using a linear function as a model, what are you assuming about the rate at which you
grow?
e. What fact in the real world sets an upper bound on the domain in which this linear model gives
reasonable answers?
44. Solve each equation to three decimal places.
a. e3 x  12
b. 4e
 x 1
 64
45. Simplify:
ln e
a.
4
3ln e2
b.
2
200
Chapter 9: Conic Sections
201
Section 9-2: Circles
QUADRATIC RELATION: A quadratic relation is a relation specified by an equation or inequality of the form
Ax2  Bxy  Cy2  Dx  Ey  F  0
where A, B, C, D, E, and F stand for constants, and where the “=” sign may be replaced by an inequality sign.
CIRCLES:
Definition: A _____________________ is a set of points in a plane, each of which is ___________________
from a _____________ point called the _________________.
Proof/Origin of the equation of a circle:
Example #1: Find the center and radius of each of the following circles.
a.
 x  3   y  2
2
2
 25
b. x2   y  4  144
2
c.

x 7

2
C: _______________ r: ___________
C: _______________ r: ___________
  y  1.1  13
2
C: _______________ r: ___________
202
Example #2: Complete the square to find the center and the radius. Then sketch the graph.
1. x2  y 2  2 x  4 y 11  0
2. x2  y 2 10 x  6 y  30  0
Example #3: Find the equation of a circle
1. with center (2, -3) and point (5, 1)
with center (4, 1) and point (-3, -2)
203
“h” and “k” for the coordinates of the center of a circle
Why use “h” and “k” for the coordinates of the center of a circle in the equation of a circle?
We all know that “h” and “k” are the variables for the coordinates of the center of a circle in a
circle equation. But what most people don’t know is why use “h” and “k”? There are 26
letters in the English alphabet. Why are these two lucky ones that get to be used? The x-axis
is the one of the two axes that is horizontal, “h” representing the x-coordinate of the center,
the x-axis is horizontal, and horizontal starting with the letter, “h”, it seemed right to past
mathematicians to make it the variable. You now probably think that the “y” coordinate
should be “v”. Wrong. The variable “v” already stands for vertex. Mathematicians then
decided to use the letter “k”. They used this sacred letter because it looked like the already
used variable “h”. This is all thanks to Pythagoras!
http://algebrahelp101.weebly.com
204
Section 9-3: Ellipses
Definition of an Ellipse: It is a set of points in a plane such that each point, the sum of its distances d 1 and d2
from the foci is constant.
Label the center, vertices, major axis, minor axis, and foci on the ellipse below.
Derivation for the standard form of an ellipse:
205
Section 9-3: Ellipses
Definition of an Ellipse: It is a set of points in a plane such that each point, the sum of its distances d1 and d2
from the foci is constant.
 x  h   y  k 
2
General Formula:
rx 2
ry 2
2
1
Sketch each ellipse. Find the coordinates of its vertices, center, foci, and lengths of its major and minor
axes.
a.
x2 y 2

1
16 9
center: __________
vertices: ________________________
foci: _________________________
Length of major axis: ___________
Length of minor axis: ___________
b. 36 x 2  25 y 2  900
center: ____________
vertices: _______________________
foci: ______________________
Length of major axis: _____________
Length of minor axis: _____________
206
c. 25  x  4   9  y  2   225
2
2
center: ______________
vertices: ___________________
foci: ___________________
length of major axis: _________
length of minor axis: _________
Part II: Each ellipse has its center at the origin. Find an equation of the ellipse.
a. Vertex (5, 0) and minor axis is 3 units long
b. Vertex (0, -8) and one focus at (0. -6)
Part III: Find the center and the vertices by completing the square.
a. 4 x 2  9 y 2  16 x  90 y  205  0
b. 49 x 2  16 y 2  98x  64 y  671  0
207
Section 9-4: Hyperbolas
Definition of a Hyperbola: It is a set of all points P (x, y) in a plane such that PF1  PF2  2a .
Vocabulary:
Label the following on the hyperbola at the right:
1.
2.
3.
4.
5.
6.
Center
Vertices
Foci
Asymptotes
Transverse Axis
Conjugate Axis
Standard Form of a Hyperbola:
Part I: Sketch each hyperbola. Find its center, vertices, foci, and slopes of its asymptotes.
a.
x2 y 2

1
9 16
Center: _________
b.
y 2 x2
 1
25 4
Vertices: _______________ Center: ____________ Vertices: _________________
Foci: __________ Slopes of Asym: ____________ Foci: ___________ Slopes of Asym: ____________
208
Part II: Sketch and find the following:
 y  1
2
 x  3

2
1
a. 25 x  16 y  400
b.
center: _______ rx = ______ ry = ________
center: _______________ rx = ________ ry = ________
Vertices: ___________ Foci: ___________
Vertices: _______________ Foci: ________________
Equations of Asymp: ____________________
Equations of Asymp: ___________________________
2
2
49
4
Part III: Find an equation of the hyperbola with center at the origin that satisfies the conditions.
a. a vertex at (8, 0) and a focus at (13, 0).
b. a vertex at (4, 0) and an asymptote with equation y  2 x .
209
Part IV: Sketch the following:
a. xy = 8
b. xy = -12
Part V: Find an equation of the hyperbola described.
a. Center at (4, 0); one vertex at (8, 0); one focus is at (10, 0).
b. Vertices are (6, 0) and (6, 10); asymptotes have slopes 
3
7
210
9-5: Discovering the Properties of a Parabola
Patty Paper Activity
a.
b.
c.
d.
Draw a straight line from one side of the patty paper to the other. This line will be called the directrix.
Place a point called the focus anywhere except on the directrix.
Fold the paper so that the focus lies on one end of the directrix, and crease the paper. Then unfold it.
Move the focus along the directrix, making folds as you go, until you come to the other end of the
directrix. You should make between 15 and 25 folds.
e. Compare the parabola formed by your creases with your classmates’ parabolas.
f. Make a conjecture about how to make a narrower or wider parabola.
g. Verify your conjecture by folding a second parabola.
h. Fold your parabola in half along its axis of symmetry.
i. Explain how the definition of a parabola is related to your folded parabola.
211
Section 9-5: Parabola
Definition: a parabola is the set of all points in a plane that are the same distance from a given point called
the __________ and a given line called the ________________.
Proof:
Standard Forms of a Parabola:
For problems 1-6, determine the direction that each parabola will open.
1. x  2 y 2  3
1. _________________________
2. y  4  x  1  6
2. _________________________
3. y  8 x 2
3. _________________________
2
4. x  3  y  9   1
4. _________________________
5. x  5 y 2
5. _________________________
2
6. y  10  x  1  7
2
6. _________________________
212
Given the equation of a parabola, determine the direction that each parabola will open. Then, find its vertex,
and axis of symmetry.
Direction
Vertex
Axis of Symmetry
7. y  3  x  4   1
__________
________
______________
8. x  2  y  1  6
__________
_________
______________
9. y  8 x 2
__________
_________
______________
10. x  3  y  9   1
__________
_________
______________
11. x  5 y 2
__________
_________
______________
__________
_________
______________
2
2
2
12. y  10  x  1  7
2
Example #1: Write the equation of a parabola given the focus at (5, 5) and directrix at y = -3.
213
Example #2: Write the equation of a parabola given the vertex at (6, 2) and directrix at x = 4.
Example #3: Given y  
1
2
 x  3  5 , find the vertex, focus, axis of symmetry, and the equation of the
8
directrix.
Vertex: ___________
Axis of symmetry: ___________
Focus: ____________
Eqn of the directrix: __________
Example #4: Given the equation y  3x 2  24 x  50 , graph it and then find the vertex, focus, axis of symmetry,
eqn. of the directrix, and a symmetric point.
Vertex: ________
Focus: _________ a.o.s. ________ Directrix: _________ Symmetric Point: ________
214
Example #5: Given x  
1 2
y  4 y  9 , find the vertex, focus, axis of symmetry, the equation of the directrix, and a
2
symmetric point. Also, graph it.
Vertex: _________
Focus: ________a.o.s. _________ Directrix: ________Symmetric Point: ____________
215
Section 9-5: Parabolas
Find the vertex, focus, axis of symmetry, equation of the directrix, and a symmetric point for each of the
following parabolas. Then, sketch the parabola.
1
1. x   y 2  2 y  8
2. y  2 x 2  20 x  9
2
Vertex: ___________ focus: _______________
Vertex: ______________ Focus: ____________
a.o.s. ___________ directrix: _______________
a.o.s. ____________ directrix: _____________
symmetric point: ___________
symmetric point: ___________
3. x  
1 2
y  4 y  15
3
4. y 
3 2
x  12 x  3
2
Vertex: ___________ focus: _______________
Vertex: ______________ Focus: ____________
a.o.s. ___________ directrix: _______________
a.o.s. ____________ directrix: _____________
symmetric point: ___________
symmetric point: ___________
216
Chapter 9: Conic Sections: Real-World Applications
1. Seismology: When an earthquake occurs, the most serious property damage usually occurs at or near the
center of the quake, called the epicenter. The damage is usually less severe as the distance from the epicenter
increases. Shock waves radiate from the epicenter in a circular pattern. The University of Southern California
(USC) is located about 4 kilometers west and about 4.5 kilometers south of downtown Los Angeles. A
seismograph on the campus indicated that an earthquake occurred, and it is estimated that the epicenter of
the quake was about 60 kilometers from the university. Assume that the origin of a coordinate plane is
located at the center of Los Angeles. Write an equation of the set of points that could be the epicenter of the
quake and draw the graph.
Equation: _______________________________
2. Air Traffic Control: The radar for a county airport control tower is located at (5, 10) on the map. It can
detect a plane up to 20 miles away. Write an equation for outside limits that a plane can be detected.
2. ___________________________
3. A cross section of the whispering chamber at the Museum of Science and Industry in Chicago is shaped
like an ellipse. In this chamber, a person standing at a focus point can hear a person standing at the other
focus point whispering, even though they are 43.42 feet apart. Find an equation of the cross section of the
whispering chamber if the length of the major axis is 47 feet and the length of the minor axis is 18 feet.
Assume that the center of the ellipse is at the origin and the major axis is horizontal.
3. _________________________
217
4. Space Science: The space shuttle travels in an elliptical orbit around Earth. The center of Earth is one focus
of the ellipse, and the high and low points of the orbit are both on the major axis. Suppose a shuttle is orbiting
Earth so that its high point is 200 miles above Earth’s surface and its low point is 100 miles above the surface.
Let the x-axis be the major axis.
a. Find an equation of the path of the shuttle, using the center of the ellipse, not the center of the
Earth, as the origin. Note the Earth’s diameter is about 8000 miles.
b. Find the equation of the path of the shuttle, using the center of Earth as the origin. (Hint: Where
would the center of the ellipse be then?)
5. A comet travels along a path that is one branch of a hyperbola. The equation of the hyperbola is
y2
x2

