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Math 89 Hybrid Course
Notes
Spring 2016
Instructor: Yolande Petersen
How to use these notes
The notes and example problems cover all content that I would normally cover in face-toface (f2f) course. If you can do every example problem, you should have reasonable
coverage of the course. My personal recommendation:
1. Look over the notes before each class. Do all the examples you know how to do easily,
and make note of the ones that are difficult for you
2. In class, take notes on the problems that are presented. If time permits, students may
request to see a few additional difficult problems.
3. After lecture, try to complete all the problems in the notes, using videos if necessary.
4. If you still have trouble, post your question in the discussion forum, where a classmate or
the instructor can answer it.
5. Check your notes with the posted completed notes.
6. Steps 1-5 are equivalent to what you would learn from previewing and attending class
every day in a f2f course. Now you should be ready to complete the homework.
Explanation of symbols:
1. Asterisks (*) indicate problems that I intend to cover in class. Some problems without an
asterisk will also be covered, if time permits. If time is tight, I may skip a * problem.
2. When an example has 2 problems with “OR”, the first problem is what I would prefer to
present in class, and the second is a similar problem with a Khan Academy video . If
time allows, I will use the first example in class; otherwise the second example with
video can be used to learn the concept.
3. Problems with a corresponding Khan Academy video available are highlighted in blue.
Problems with a corresponding Petersen video will be highlighted in yellow
Problems with a corresponding Souza video will be highlighted in green
1
R. 1 Introduction to Algebraic Expressions
Topics include:





Sets of numbers
o Natural #’s: {1, 2, 3, ….}
o Whole #’s: {0, 1, 2, 3, ….}
o Integers: {… - 3, - 2, - 1, 0, 1, 2, 3, ….}
o Rational #’s: {n| n = a/b, where a and b are integers, b  0}
Order of Operations
1. Parentheses/Grouping Symbols (including radicals, absolute value, etc.)
2. Exponents
3. Multiply & Divide
4. Add & Subtract
3  19
Example a: Simplify 11  2
7
3 1
Laws/Properties:
o Commutative
 Addition: a + b = b + a
 Multiplication: ab = ba
o Associative
 Addition: (a + b) + c = a + (b + c)
 Multiplication: (ab)c = a(bc)
o Distributive: a(b + c) = ab + ac
o Identity
 Addition: 0 + a = a  additive identity is ______
 Multiplication: 1 x a = a  multiplicative identity is ______
o Inverse
 Addition: a + (-a) = 0  additive inverse is _____
 Multiplication: a x ( 1/a) = 1  multiplicative inverse is _____
o Multiplication by 0: 0 x a =
o Multiplication by -1: -1 x a = -1(a) =
Equivalent fractions  both reduced forms will generally be accepted on tests
100
4 50
2
 16 
 16
6
6
3
3
Combining like terms;
Ex b Simplify: (4x2 + x – 5) – 2(x2 – 7x – 3)
2
R. 2 Equations, Inequalities, and Problem-Solving
Solving Equations - Isolate the Variable
Use the Add. & Mult. Prop. of Equality to “undo” addition & multiplication
Ex a Solve: 4(1 – 3x) = 9 – 7x
3 possible outcomes:
1. One solution (conditional) Solve: 3x + 1 = 7  x = 2 (one solution)
2. No solution (contradiction) Solve x + 3 = x  3 = 0 (no solution)
3. Infinitely many sol. (identity/dependent system) Solve x = x  0 = 0 (inf. sol.)
*Solving Inequalities
Algebraic
 x>3

x<-2

80 < x < 90
Set Builder Notation
Interval Notation
Goal in solving: Isolate x
Caution 1: If you multiply or divide by a negative number, the inequality symbol
_________
Caution 2: Subtracting is not the same as multiplying by a negative #.
Ex b Solve and graph on a number line, giving your solution in set builder AND
interval notation: 2y – (4y – 3) > 10
3
R.3 Introduction to Graphing
3 Forms of Linear Equations
1. General Form: Ax + By = C, e.g. 3x + 2y = 12
2. Slope-Intercept: y = mx + b, e.g., y = -3/2x + 6 (m & b useful landmarks)
3. Point- Slope : y – y1 = m(x – x1) (not a final answer form), e.g (y - 3) = -3/2(x - 2)
x-intercept: point where the line crosses the x-axis (y = 0)
y-intercept: point where the line crosses the y-axis (x = 0)
Ex a
Graph the equation: 9x + 16y = 72 by calculating and plotting intercepts
Slope = m =
m 
rise
vertical change

run horizontal change
y 2  y1
where (x1, y1) and (x2, y2) are the coordinates of a point.
x 2  x1
Generic slope concepts:
Positive slope: line _________;
Negative slope: line __________
Steep lines have slopes with _______________, “gradual” lines have
_______________
*Ex b Draw a rough sketch of the lines with the given m and b (in <30 seconds):
4
Graphing a Line using Slope-Intercept Form (anchor and count)
1. Use b to anchor a point on the y axis
2. Use m to determine how many vertical and horizontal units to count for another
point.
2
Ex c
Graph the equation: y =  x  3
3
Vertical and Horizontal Lines
Ex d
Find the intercepts, slope, and graph of the equation y = - 3
Ex e
Find the intercepts, slope, and graph of the equation x = 2
Finding the Equation of a Line
When you see "Find the equation of the line that…" use Point-Slope Form:
Point – Slope Form – useful tool for intermediate work, but not for final answer
(y – y1) = m(x – x1), where
m = slope
(x1, y1) = coordinates of a point
x & y remain as variables
5
*Ex f Find the equation of the line thru (-1, -4) and (3,2) in slope-intercept form.
To get an answer in General Form (Ax + By = C):
1. Get rid of fractions
2. Get x and y terms on one side, number on other side
3. Get