 1 . Find the coordinates of the vertices and foci and the equations of the asymptotes.
225 400
Vertices: ______________________
Foci: _________________________
Eqns. of Asymptotes:
______________________________
FYI…cool info: During World War I and II, the Long Range Navigational system (LORAN) was developed and
used. This system is based on the shape of a hyperbola. Two stations send out different signals at the same
time. A ship receives these signals and notes the difference between the time it received one of the signals
and the time it received the other. This information is used to locate the ship on a hyperbola with foci located
at the two stations. Another set of signals can locate the ship on another hyperbola, and the ship’s location is
the intersection of the two hyperbolas. Since this system does not rely on land sightings, it can be used for
successful navigation at night or for long missions over the ocean.
218
Section 9-6: Equations from Geometric Definitions
1. For each point, its distance from the fixed point (5, 0) is twice its distance from the fixed point (-5, 0).
2. Each point is equidistant from the point (4, -2) and the line y = 5.
3. Use the geometric definition of an ellipse to show that the ellipse with foci (3, 0) and (-3, 0) and
major axis 10 units long is 16 x 2  25 y 2  400 .
219
Chapter 9: Conic Sections and Key Characteristics
Circle
(x - h)2 + (y – k)2 = r2
C
Center (h,k)
r
radius = r
B
CoV1
V2
F2
C
Ellipse
V1
F1
( x  h) 2 ( y  k ) 2

1
2
2
a
b
CoV2
c2 = a2 – b2
center (h,k)
a = largest #
Major Axis in x direction
V1
Ellipse
F1
CoV2
C
CoV1
( x  h) 2 ( y  k ) 2

1
2
2
b
a
F2
V2
c2 = a2 – b2
center (h,k)
a = largest #
Major Axis in y direction
220
Hyperbola with a
horizontal transverse axis
Hyperbola with a
vertical transverse axis
F(h, k+c)
vertex: (h, k+a)
(h, b+k)
vertex
(h-a,k)
vertex
(h+a,k)
(h-b, k)
focus
(h-c,k)
focus
(h+c,k)
center
(h,k)
(h+b, k)
Center
(h, k)
(h, k-b)
F(h,k- c)
(x  h)2 (y  k)2

1
2
2
a
b
c2 = a2 + b2
Center (h,k)
Note: a is always first, not necessarily largest
like ellipses
vertex: (h, k-a)
(y  k)2 (x  h)2

1
2
2
a
b
c2 = a2 + b2
Center (h,k)
Note: a is always first, not necessarily largest
like ellipses
8
8
6
6
4
Focus
4
Focus
Vertex
2
Vertex
2
Directrix
directrix
-10
-4
-2
2
4
-5
6
5
10
8
-2
-2
-4
( x  h)  4 p ( y  k )
-4
2
( y  k )2  4 p( x  h)
-6
-6
Vertex (h,k)
Concave up/down
Vertex (h,k)
Concave left/right
-8
-8
p = distance of focus or directrix from vertex
P = distance of focus or directrix from vertex
221
Chapter 9: TI-Calculator Activity on Conics
Due: _________________________
Assignment: You are to explain your assigned conic and the effect that the xy-term has on the graph.
Discuss any key distinctions and/or unique properties of that conic. Be creative!! Show all work, steps, and
method(s) used to find the unique properties and to graph on a separate piece of paper. Finally, use the
graph link in the Math Lab to print out your graph from the TI-calculator.
Last Name begins with:
A-G
x 2  xy  y 2  1
H-R
x 2  2 xy  y 2  4
S-Z
x 2  xy  y 2  1
222
Section 9-7: Identifying Quadratic Relations
Circle:
Ellipse:
Hyperbola:
Parabola:
________ coefficients
_________ coefficients
_______coefficients
only ________
________ signs
_________ signs
_______signs
squared term!!
1. x 2  3 y 2  9
2. 3 x 2  3 y 2  9
3. 3 x 2  3 y 2  9
4. x 2  9 y  9
5. x 2  5 y 2  3 x  4 y  10  0
6.  x  5   4  y  1  9
2
2
7.  x  3  y  9
2
8. xy  8
 y  1
9.
4
2
 x  5

2
1
25
10. 3 x  y  9
11. 2  x  4   2 y 2  7
2
12. 15 x 2  13 y 2  3 x  9 y  8  0
13.  x  2    y  8   0
2
2
14. 9  y  1  x  2
2
223
Chapter 9: Conics Review Worksheet #1
Identify the shape of the graph, transform it into standard form, sketch the graph, and find the center,
vertices, foci, and/or equations of the asymptotes when appropriate.
1.
9x2  4 y2 18x 16 y 11  0
2.
x2  y2  2x 10 y 10  0
3.
x  2 y2 12 y 15
4.
25x2 16 y2  50x 128 y 169  0
224
Find the solution set.
5.
7.
7 x 2  9 y 2  31
5 x 2  9 y 2  161
x 2  7 y  16
3x 2  2 y  25
6.
8.
x2  y 2  25
x  y 1
x 2  y 2  16
x 2  2 y  17
225
9. Write an equation of the graph which is the set of point such that each point of the graph is equidistant
from the point (3, 4) and the line x = -1.
10. Write the equation of a graph whose path of points moves so that the sum of its distances from the
points F(-1, 5) and F( 9, 5) is 24.
11. Write the equation of a hyperbola that has the graphs of y 
4
4
x and y 
x as its asymptotes and
7
7
the point P(14, 0) as one vertex.
226
Chapter 9: Conics Review Worksheet #2
1. Write the equation of each equation described below:
a. The endpoints of the major axes are at (0, 10) and (0, -10). The foci are at (0, 8) and (0, -8).
b. The foci are at (12, 0) and (-12, 0). The endpoints of the minor axes are at (0, 5) and (0, -5).
c. The center is at (1, 4), one focus is at (5, 4), and one vertex is at (8, 4).
2. Write the equation of each hyperbola described below.
a. The center is at the origin with a horizontal transverse axis 8 units long and one focus is at (6, 0).
b. The endpoints of the transverse axis are at (5, 7) and (5, 1) and the endpoints of the conjugate
axis are at (12, 4) and (-2, 4). Find the slopes of the asymptotes.
c. The center is at the origin with a vertex at (0, 2) and an asymptote with equation y 
2
x.
3
227
3. Write each equation in standard form and identify the shape of the conic.
a. 25x 2  16 y 2  50 x  128 y  169  0
b. 4 x 2  4 y 2  40 y  5  0
4. Solve each system algebraically.
a.
c.
3 y 2  x 2  12
y  x2
b.
y  x2  2x  3
y  x  9
2 x 2  y 2  22
x 2  y 2  10
228
5. State whether the graph of each equation is a circle, an ellipse, a parabola, or a hyperbola.
x2 y 2

1
9 25
a. _________________________________
b. y   x  4   3
b. _________________________________
c. x 2  6 x  4 y 2  8 y  1
c. _________________________________
d. 9 x 2  36 x  4 y 2  24 y  36
d. _________________________________
a.
2
e.
x2  4 y 2  4
f. 2 x  2 y  4  0
2
2
g. x  4 x  3  y  0
2
e. ______________________________
f. ______________________________
g. ______________________________
6. Write an equation of a conic section with the given characteristics.
a. circle with center (1, -2) and diameter 12
b. hyperbola with vertices (0, 2) and (4, 2), foci (-1, 2) and (5, 2)
c. ellipse with center (2, -5), one end of each axis (2, -9) and (-3, -5)
d. parabola with vertex (1, -2), x-intercept 3, and opens to the right
229
Chapter 10: Discovery Activity about Higher Degree Functions
REVIEW:
Any imaginary roots always occur in ____________________
If a polynomial is of degree n, then there are _________________ roots.
If “3 + 2i” is a root, then _________ is automatically another root.
What is “end behavior” of a function?
Part I: End Behavior of a Function
1. Graph on your calculator:
y  4 x2  3
Describe its “end behavior”?
2. Graph on your calculator:
y  x3  2x  2
Describe its “end behavior”?
3. Graph on your calculator:
y  x4  2 x2  2
Describe its “end behavior”?
4. Graph on your calculator:
y  x5  3x3  2
Describe its “end behavior”?
SUMMARY:
5. Which of the graphs above (#1,2,3,4) are odd degree functions?
6. Which of the graphs above (#1,2,3,4) are even degree functions?
7. What do you notice about the “end behavior” of even degree functions?
8. What do you notice about the “end behavior” of odd degree functions?
Take each of the functions in #1,2,3,4 and make the a-term negative. What do you notice happens to the graph of
each of the functions? Can you describe the pattern?
230
Part II: Identifying the degree of a function from its graph.
1. Sketch the graph of y  x3  2x  2 below.
a. What is the degree of this function?
b. How many turns are there on the
graph?
2. Sketch the graph of y  x4  2x2  2
a. What is the degree of this function?
b. How many turns are there on the
graph?
SUMMARY:
What do you notice about the degree of the function and the number of turns that appear on the graph?
231
Part III: Identifying the types of zeros
1. Find the zeros or (roots) of
y  x2  7 x 10 by factoring.
Sketch the graph below and label the zeros.
a. Degree of the function:
b. Number of turns:
c. Total number of zeros:
d. Total number of real zeros:
e. Total number of imaginary zeros:
2. Find the zeros or (roots) of y  x2  4x  4 by factoring. Sketch the graph below and label the roots.
a. Degree of the function:
b. Number of turns:
c. Total number of zeros:
d. Total number of real zeros:
e. Total number of imaginary zeros:
3. Find the zeros or (roots) of y  x3  x2  8x 12 by synthetic division. Sketch the graph below and label
the roots.
a. Degree of the function:
b. Number of turns:
c. Total number of zeros:
d. Total number of real zeros:
e. Total number of imaginary zeros:
232
Part IV: For each of the following functions, state its degree, even or odd, total number of zeros, real
zeros, and imaginary zeros.
a.
Number of turns:
Degree of function:
Total number of zeros:
Total number of real zeros:
Total number of imaginary zeros:
b.
Number of turns:
Degree of function:
Total number of zeros:
Total number of real zeros:
Total number of imaginary zeros:
c.
Number of turns:
Degree of function:
Total number of zeros:
Total number of real zeros:
Total number of imaginary zeros:
233
Part V: Sketch the following without using your calculator! Check end behavior (even/odd), # of bumps,
bounces, etc.
a.
y   x  4
c.
y   x  1  x  3
2
3
e. For the function
b.
y   x  2  x 1  x  5
d.
y   x  x  2   x  3
2
f  x   x3  5 x 2  9 x  5 , find all roots, both real and imaginary.
234
Section 10-2: Complex Number Review
Review: “Powers of i’s”: Complete the table.
Power
i1
i2
Value
i
-1
i3
i4
i5
i6
i7
i8
i
i2 
i4 
i3 
Part I: Simplify in terms of i.
b. i 125
a. i 21
c. i102
Part II: Plot on a complex number plane.
a. 4 + 2i
b. -3 + 8i
c. -5 – i
Part III: Given z1=3 + 4i and z2 = 1 – 5i. Find the following:
a. z1 + z2
b. z1 - z2
d.
z1
z2
c. z1z2
e. z1
235
Section 10-3: Quadratic Equations from Their Solutions
Solve: x 2  7 x  12  0
Given the solutions: -3 and -4, write the quadratic
equation.
Put the following equations into the same form:
( x  s1 )( x  s2 )  0
ax 2  bx  c  0
What do you notice about the sum and product of the roots for a quadratic equation?
Product of roots s1s2 
c
a
Sum of roots s1  s2  
b
a
236
Find equations with these solutions.
1. 3 and 8
2. 2  3
3. 1  2i
4. –6 and
5. 7  i 3
6. 0 and –5
4
3
Solve over the set of complex numbers:
7. x 2  6 x  12  0
8. 2 x 2  7 x  8  0
Factor in the set of complex numbers:
9. x 2  2 x  5
10. 2 x 2  5 x  3
237
Section 10-4: Graphs of Higher-Degree Functions—Synthetic Substitution
Find the roots (both real and complex) of each polynomial equation.
1. P  x   x3  2 x2  13x  10
2. P  x   3x4  x3  22 x2  24 x
3. P  x   x3  3x2  4 x  12
4. P  x   3x4  11x3  14 x2  7 x  1
238
Section 10-6: Mathematical Models
1. Lumber Problem: Woody Forester has the job of figuring out how much lumber can be obtained from
various sizes of monkey puzzle trees. From sawmill records, he finds the following numbers of boardfeet of lumber can be cut from trees of the given diameters.
Diameter (feet) Lumber (board-feet)
1
10
2
99
3
324
4
745
He figures that since board-feet is a cubic measure, a CUBIC function would be a reasonable mathematical
model.
a. Find the particular equation expressing board-feet in terms of diameter.
b. How much lumber can be obtained from a tree with a trunk five feet in diameter?
c. Woody finds that the function in part a has one integer zero. What is that zero? Find all other
zeros.
d. Draw the graph of this function.
e. According to this mathematical model, what is the smallest diameter tree that will produce usable lumber?
239
Chapter 10 Review: Higher-Degree Functions
1. Using your calculator, find the zeros (roots) of P( x)  x3  4x2  5x  8 . Approximate to the nearest
tenth.
2. Review of conics: Graph each of the following on the grids below.
a.
c.
1
y  x2  x  3
8
 x  1
4
2
 y  5

9
b.
2
1
d.
 x  3   y  2
2
 y  2
2
2
16
2
 x  1

9
 25
1
240
3. Find a quadratic equation with the given roots below:
b. 3  2 5
a. 5  i
c. 3 and
4
7
d.
1
2
and
3
5
4. Evaluate:
a. i10
b. i 22
c. i 304
d. 4  i
5. Sketch below each function described below:
a. quartic function with no real zeros
b. cubic function with exactly 1 real zero
c. cubic function with no real zeros
d. quartic function with 3 real zeros and 2 imaginary zeros
6. Find the all zeros of f ( x)  x4  x3  2x2  4x  24 . Show work in finding the zeros!
241
Section 11-1: Introduction to Sequences
Review of HW: pg. 563; #1-15 odd
3.
1 1 1 1 1 1
, , , , , ,...
3 5 7 9 11 13
t7 =
t8 =
tn =
7. 3, 6, 12, 24, 48, 96, …
t7 =
t8 =
tn =
9. 32, -16, 8, -4, 2, -1, …
t7 =
t8 =
tn =
242
Section 11-2: Arithmetic Sequences
3,7,11,15,19,...
Part I: Arithmetic Sequence: is a sequence in which one terms equals a constant ________________ to the
preceding term.
Can you find the “pattern” in an arithmetic sequence that is the Term Value Formula for an arithmetic
sequence?
t1 =
t2 =
t3 =
t4 =
t9 =
tn =
Example #1: Find the 29th term of 7, 11, 15, …
Example #2: Find out which term the given number 215 is in the arithmetic sequence if t1 =7 and d = 4.
243
Section 11-3: Geometric Sequences
3,6,12,24,48,...
Geometric Sequence: is a sequence in which each terms equals a constant __________________ by the
preceding term.
Can you find the “pattern” in the geometric sequence that is the Term Value Formula for a geometric
sequence?
t1 =
t2 =
t3 =
t6 =
tn =
Example #1: Find the eighth term of 54, 18, 6, …
Example #2: Find out which term 4374 is in the geometric sequence with t1 =2 and r = 3.
244
Section 11-3: Arithmetic and Geometric Means
Part III: Arithmetic Means (average!!)
Example #1: Find one arithmetic mean between 4 and 16.
4, ___________ , 1 6
Example #2: Find two arithmetic means between 4 and 16.
4, ___________, ____________, 16
Part III: Geometric Means (geometric ratios!!)
Example #1: Find one geometric mean between 4 and 16.
4, ___________, 16
Example #2: Find two geometric means between 2 and 16.
2, ___________, _________, 16
Example #3: Find three geometric means between 3 and 48.
(both real and imaginary!!)
3, _________, ___________, _________, 48
245
Sections 11-3: Arithmetic & Geometric Means
Extra Practice WK
1. Find three arithmetic means between 52 and 110.
2. Find two geometric means between 3 and 81. (both real and imaginary)
3. Find three geometric means between 4 and 324. (both real and imaginary)
246
“Means to an End” Activity
In problem-solving situations that are based on the calculation of an average, the choice of which
average to use—arithmetic or harmonic—is critical. The results obtained from each average can be
significantly different.
ab
2
and the harmonic mean is
.
1 1
2

a b
To provide consumers with a standard to compare fuel economy for new cars, the Environmental Protection
Agency (EPA) requires that estimates of fuel consumption be attached to the window of each new car. For
example, the EPA estimates that a certain car with a 1.9 liter engine and a 4-speed transmission can travel 24
miles per gallon (mpg) in the city and 34 miles per gallon on the highway.
For any two numbers a and b, the arithmetic mean is
The Fuel Economy Guide, published by the U.S. Department of Energy, is an aid to consumers who are
considering the purchase of a new vehicle. The guide estimates the fuel consumption, in miles per gallon, for
each vehicle available for the model year. In the Fuel Economy Guide, the consumer is advised as follows:
Please be cautioned that simply averaging the mpg for city and highway driving and that looking up a
single value in estimating may result in inaccurate estimates of the annual fuel cost.
*****************************************************************************************
Exercise #1:
1. For the car described above, find the arithmetic mean of the fuel consumption in miles per gallon for city
driving and for highway driving.
2. Find the harmonic mean for the city driving rate and highway driving rate.
3. Examine each mean. Explain why there is a warning about “simply averaging” the city rate and the highway
rate when finding the annual fuel cost.
247
Exercise #2:
The number of miles that a person drives in the city may not be the same as the number of miles that the
person drives on the highway. Therefore, to find the average annual fuel consumption for the car, you need
to use a weighted harmonic mean.
Let d1 represent the number of miles driven in the city in one year.
Let d2 represent the number of miles driven on the highway in one year.
Then, the number of gallons of fuel used per year for each type of driving is as follows:
 gallon  d1
City: (d1 miles) 
gallons

 24 miles  24
 gallon  d 2
Highway: (d2 miles) 
gallons

 34 miles  34
A rational function, a, for the average annual fuel consumption for the car described on the previous page can
be expressed in terms of the total miles driven.
a(d) =
d  d2
total dis tan ce
 1
total number of gallons d1 d 2

24 34
1. Suppose that you purchased this car and drove it 12,000 miles in one year---8,000 miles in the city and
4,000 miles on the highway. Find the average annual fuel consumption for this car.
2. Determine the total fuel cost for the year if gasoline costs $2.78 per gallon.
248
Exercise #3:
Jennifer is considering a strategy for an upcoming 2-mile bicycle race. During practice she maintains a speed
of 20 miles per hour for the first mile, but fatigue reduces her speed to 10 miles per hour for the second mile.
1. Explain why Jennifer’s average speed over these 2 miles is not the same as the arithmetic mean of 20 miles
per hour and 10 miles per hour.
2. Find Jennifer’s average speed for these 2 miles.
3. Determine the speed at which Jennifer must travel during the second mile if she rides 20 miles per hour
during the first mile and she wants her average speed for the entire 2-mile trip to be 15 miles per hour.
249
Section 11-4: Introduction to Series
Evaluate:
6
1.
4k  7

k 1
2. Write S n using sigma notation: S15 for 4 + 10 + 16 + …
250
Section 11-5: Arithmetic and Geometric Series
Formula derivation for an arithmetic series:
Formula derivation for a geometric series:
Practice:
1. Find S18 for 5 + 8 + 11 + …
2. Find S8 for 3 + 9 + 27+…
251
Section 11-6: Convergent Geometric Series
An infinite geometric series is a geometric series with infinitely many terms.
A partial sum of an infinite series is the sum of a given number of terms and not the sum of the entire series.
Find the sum of the infinite geometric series, if it exists.
4  2 1
1 1 1
   ...
2 4 8
Sum of an Infinite Geometric Series: If a geometric sequence has a common ratio r and |r| < 1, then the
sum, S, of the related infinite geometric series is as follows:
S
Tell whether the geometric series converges. If so, find the value to which it converges.
1. t1  5,
r
1
2
2. t1  93,
r
3
4
Find the sum of each infinite geometric series, if it exists.
3.
9
3
1
1
    ...
17 17 17 51

 1
5.   
k  0  10 
k
4.
2 12 72 432
 

 ...
5 5 125 625
 2 
6.  

j 0  3 

j
Write an infinite geometric series that converges to the given number.
3. 0.29292929292929…
4. 0.43535353535…
252
Sections 11-1 to 11-6: Review
Tell whether the sequence is arithmetic, geometric, harmonic or neither. If arithmetic, find the common
difference. If geometric, find the common ratio.
1. 5, 10, 15, 120,…
1. ______________________
1 1 1 1
, , , ,...
3 6 12 24
2. ______________________
3. 0, -20, -40, -60, …
3. ______________________
1 1 1 1
, , , ,...
5 6 7 8
4. ______________________
5. 4, -6, 10, -14, …
5. ______________________
2.
4.
Find the specified term.
6. Fifteenth term of 20, 15, 10, 5, …
7. Tenth term if
6. ______________________
1 1
, ,1, 2,...
4 2
7. ______________________
8. Sixty-first term of the sequence for which t1 = 4 and r = 0.95
8. _____________________
9. Ninety-fifth term of sequence for which t1 = -14 and d = -3
9. _____________________
Find the specified number of arithmetic means between the given numbers.
10. three, between -8 and 72
11. Four, between 46 and 11
253
Find the specified number of real geometric means between the given numbers.
12. Two, between 3 and 81
13. Three, between 162 and 32
Find the specified number of geometric means if the common ratio is both real and imaginary.
14. Three, between 6 and 96
Evaluate.
4
5
15.
 k 2 1
16.
k 1
 1

k 1
k
3k
Write Sn using sigma notation.
17. S20 for 1 + 3 + 5 + …
18. S10 for
1 1 1
   ...
4 8 12
Tell whether the geometric series converges. If so, find the value to which it converges.
19. t1  29 and r  
2
3
20. t1 = 59 and r = 1.2
254
Section 11-7: Sequences and Series as Mathematical Models
1. Tarzan Problem
Tarzan jumps onto his grapevine and pushes off. On the first swing, he goes a distance of 50 meters. As he
swings back and forth, each subsequent distance is 75% of the previous one.
a. What kind of sequence do Tarzan’s swinging distances form?
a. ___________________
b. Write an equation expressing swinging distance as a function of the swing number.
b. ___________________
c. How far does Tarzan go on the 6th swing?
d. After how many swings will Tarzan be going no more than 0.2 meters?
c. ___________________
d. ___________________
2. Walking Problem
You start a walking program with a basic walk of 12 miles per week. Each week you increase your distance
by 5% of the amount of the week before.
a. What is the distance you walk in the fifth week?
b. What total distance have you walked in five weeks?
a. _________________
b. _________________
255
3. Flag Problem
Assume that the flag flying in front of your school loses 5% of its color each month due to fading.
a. What amount of color was in the flag when it was new?
a. __________________
b. After one month, how much color remains? After two months? After 3 months?
b. __________________
c. How much color would be left after 12 months?
c. __________________
d. If the school kept the flag in use until it had 25% color, how many months
would the flag be able to fly?
d. __________________
e. Would it ever lose all of its color? Explain.
e. ___________________
4. A car problem
A car depreciates in value each month that you own it. Assume that you paid $15,000 for a used car on
January 5, 2008. The value of the car depreciates by .06 of its present value each month.
a. What is its value on February 5, 2008?
a. ____________________
b. What is its value one year after purchase?
b. ____________________
c. What is the value when the 3-year warranty expires?
c. _____________________
d. When will the car be worth nothing?
d. _____________________
256
Section 11-8: Factorials
Evaluate each expression.
1. 5!
2. 10!
4.
12!
6!
5.
7.
 n  1!
 n  1!
8.
3. 13!
3!6!
8!
n ! n  3!
 n  1! n  2 !
6.
10!
7!3!
9.
 n  1! n  4 !
 n ! n  3!
Write in factorial form.
10. 15 14 13 12
12.
6543
13 12 11
11. 25 24 23 12 1110
13.
55 54 53 17 16
8765
Assume a and b are positive integers. Decide whether each statement is true or false. If it is true, explain
why. If it is not true, give a counterexample.
14. a! b!  b! a!
15.  ab !  a !b!
b!
16.  a !  a  
17.  a  b !  a ! b!
b
257
Section 11-9: The Binomial Formula
When you expand a binomial to a power it becomes a series of terms. There are several patterns which show
up in this series. You will be responsible for knowing these patterns and the binomial theorem!
Example #1:
Expand  x  y  :
x3  3x2 y  3xy 2  y3
Expand  x  y  :
x4  4 x3 y  6 x2 y 2  4 xy3  y 4
Expand  x  y  :
x5  5x4 y  10 x3 y 2  10 x2 y3  5xy 4  y5
3
4
5
Question #1: What do you notice about the pattern followed by the powers of x?
Question #2: What do you notice about the pattern followed by the powers of y?
Question #3: What do you notice about the degree of each term?
Question #4: What do you notice about the number of terms that will be in the series?
Example #2: Expand  a  b  and leave in factorial form.
5
258
Example #3: Expand  5c  4d  and leave in factorial form.
3
**Look for patterns between the term number and value in examples #2 and 3.
Example #4: Find the 17th term in  m  p  . Simplify!
25
Example #5: Find the 10th term in  2 x  3 y  . You may leave the coefficient in factorial form unless
17
instructed otherwise.
Example #6: Find the term that contains y 6 in  a  y  .
8

Example #7: Find the term that contains x12 in x 2  y3

18
.
259
Chapter 11: Extra Practice Problems
The following are challenging problems that allow you to extend your understanding of the basic concepts
regarding sequences and series. You must show an algebraic approach to solving each problem to
demonstrate mastery and synthesis of understanding. Good Luck!
1. Find the sum of the odd counting number less than 100.
2. What is the tenth term of the harmonic progression
1 1 1
, , ,...?
3 8 13
3. The fourth term of an arithmetic progression is –17 and the eighth term is –30. Find the sum of the
first five terms of the progression.
4. The sum of the first three terms of an arithmetic progression is 27. If the third term is 13, what are the
other terms?
260
5. If –86 is the sum of the first n terms of the geometric progression –2, 4, -8,…, what number is
represented by n?
6. What is the sum of the fifth through the fourteenth terms, inclusive, of the geometric progression
whose first term is 12 and whose common ratio is 2 ?
7. The third term of a geometric progression is 528 and the sixth term is 7 2 . What is the seventh term?
8. The first three terms of a sequence of four numbers form an arithmetic progression and the last three
terms form a geometric progression. If the sum of the first and third terms is 28 and the sum of the
second and third terms is 10, what are the numbers?
9. The sum of the first eight terms of an arithmetic series is 32 and the sum of the first eighteen terms is
162. Find the first term and the common difference.
261
Chapter 11 & 12: Partner or Group Project
Outcome:


Group designs a 15-20 minute lesson plan that will introduce the related topic to the class in a clear and
concise manner.
Each group is individually accountable for making the group presentation successful by explaining their
topic to the class.
Activity:
1. Research your assigned topic together at the library or via the internet. Take notes and outline pertinent
ideas/concepts about your topic.
2. Research for any additional materials that may assist in your lesson plan and presentation in the math lab
or by consulting a math teacher.
3. Create visual aids (posters, transparencies, worksheets, and/or a power point presentation, Prezi
presentation etc.) to help explain and present your topic to the class.
Project:
1. Using your research and new understanding of your chosen topic, present your material in a clear and
creative manner that engages the interest of the class and best describes the key ideas of the topic.
2. Your presentation must include at least one component that involves everyone in the class (such as a
hands-on activity, game, worksheet/puzzle, etc.) Be creative!! Your presentation on any mathematical
topic should be more than just a list of facts. It should also convey the elements of discovery and
excitement. Imagine what it must have felt like to be amongst the first to experience the development of
a new idea, theory, or puzzle. Include worked-out examples and diagrams or photographs. Investigate the
social and historical events surrounding your topic. Also investigate the economic and/or scientific needs
that might have promoted each the discovery or use of your topic. Finally, convince the class that your
topic was both meaningful and relevant to the field of mathematics and discovered or solved by real
people.
3. Complete the attached lesson plan, attaching any research, web-site addresses, printouts, sources used
to gather the material for your presentation. If you do not have neat handwriting, then the lesson plan
should be typed!!
4. Turn in any worksheets/materials that need to be xeroxed at least one day prior to your presentation.
5. Turn in your final lesson plan to me prior to beginning your presentation to the class.
6. Remember: Your group project/presentation should not exceed 20 minutes in length!
262
Conclusions: Grading for each pair/group will be done as follows:
1. Final Draft of Lesson Plan



Neatness/legible
All components answered in complete sentences
All components done as described
2.




Presentation
Introduction: Described origin or historical significance about topic
Explained topic/how to solve it
The topic was presented in a manner easily understandable to the class
At least one visual aid was used to present the topic
What more recent discoveries have been made that relate to your topic?
Give examples.
3. Class activity
 All students are actively engaged in the activity
 All students are able to show/demonstrate a level of understanding
of what the topic is about
 (Extra Credit): Calculator programs/applications
5 points
5 points
5 points
5 points
5 points
5 points
10 points
5 points
20 points
10 points
(3 points)
4. Evaluations
 Class evaluations
5 points
TOTAL POINTS POSSIBLE
80 points
*****************************************************************************************
Topics:
Fibonacci and the Fibonacci Sequence
Cardan’s rings
History of Pi
Golden Ratio
TVM Solver (TI-Calc)
History of Zero
Pascal’s Triangle
M.C. Escher & tessellations
Soma Cubes
Tower of Hanoi
The Mobius strip
Fractals
The Seven Bridges of Konigsberg
Bachet’s weighing problems
Slide Ruler
The abacus, including the Chinese suan pan
And the Japanese soroban
Sierpinski’s Triangle
Napier’s rods
263
Final Group Evaluation Form
Names: _________________________________
Topic: ________________________
1. Final Draft of Lesson Plan



Neatness/legible
All components answered in complete sentences
All components done as described
2.




Presentation
Introduction: Described origin or historical significance about topic
Explained topic/how to solve it
The topic was presented in a manner easily understandable to the class
At least one visual aid was used to present the topic
What more recent discoveries have been made that relate to your topic?
Give examples.
3. Class activity
 All students are actively engaged in the activity
 All students are able to show/demonstrate a level of understanding
of what the topic is about
 (Extra Credit): Calculator programs/computer website
4. Evaluations
 Class evaluations
TOTAL POINTS POSSIBLE
5 points
5 points
5 points
5 points
5 points
5 points
10 points
5 points
20 points
10 points
(3 points)
5 points
80 points
*********************************************************************************
Additional comments:
264
Lesson Plan
Topic: _________________________________
(print neatly or type it!)
Group Names: ________________________________
The objective of our lesson is to
Materials Used/Needed for our lesson:
Introduction to our lesson:
Historical explanation of our topic:
Our class activity and/or methods used to teach our lesson:
Why we chose the visual aid(s) that were used in this lesson:
Sources used/cited for this presentation:
265
Class Evaluation Form
Topic: ______________________________
Needs
Improvement
Very Well
Done
1. The clarity of the speaker’s voice.
1
2
3
4
5
2. The information presented.
1
2
3
4
5
3. Topic easily understandable.
1
2
3
4
5
4. Overall group work.
1
2
3
4
5
5. What were the strengths of this group’s presentation?
6. Any improvements/suggestions/final comments to the group?
266
Section 12-1: Introduction to Probability
Suppose that two dice are rolled, a black one and a white one. The possible outcomes are shown below.
Find the probability of each of the following events:
1. The total is 8.
2. The total is at least 8.
3. The total is less than 8.
4. The total is at most 8.
5. The total is 6.
6. The total is 11.
7. The total is between 6 and 11, inclusive.
8. The total is between 6 and 11.
9. The total is between 5 and 12, inclusive.
10. The total is 14.
11. The numbers are 1 and 3.
12. The black die is a 4 and the white die is a 6.
13. The black die is a 4 or the white die is a 6.
267
Section 12-2: Words Associated with Probability
New vocabulary terms for understanding probability:
A random experiment: _________________________________________________________________
An outcome: _________________________________________________________________________
An event: ____________________________________________________________________________
Equally likely: _________________________________________________________________________
Sample Space: ________________________________________________________________________
Probability: ___________________________________________________________________________
General notation for probability:
PE 
nE
nS 
268
Section 12-3: Two Counting Principles
Activity: A pizza shop offers a special price on a 2-topping pizza. You can choose 1 topping from each of the
following groups:
 provolone cheese or extra mozzarella cheese
 pepperoni, sausage, or ham
1. Begin a tree diagram with the two cheese choices in the space above.
2. From each cheese choice, extend a line for each meat choice.
3. How many possible different combinations of two toppings are possible?
4. If the special included a third topping of either onions or green peppers, how would you extend your
diagram to show the additional possibilities? How many total 3-topping combinations are there?
Fundamental Counting Principle #1: If there are m ways that one event can occur and n ways that another
event can occur, then there are ________________________ ways that both events can occur.
n(A and then b) = n(A) x n(B)
5. How many possible different combinations are possible if you choose a cheese or a meat topping? Draw a
new tree diagram to illustrate these choices below.
Fundamental Counting Principle #2: If there are m ways that one event can occur or n ways that another
event can occur, then there are __________________________ ways that either event can occur.
n(A or B) = n(A) + n(B)
269
Section 12-4: Probabilities of Various Permutations
What is a permutation? __________________________________________________________
(linear permutation)
Example: Find the number of ways to listen to 5 different songs on your i-Phone from a selection of 7 songs.
Formulae for finding the Permutation of n objects taken r at a time: ____________________________
How to evaluate a permutation on your calculator:
1. ______________________________________________________
2. ______________________________________________________
3. ______________________________________________________
Practice problems:
1. Find the number of ways that you can arrange 9 letters taking 4 letters at a time.
2. You have a homework assignment of 18 problems. How many ways could you work all 18
problems? How many ways could you work 16 of the 18 problems?
3. Find the number of permutations in the following words:
a. LETTER
b. ALGEBRA
c. MATHEMATICS
270
Section 12-5: Probabilities of Various Combinations
Comparing Combinations and Permutations:
Consider a state lottery in which 3 numbers from 0 to 9 are selected. The numbers are not repeated.
A lottery player can choose whether to play exact match or any-order match.
1. A person selects the numbers 8-4-1 and plays exact match. Write all of the ways that winning
number can be drawn.
2. A person selects the numbers 8-4-1 and plays any-order match. Write all of the ways that winning
numbers can be drawn.
3. Which has more ways to win: exact match or any-order match?
4. Explain why the prize is greater for winning with an exact match.
What is a combination? ____________________________________________________________________
Formulae for Combinations of n objects taken r at a time: ________________________________________
How to evaluate a combination on your calculator?
Practice problems:
1. How many ways are there to choose a committee of 2 people from a group of 7 people?
2. How many different 12-member juries can be chosen from a pool of 32 people?
3. A bag consists of 5 white marbles and 3 green marbles. You choose 4 marbles. Find the probability
of selecting each combination:
a. 2 green and 2 white marbles
b. 1 green and 3 white marbles
c. at least 3 white marbles
d. the one marble with a crack in it
271
Section 12-6: Properties of Probability
Example #1: Calvin and Phoebe visit the Children’s Memorial Hospital Ward. The probability that Calvin will
catch the chicken pox as a result of the visit is 0.13 and the probability that Phoebe will catch the chicken pox
is 0.07. Find the probability that
a. both catch the chicken pox
b. Calvin does not catch the chicken pox
c. Phoebe does not catch the chicken pox
d. Calvin and Phoebe both do not catch the chicken pox
e. At least one of them catches the chicken pox
Example #2: Terry Torrey has the following probabilities of passing various courses: Math, 90%; English 80%;
and Biology, 75%. What is his probability of
a. passing all 3?
b. failing all 3?
c. passing English only?
d. passing English and Biology only?
e. passing at least one?
f. passing exactly one?
272
“Let’s Make a Deal” Controversy
You have reached the final round of a TV game show called “Let’s Make a Deal.” Behind one
of the three numbered doors is a new car. Behind each of the other two doors is a goat.
You have chosen door number 1. The host, Monty Hall, knows what’s behind each door. He
opens door number 3 to show you a goat. Then he pops the question: “Do you want to
change your mind?” What would you do? Would you stick with door number 1 or switch to
door number 2?
Many contestants faced such a dilemma on the show, which ran for over 25 years. In 1990, a question based
on this situation was submitted to a columnist, Marilyn vos Savant, who is reported to have the highest IQ in
the world. Here’s what she said when asked the above question in her weekly newspaper column:
“Yes, you should switch. The first door has a one-third chance of winning, but the second door has a
two-thirds chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you
pick door No. 1. Then the host, who knows what’s behind the doors and will always avoid the one with the
prize, opens them all except door #777,777. You’d switch to the door pretty fast, wouldn’t you?”
Nearly a year later, an article in the New York Times reported that Marilyn vos Savant had received about
10,000 letters in response to her answer. Most of the letters disagreed with her. Many came from
mathematicians and scientists with arguments like this:
“You blew it! Let me explain: If one door is shown to be a loser, that information changes the
probability of either remaining choice—neither of which has any reason to be more likely—to ½. As a
professional mathematician, I’m concerned with the general public’s lack of mathematical skills. Please help
by confessing your error and, in the future, being more careful…
Activity:
1. Before you analyze the problem in detail, explain which strategy you think is best and why.
2. Make three cards (2 goats, 1 car) to model the situation. Try 10 games in which you stick with your choice.
Then try 10 games in which you switch. Compare the results.
3. Make a table with column headings as: door with car, door you choose, door you are shown, result if you
switch, result if you stick. Complete it using all possible options. According to your table, should you
switch? Explain.
4. Suppose that you wrote the response for Marilyn vos Savant’s column. How would you answer the
question about the game show?
273
Section 13-2: Measurement of Arcs and Rotation
Introduction to Trigonometric and Circular Functions
I. Measurement of Arcs and Rotations
a) Positive angles have an initial side on the positive x-axis and are measured counter-clockwise.
Examples: 45
90

220
0
180
90

0
180
270
270
b) Negative angles have an initial side on the positive x-axis and are measured clockwise.
-270
Examples: -60
-225
-270
0
-180
0
-180
-90
-90
c) Coterminal angles are angles that share the same terminal side.
Example: 220, 580, -140, -500
90

0
180
270
274
d) Reference angle is the acute angle or arc between the terminal side and the x-axis.
Examples: 220 has a reference angle of 40
-60 has a reference angle of 60
90

-270
0
180
0
-180
270
-90
Practice: Draw each angle and find its reference angle
1) 120
2) 285
90

0
180
90

0
180
270
270
e) Radians are arc length. They can be measured the same way as an angle, but they represent a piece of
the circumference of the circle.
Examples:

4
2
3

2


2
0, 2

0, 2
3
3
2
2
275
f) Converting degrees to radians, and radians to degrees.
Degrees to radians
degree 
Radians to degrees

180
radian 
Examples: Convert 60 to radians
60


=
180 3
Convert
180


to degrees
6
 180

= 30
6

Part I: Convert the following angle measures from degrees to radians.
1. 120
2. 4180
3. 315
Part II: Convert the following angle measures from radians to degrees.
4.
5
6
5. 3
6.

2
276
Mark the radian measures on the unit circle below.
277
Section 13-3: Definitions of Trigonometric and Circular Functions
Evaluating trigonometric and circular functions
a) Trigonometry is based on the study of the unit circle. The unit circle has center at (0,0) and a radius of 1
unit.
(0,1)
(1,0)
(-1,0)
(0,-1)
cos =
adj x x
  x
hyp r 1
sec =
hyp r 1
1
  
adj x x cos 
sin =
opp y y
  y
hyp r 1
csc =
hyp r 1
1
  
opp y y sin 
tan =
opp y sin 
 
adj x cos 
cot =
adj x cos 
 
opp y sin 
Therefore, any point on the unit circle can be expressed as
(x,y)  (cos, sin)
Know these special right triangles!
30
45
2x
x 3
x 2
x
60
x
45
x
Using the trig ratios, define sine, cosine, and tangent. (SOHCAHTOA)
sin  =
r
cos =
y
θ
x
tan  =
 x, y    r cos , r sin 
Label the positive/negative values of the trigonometric functions that live in each quadrant.
How can you remember this?
278
Reciprocal trigonometric functions
sec 
csc  
cot  
Part I: Find the six trigonometric functions for 60
(0,1)
(1,0)
(-1,0)
sin 60 
csc60 
cos 60 
sec60 
tan 60 
cot 60 
(0,-1)
Part II: Find sin  , cos , and tan  and their reciprocal functions.
a) The angle θ intersects a circle at the point (4, 7)
sin  
csc 
cos 
sec 
tan  
cot  
b) The angle x intersects a circle at the point (-5, -8)
sin  
csc 
cos 
sec 
tan  
cot  
c) The angle x intersects a circle at the point (6, -2)
sin  
csc 
cos 
sec 
tan  
cot  
279
Part III: Label the four points on the unit circle. (These points are crucial for you to know!)
Part IV: Using what we learned in parts one and two, find the following without a calculator!
a. sin 90
b. cos 270
c. sin 2
d. cos   
e. sin180
f. cos

2
Part V: Name each quadrant described. (Think of the circle and what sin, cos, and tan represent.)
a) sin θ> 0 & tan θ< 0
b) cos θ< 0 & sin θ> 0
c) sec θ< 0 & csc θ< 0
d) sin θ <0 & cot θ> 0
Part VI: Solve the equations below, where θ is in radians.
a) cos θ = 1
b) sin θ = 0
c) sin θ = -4
d) tan θ does not exist
280
Part VII: State whether the values are positive, negative, or neither.
2
3
11
6
a. cos5
b. sin
d. cos300
 
e. cos   
 2
f. sin 48
g. cos 45
h. sin  315
i. cos90
 7 
j. sin 

 4 
k. tan 60
l. tan 285
 7 
m. sin  

 3 
n. cos  120
o. tan90
c. sin
Part VIII: Special Right Triangle Review
281
Find the six trigonometric functions of the following values.
a. 30
b. 135
sin 30 
csc30 
cos30 
sec30 
tan 30 
cot 30 
sin135 
csc135 
cos135 
sec135 
tan135 
cot135 
c. 120
Part IV: Find the exact values of the following functions.
a. sin 45
b. cos120
c. tan 45
282

d. cos(30)
 3 
e. sin  
 4 
f. cos
g. tan 60
h. sin180
 
i. tan   
 2
j. sec135
k. csc225
l. sin 
m. cot  150
n. cos
3
2
2
o. csc300
283
Section 13-1 to 13-3 Review
1. Find the least positive co-terminal angle to 380 .
1. ___________________
2. Find the greatest negative co-terminal angle to 140 
2. ___________________
Convert the following to degrees:
3.
3
4
4.

5
5.
7
10
Convert the following to radians:
6. 220
7.  80
Find the exact values of
8. cos 60
11. sin

4
9. sec 300
10. tan
12. cot180
13. csc
5
4
7
6
284
Evaluate the given expression, leaving the answer in simplest radical form.
14. tan120  cot  30
15. sin300 csc300
16. 20sin 60 cos 240
17. cos180 cos 45  sin180 sin 45
18. cos 2 150  sin 2 150
19.
20. sin
2

2

cos  cos
sin
3
6
3
6
sin120
cos120
21. csc 2   tan 2 
285
Section 13-4: Approximate Values of Trigonometric and Circular Functions
Use a calculator to find each value. Round your answers to three decimal places.
13
5
3. cot
 10 

 7 
6. tan  27452
1. sin 347
2. cos
4. sec  84 
5. csc  
7.  sec18 tan  67
8.  csc
10. Tan 112
11. csc  Sin
13. tan 72
14. sec1212'


 
13 
 cot

8 
8 


3 

5 
5
8
9. Cos7911'
12. tan16723'
15. csc6514'
286
Section 13-3: The Wrapping Function
For many students, the conversion and understanding between degrees and radian measure, or vice
versa can seem quite arbitrary. The process is more than memorization and application of a formula. It is
important to clearly understand why mathematicians needed an alternative unit of measure. By using an
inexpensive paper plate and a strip of adding-machine tape, you can solve the mystery of using radians as
angle measures.
Mathematicians have used the term radian for only a little more than a century. It developed as a
contraction, combing the words radi-al and an-gle. The purpose of the unit was to serve as a standard
measure for angles coming from, or radiating from, the center of a circle (Whitaker 1994). Wheelwrights
commonly used the term radian to relate the length of a spoke of a wheel to its circumference (McGinty,
Mutch, and Van Beynen 1985). This correspondence allowed the angle measure to be determined by the
linear measure of the corresponding arc. Consequently, all units of measure in the circle could then be linear.
The wheelwright’s use of the radian constitutes the basis for a concrete development of the radian measure.
The activity presented below will help you to experience the concept of radian measure. Although the activity
focuses on radian measurement, the activity also connects many geometric concepts regarding the circle.
Materials Needed:
Paper Plate
Adding-machine tape
Scissors
Activity:
(Students should work in pairs)
1. Each student should be given a paper plate and a strip of adding-machine tape that is longer than the
circumference of the plate. For best results, the paper plate should be flattened. (See figure 1 below).
2. One student of each pair should wrap the adding-machine tape around the paper plate as a representation
of the plate’s circumference. Allow the adding-machine tape to overlap, and then the other student
should cut through the overlapping section with scissors so that the ends just meet. Each pair of students
should repeat this process, reversing roles.
3. You will then have a length of adding-machine tape that is exactly the length of the circumference of the
circle.
4. Compare the length of the adding-machine tape, that is, the circumference, with the diameter of the
circle. What do you notice? Record your observations below:
5. Notice that  is the ratio of the circumference of a circle to the diameter of the circle, a ratio that will
always be a little more than three. Remember the wheelwright’s use of the term, radian, as a relationship
between the spoke of a wheel and the circumference of the wheel.
6. Now, fold the paper plate into four equal sections by folding in half, then in half again. When the plate is
unfolded, the creases act as spokes of the wheel.
7. Next, mark your adding-machine tape in “spoke” units by using one of the creases as a guide. You should
finish with your tape measured with six markings, the sixth coming slightly before the end of the tape.
8. Notice the relationship between the formula for the circumference of the circle to the paper plate and the
adding-machine tape. The tape is just a little more than six, or 2  , radii. You should think of the radius as
a unit radius, like the unit circle. The radius could also be considered as one spoke of a wheel.
287
9. On your paper plate, you should mark the end of one spoke as 0. Then use the adding-machine tape to
mark one radius along the edge of the plate. The central angle associated with the arc whose length is one
radius is considered to have an angle measure of one radian. How many spokes, or radii, will you need to
measure the circumference? ________Mark the 2  on your plate as well.
10. Now switching ideas to radians as a measure of an angle, you will need to begin comparing the degree
measures of familiar angles with their corresponding radian measures.
11. Fold your adding-machine tape in half and label this crease as  and as 180. Now label where this
distance of  “spokes”, or radii, along the circumference will occur on your paper plate.
12. You should begin to explore other angle measures with corresponding radian measures with
corresponding radian measures of the arcs intercepted by those angles.
13. Create folds halfway between each of the current folds in both your paper plate and the adding-machine

tape. Label the increments of
respectively on both the paper plate and on your adding-machine tape.
4



14. For and
radians, fold the section between 0 and
into three equal sections, the folds for
6
3
2


and
may lack some precision because of the difficulty in folding the plate. You may opt to fold your
6
3
adding machine tape first in half, then in thirds, and then in half again to use the folds from the tape to
mark the paper plate more accurately. You can decide which method works best for you.


15. How can you find other radian measures such as multiples of and
?
8
12
Describe your method to exploring and finding these measures on your paper plate and on your addingmachine tape.
288
13-5: Graphs of Trigonometric and Circular Functions

Graph y  sin x
(0,1)

2
 (-1,0)
(1,0) 0, 2
3
(0,-1)
2
x
sin x
0

2

3
2
2

Graph y  cos x
(0,1)

2
 (-1,0)
(1,0) 0,
2
3
2
1)
(0,-
x
cos x
0

2

3
2
2
289
Definitions
A cycle is the shortest repeating portion of a periodic function. (One full sine, cosine, or tangent graph)
A period is the horizontal length of each cycle of a periodic function. (How long does it take before it
“repeats” itself!)
The amplitude is the distance from the graph’s axis (middle of the graph) to a high or a low point. (The
distance from the sinusoidal axis to either the maximum value or the minimum value.)
The vertical shift moves the sinusoidal axis up or down from the x-axis.
The general equation for a sinusoidal function is:
y = C + AcosB(x – D) or y = C + AsinB(x – D)
A is the amplitude
C is the vertical shift
B is the number of cycles the sinusoid makes in 2 units
D is the phase displacement or horizontal shift

Period p =
For the graph of y = sinx, state the period:___________ and the amplitude:_____________.

For the graph of y = cosx, state the period:___________ and the amplitude:_____________.
Practice:
a) Graph: y  sin x
y  sin 2 x
y  sin 3 x
y   sin x
amplitude:________
amplitude: _______
amplitude: _______
amplitude: _______
period:______________
period: _____________
period: _____________
period: _____________
290
b) Graph: y  sin x
y  2sin x
y  3sin x
y  2sin 3 x
c) Graph: y  cos x
y  cos 2 x
y  cos 4 x
y   cos x
d) Graph y  cos    30
y  3cos 2   30
amplitude:________
amplitude: _______
amplitude: _______
amplitude: _______
period:______________
period: _____________
period: _____________
period: _____________
amplitude:________
amplitude: _______
amplitude: _______
amplitude: _______
period:______________
period: _____________
period: _____________
period: _____________
amplitude:_______
period:___________ horizontal shift:_________
amplitude: ______
period: __________ horizontal shift: ________
291


e) Graph y  2sin  x  
2

amplitude:_______
period:___________ horizontal shift:________

To graph a trigonometric function on your calculator, follow these steps

To graph this circular function on your calculator, follow these steps
292
Section 13-6: General Sinusoidal Graphs
For each function, find the vertical shift, phase shift, amplitude and period. Then, sketch the graph.
1. y  3  2sin 2   10
VS: ________ HS: __________ Amp: _________ Per: ___________
2. y  5  3cos 4   60
VS: ________ HS: __________ Amp: _________ Per: ___________
1
x  
4
VS: ________ HS: __________ Amp: _________ Per: ___________
3. y  1  5cos
293
4. y  4  2sin
5. y  2  5cos
1
 x  3 
2

4
 x  4
1

6. y  1  6sin 2  x  
2

VS: ________ HS: __________ Amp: _________ Per: __________
VS: ________ HS: __________ Amp: _________ Per: ___________
VS: ________ HS: __________ Amp: _________ Per: ___________
294
Section 13-6: General Sinusoidal Graphs
Graph each of the following for at least one full cycle. Find the amplitude, period, horizontal (phase) shift, and
the vertical shift.
1. y  2  2sin 3


2. y  3  4sin 2  x  
2



3. y  5  2 cos 3  x  
6

4. y  1  6cos8   20
1

5. y  1  4 cos  x  
2
2


6. y  3  sin 2  x  
4

295


7. y  4  sin 2  x  
8

8. y  tan 2 x
9. y  2 cos  x
10. y  3sin   x  4
11. y  2  5cos

4
 x  4
12. y  8  4cos   x  0.5
296
Section 13-6: Reciprocal Function Graphs
y  csc x
y  sec x
y  cot x
For each of the following functions, find the phase shift, vertical shift, amplitude and period. Then, sketch
the graph.
1. y  2  4 csc

5
 x  3
2. y  3  2sec3   20
VS: _________ PS: __________ Amp: __________ Per: _________
VS: __________ PS: __________ Amp: ___________ Per: _________
297
3. y  10  20sec
4. y  6  7 csc

4

3
 x  1
 x  2
5. y  1  3csc12   30
VS: __________ PS: __________ Amp: ___________ Per: _________
VS: __________ PS: ___________ Amp: __________ Per: _________
VS: __________ PS: __________ Amp: __________ Per: ___________
298
6. y  3 tan 4
VS: __________ PS: ___________ Amp: __________ Per: __________


7. y  tan 3  x  
6

VS: __________ PS: ___________ Amp: __________ Per: __________
8. y  cot 2 x
VS: __________ PS: ___________ Amp: __________ Per: __________
9. y  cot  x   
VS: __________ PS: ___________ Amp: __________ Per: __________
299
Section 13-6: Tangent & Reciprocal Function Graphs
Sketch one complete cycle for each of the trigonometric or circular functions listed below. Identify the
amplitude, vertical shift, phase shift, and period for each function. (For x, graph in radians and for  in
degrees)
1. y  tan x
2. y   tan x
3. y  tan 2 x
4. y   tan 3 x
5. y  2 tan  x  45


6. y  tan 3  x  
6

300
7. y  4 csc 3
8. y  2sec 5
9. y  4  2csc3   30


10. y  1  3sec 4  x  
4

11. y  3cot 6 x
12. y  2cot   90
301
Section 13-7: Equations of Sinusoids From Their Graphs
Write the particular equation of the sinusoid graphed.
1.
A = _________ Period = _________
VS = ________ HS = __________
2.
A = __________ Period = ________
VS = _________ HS = __________
3.
A = __________ Period = ________
VS = _________ HS = __________
302
Section 13-8: Sinusoidal Functions as Mathematical Models
1. EMPLOYMENT: The number of people employed in a resort town can be modeled by the function
 x 
g  x   1.5sin 
 1  5.2 , where x is the month of the year (beginning with 1 for January) and g  x  is the
 6

number of people (in thousands) employed in the town that month.
a. What type of resort might this be? Explain.
b. About how many people are permanently employed in the town?
c. Find two months when there are about 4500 people employed in the town.
d. If a major year-round business in the town were to close, which one of the constants in the function
model would decrease?
2. FERRIS WHEEL PROBLEM: The world’s largest Ferris Wheel, as of 1998, is the Cosmoclock 21 in Yokohama
City, Japan. Its center is 344.5 feet above the ground and it has a diameter of 328 feet. The center of the
Cosmoclock 21 is located at the origin of the coordinate plane. Assume that a point, P, begins its rotation at
(164, 0) and that it rotates in a counterclockwise direction. You find that it takes you 6 seconds to reach the
top and that the wheel makes a resolution once every 20 seconds.
a. Sketch the graph of this function.
b. What is the lowest point that you reach on the wheel?
c. Write the particular equation of this function.
d. Predict your height when t = 12 seconds.
e. Predict your height when t = 15 seconds.
303
Section 13-9: Inverse Circular Functions
Use a calculator to find the value of  , in degrees, for each trigonometric function. Round your answer to
three decimal places.
1. sin   0.4899
2. cos  0.8258
3.   C sc1 3
4.   cot 1 0.1432
5. tan   1.7
6.   Sec 1 4.2
Use a calculator to find the value of x, in radians, for each circular function. Round your answer to three
decimal places.
7. cos x  .234
8. x  Arc csc0.765
9. x  Arc tan 2.9143
10. x  Arc sin 0.497
304
Section 13-9: Inverse Circular Relations
relation

y  sin 1 x
y  sin x
y  arcsin x

2
function
y = Sin-1 x
1



2

2
0
-1

2

y = Arcsin x
-1
Range



2
 y

2
0
1
0
1
0
1

2

2

relation

2
y  cos1 x
y  cos x
y  arccos x
function
-1
-1
1
y = Cos x
y = Arccos x



2

2
0
-1

2

Range


2

2
0  y 
relation

y  tan 1 x
y  tan x
y  arctan x

2
function
1
y = Tan-1 x



2

2
0
-1

2

y = Arctan x
-1
Range



2
 y

2

2

2
305
relation

y  csc1 x
y  csc x
y  arc csc x

2
function
y = Csc-1 x
1
-1
y = Arccsc x



2


2
0
2
-1


Range


2
 y

2
0
1
0
1
0
1

2

2
and y  0
relation

y  sec1 x
y  sec x
y  arcsec x

2
function
y = Sec-1 x
1
y = Arcsec x



2


2
0
2
-1


Range
0  y 
and y 
-1

2

y  cot 1 x



2

2

2
y  arc cot x
1
function
0
-1
2

2
relation
y  cot x


2

y = Cot-1 x
-1
y = Arccot x

Range

2
0  y 

2
306
Section 13-9: Inverse Circular Functions
307
Section 13-9: Inverse Trigonometric Functions
Find:
1
a.   sin 1  
2
a. __________________________________
1
b.   Sin 1  
2
b. __________________________________

2
c.   cos 1  

 2 
c. __________________________________

2
d.   Cos 1  

 2 
d. __________________________________
e. x  arctan 3
e. __________________________________
f. x  Arc tan 3
f. ___________________________________
308
Section 13-9: Inverse Trigonometric Functions
Find each value.

1. Cos 1  


 2

 2 
3

2 


3
8
5. sin  Sin 1 
4. Arc cos1



7. tan  Cos 1 
3

2 
8. sec  Cos 1


12 

13 
11. Sin 1  cos
10. cot  Arc sin



13. sin  Arc tan



2. Sin 1 

3

3 





2

9

3. Arc tan  
3

3 


3
5
6. cos  Sin 1  
9. csc  Arc tan  1



3
12. Cos 1  tan

15. csc  Sin 1
14. cot Sin 1 0


3 

4 
9

10 
309


16. sec  Cos 1

4

5
 1 


19. sin  Arc sin    
2



22. Arc cos  sin
2 

3 


17. Cos 1  Sin


6

20. cos Tan 1 3



5 

23. cos  Arc tan    
12




15 

17 


3
5
18. sin  Cos 1
21. tan  Sin 1 

24. sin Cos 1


2
 Tan 1  1 
2

310
Section 13-10 & 13-11: Inverse Circular Relations
For each of the equations below, transform it so that  or x is in terms of y. Then, find the first three positive
values of  or x for which y = 4.
1. y  2  4cos   10
2. y  4  2 cos
1
  360 
2
3. y  3  6cos5   20
4. y  1  5cos

4
 x  0.8 
(Hint: use your calculator in radian mode and round to three decimal
places!)
311
Chapter 13 Review
I.
Find the measure of the reference angle of each of the following:
a.
II.
7
6
Find the exact value of each of the following functions (if possible):
a. cot180
d. csc
III.
c. 75
b. 235
4
3
b. sin

6
e. tan 270
c. sec
3
4
f. cos
11
6
Graph the following functions and find the amplitude, period, horizontal (phase) shift, and vertical
shift.
a. y  1  sin12   6
b. y  2  6 cos

4
 x  3
A = _________ Period: __________
A = ___________ Period: ____________
HS: ________ VS: _________
HS: _________
VS: ___________
312
IV.
Find the least positive angle measurement that is co-terminal.
a. 74
V.
43
5
c. 1426
Find the greatest negative angle measurement that is co-terminal.
a.
VI.
b.
85
6
25
4
b. 920
c.
  
a. csc 

 6 
b. cot 210
c. sec270
d. sec  225
e. csc330
f. cot
Find each exact value.
9
4
VII. Use a calculator to find the value of  , in degrees, for each trigonometric function. Round your
answer to three decimal places.
a. sin   0.4899
b.   cos 1 0.8258
c. sec x  1.733
d. x  Arc cot 0.1223
VIII. Use your calculator to find the following values. (Use parentheses!!)
3cos 65  tan11
a.
cot 5
sec
b.
3
5
4

sin
 cot
5
5
c.  sec18 tan  67
313
Poster Project
It’s finally here….your extra credit opportunity! Please read carefully as your bonus points
depend on it.
Your mission: To help Mrs. Gapinski make her room a little more mathematically
appealing by making a math poster.
Due Date: _________________________________ No late posters will be accepted!
Bonus Points will be awarded on the following:





Poster is at least 11 x 17 (inches) in size
Poster is colorful
Poster includes a saying or quote that is both positive and math related
(you may make up the quote or cite someone else)
Poster is neat and easy to read
Work shows thoughtfulness and effort
314
Section 14-1: Properties of Trigonometric Functions
I. Reciprocal Properties
1
= ________
sin x
1
1
= _________
= __________
cos x
tan x
csc x = ________
sec x = _________ cot x = __________
Thus, (sin x)(csc x) = _______
(cos x)(sec x) = _______
(tan x)(cot x) = _______
II. Quotient Properties
tan x = __________
cot x = _____________
III. Pythagorean Identities:
From the unit circle:
cos 2 x  sin 2 x  1
1  tan 2 x  sec 2 x
cot 2 x  1  csc 2 x
315
IV. Use the trigonometric identities and your knowledge of reciprocal trigonometric functions to
transform the following:
“Rule of Thumb”: 1. Write expressions in terms of sine and cosine to simplify.
2. Substitute in the trigonometric identities where appropriate.
a) csc x  tan x to sec x
c) sin  + cot   cos 
b) csc x  tan x  cos x to 1
to csc 
d) cos2x – sin2x
to 1 – 2sin2x
316
e) sec  - cos 
g)
cos 
sec   tan 
to sin   tan 
to 1 + sin 
cot 2 x
f)
csc x
to csc x – sin x
h) cot  + tan 
to sec   csc 
317
Section 14-2: Trigonometric Identities
I. Prove that each equation is an identity.
Steps in proving identities:
1. Work with the more complicated side to simplify.
2. Perform algebraic operations such as:
a. add fractions
b. factor
c. multiply by a clever form of 1.
d. distribute, square, or multiply polynomials
3. Write expressions in terms or sine and cosine
4. Substitute one of the three trigonometric identities where appropriate.
5. Keep looking at the answer to make sure you are on the right track.
Directions: Prove that each equation is an identity. Show all steps CLEARLY in a vertical proof!
a) sec x(sec x – cos x) = tan2x
b) (sec  + 1)( sec  - 1) = tan2 
c) cos4x – sin4x = 1 – 2sin2x
d)
cos x
cos x

 cot 2 x
sec x  1 tan 2 x
318
e)
sec2 x  6 tan x  7 tan x  4

sec2 x  5
tan x  2
f) (1  cos2 )(cot )  sin  cos 
g)
sin x cos x csc x


cos x sin x cos x
h)
i)
sec2 x  1
 sec2 x
2
sin x
j) tan 2 x  sin 2 x  sin 2 x tan 2 x
csc x cos x

 tan x
cos x sin x
319
m)
1  sin 2 x
 sin x
csc x  sin x
n)
o) (tan 2   1)(cos2   1)  tan 2   2
p)
1
1

 2sec 2 
1  sin  1  sin 
r)
q)
sec x  cos x
 tan 2 x
cos x
cot   1 1  tan 

cot   1 1  tan 
(1  cos2 )(1  cos2 )  2sin 2   sin 4 
320
Section 14-2: Trigonometric Identities #2
Prove the following trigonometric identities in a vertical proof.
tan x sin x
 sec x  1
1  cos x
1. sin 3 x cos 2 x  sin 3 x  sin 5 x
2.
1
3. sec x  tan x 
sec x  tan x
tan 2 x  6 tan x  5 tan x  5
4.

sec2 x  2
tan x 1
5. tan 2 x sin 2 x  tan 2 x  cos 2 x  1
321
FORMULA SHEET
Composite-Argument
Double-Argument
cos  A  B   cos A cos B  sin A sin B
sin 2 x  2sin x cos x
cos  A  B   cos A cos B  sin A sin B
cos 2 x  cos 2 x  sin 2 x  1  2sin 2 x  2cos 2 x  1
sin  A  B   sin A cos B  cos A sin B
tan 2 x 
sin  A  B   sin A cos B  cos A sin B
2 tan x
1  tan 2 x
tan A  tan B
1  tan A tan B
tan A  tan B
tan  A  B  
1  tan A tan B
tan  A  B  
Half-Argument
1
sin x  
2
1
cos x  
2
1
tan x  
2
Linear Combination of Sine and Cosine
1
1  cos x 
2
1
1 cos x 
2
1  cos x
sin x
1  cos x


1  cos x 1  cos x
sin x
A cos x  B sin x  C cos  x  D  ; where
C  A2  B 2 , cos D 
A
B
, and sin D 
C
C
Sum and Product
2 cos A cos B  cos  A  B   cos  A  B 
2sin A sin B   cos  A  B   cos  A  B 
2sin A cos B  sin  A  B   sin  A  B 
2 cos A sin B  sin  A  B   sin  A  B 
1
1
 x  y  cos  x  y 
2
2
1
1
cos x  cos y  2sin  x  y  sin  x  y 
2
2
1
1
sin x  sin y  2sin  x  y  cos  x  y 
2
2
1
1
sin x  sin y  2 cos  x  y  sin  x  y 
2
2
cos x  cos y  2 cos
Law of Sines
Law of Cosines
Area
sin A sin B sin C


a
b
c
a2  b2  c2  2bc cos A
1
A  bc sin A
2
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Section 14-3: Properties Involving Functions of More Than One Argument
I. Even and Odd Functions
Even Functions
Odd Functions
f  x  f  x
f  x   f  x
II. Co-Function Properties: Functions of Complementary Arcs
III cos( A  B)


cos   x  
2



sin   x  
2



cot   x  
2



tan   x  
2



csc   x  
2



sec   x  
2

Proof:
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IV. cos( A  B)
V. sin( A  B)
VI. sin( A  B )
VII. tan( A  B )
VIII . tan( A  B)
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IX. Find the exact values for the following:
1. sin15
2. cos 75
3. tan 75
X. Prove that the given equations are identities.



1. sin x  30  cos x  60
  cos x
2.


2 cos  x    cos x  sin x
4

325
Section 14-4: Multiple Argument Properties
I. sin 2A
II. cos 2A
III. tan 2A
326
Demonstrate the double-angle properties:
1. cos 2A; A = 30°
2. tan 2A; A =

4
3. Calculate sin 2A, cos 2A, and tan 2A for the angle or the arc described.
a. sin A 
12
, Quadrant I
13
4
b. cos A   , Quadrant II
5
c. tan A  
5
, Quadrant IV
12
327
4. Prove the identity: csc2  cot 2  tan
328
Section 14-5: Half-Angle Formulas
Half-angle derivations:
sin
1
x
2
1
cos x
2
tan
1
x
2
I. Verify that the half-argument properties actually work by substituting the given measure of the angle or the
arc into the formula and showing that you get the right answer.
a. cos
1
x
2
x  30
b. sin
1
x
2
x=

4
c. tan
1
x
2
x
3
2
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II. Calculate sin ½ x, cos ½ x, and tan ½ x for the angle described.
a. cos x 
3
5
630  x  720
b. sin x 
5
13
180  x  360
III. Prove the identity:
sin x
1  cos x

 2 csc x
1  cos x
sin x
330
Section 14-6: Sum and Product Properties
I. Find the exact value of the expressions:


1
3
1. cos  2 Arc sin 


2. sin  Tan 1
1
1
 Tan 1 
2
3
II. Sum and Product Properties
a. Prove: sin  A  B   sin  A  B   2sin A cos B
b. Prove: cos  A  B   cos  A  B   2cos A cos B
331
Transform the indicated product to a sum or difference of sines or cosines of positive arguments.
1. 2cos73 sin 62
2. 2cos53 cos 49
Transform the indicated sum or difference as a product of sines and cosines of positive arguments.
3. cos 47  cos59
4. sin 9 x  sin11x
Prove that the given equation is an identity.
sin5x  sin3x  4sin 2x cos 2x cos x
332
Section 14-7: Linear Combination of sine and cosine
Transform the given expression to the form C cos (x – D), assuming that the functions are trigonometric and
circular.
1.
3 cos x  sin x
2.
6cos x  6sin x
3. 15cos3x  8sin 3x
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Section 14-9: Trigonometric Equations
Solve the following equations for 0  x  2 .
1. sin x  cos 2x
2. cos 2x  cos x  2  0
3. cos 2x  cos x  0
4. sin 2 x  2 sin x  0
Solve for the indicated domain.
5. tan 2   tan   0
  [90, 90]
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14-9: Trigonometric Equations
Find the exact solutions of each equation for 0    360 .
1. 2 cos 2   cos   1
2. sin   sin  cos  0
3. 2 tan 2   sec 2 
4. 2 cos 2   sin   1  0
Find the exact solutions of each equation for 0  x  2 .
5. 2sin
x
1  0
2
7. 2cos3x  1  0
6. cos x  sin x  0
8. 1  cos 2 x  cos 2 x  2 cos x  1
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Chapter 14 Exploration Lab
This exploration lab provides you with an opportunity use the various trigonometric formulas that you’ve
learned in this chapter through real-world applications. Show all work to receive full credit. You may use
your calculator on it. Good luck!
Exploration #1: Sum or Difference of Angles
In Earth’s northern hemisphere, the day with the most hours of sunlight occurs around June 22, and the day
with the fewest hours of sunlight occurs around December 22. Suppose that E is the amount of light energy
reaching a square foot patch of ground when the sun is directly overhead. When the sun is not overhead, the
amount of light will depend on the angle that a ray of sunlight makes with the horizon.
On June 22, the maximum amount of light energy falling on a square foot of ground at a certain location is
given by E sin 113.5    where  (the Greek letter phi) is the latitude of the location. How would the
amount of light energy that you receive compare with the amount received by other parts of Earth?
Use the formula for sin     to find the light energy that falls on a square foot of ground in each of the
following cities on June 22. Express your answer in terms of E, the energy from an overhead sun. Check your
answer by substituting directly into the expression given above.
a. Anchorage, Alaska (latitude: 61.2  N)
b. Key West, FL (latitude: 24.6  N)
c. Highland Park, IL (latitude: 42.181  N)
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Exploration #2: Double-Angle & Half-Angle Formulas
A plan that travels at the speed of sound (about 740 miles per hour) is said to be traveling at Mach 1. The
Mach number, named after the Austrian physicist Ernst Mach (1838-1916) is defined as the ratio of the speed
of the plane to the speed of sound. On October 14, 1947, Charles (Chuck) Yeager became the first person to
fly an aircraft faster than Mach 1. In the process, his Bell X-1 rocket airplane created a sonic boom.
When a plane travels at a Mach number greater than 1, a sonic boom is created by sound waves forming a
cone that intersects the ground in the outline of a hyperbola. If  is the measure of the angle at the vertex of

1
the cone, then the Mach number is related to  by the equation sin 
, provided that M > 1.
2 M
a. A plane traveling at supersonic speed sends out sound waves that form a cone with a vertex angle of 60  .

Find the speed of the plane if Mach 1 is about 740 mph. (Find the value for sin first. Then, set that value
2
equal to 1/M and solve for M. Multiple the number of M that you found by 740 mph to get the speed of the
plane.)
b. The Mach number for a certain plane traveling at supersonic speed is 1.4. Find the measure of the vertex
angle of the cone formed by the sound waves that the plane sends out.
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Section 15-1: Right Triangle Problems
Solve each triangle.
1.
2.
48
24
55 
7.9
3. In ABC ,
C is a right angle. Find the remaining sides and angles. Round your answers to the
nearest tenth.
a. b = 5, c = 10
b. a = 8.1, b = 6.2
4. Ballooning: From a hot-air balloon 3000 ft above the ground, you see a clearing whose angle of
depression is 20 . Find your horizontal distance from the clearing.
338
Section 15-2: Law of Cosines
I. Derivation of the Law of Cosines
y
A
C (0,0)
B (a,0)
x
When do we use the Law of Cosines? ________________________________________________________________
Law of Cosines
II. Application of the Law of Cosines
1. In ABC, a = 7, b = 3, and mC = 130, find c.
2. In DEF, d = 5, e = 10, and f = 12. Find mF.
339
3. In ABC, a = 6, b = 14, and c = 22. Find mC.
4. Two airplanes leave an airport at noon, the first headed due north and the second headed 37 east of north.
How far apart are the two airplanes at 2:00 p.m. if the first airplane is traveling at 125 mph and the second is
traveling at 158 mph?
340
Section 15-3: Area of a Triangle
How do we find the area of a right triangle?
How do we find the area of non-right triangles?
Example #1: A triangle has sides of lengths 12 in. and 15 in., and the measure of the angle between them is
24 . Find the area of the triangle to the nearest tenth.
Example #2: A triangle has sides of lengths 10 m, 12 m, and 13m. Find the area of the triangle to the nearest
tenth.
Example #3: Two sides of a scalene triangle are 9 m and 14m. The area of the triangle is 31.5m 2. Find the
measure of one of the angles of the triangle to the nearest tenth of a degree.
341
Section 15-4: Oblique Triangles--Law of Sines
Derivation of the Law of Sines:
When do we use the Law of Sines? __________________________________________________________
Examples:
1. In RST , m R  78 , m T  39 , and TS = 19 in. Find RS.
2.
342
Section 15-5: Law of Sines—Ambiguous Case
Example: Given m A  35 , a = 11, and b = 15. Find m B .
*************************************************************************************************
Multiple Choice Question:
You can construct a triangle with compass and straightedge when given three parts of the triangle (except for
three angles). Which of the following given sets could result in an ambiguous case?
a. Given: three sides
b. Given: two sides and an included angle
c. Given: two sides and a non-included angle
d. Given: two angles and a non-included side
*****************************************************************************************
Practice Exercises: In each ABC , find the measures for B and C that satisfy the given conditions. Draw
diagrams to help you decide whether two triangles are possible. Remember that a triangle can have only one
obtuse angle!
1. m A  62 , a = 30, and b = 32
2. m A  43 , c = 16 mm, and a = 24 mm.
343
3. m A  23.6 , a = 9.8, and b = 17
4. m A  155 , a = 12.5, and b = 8.4
5.
344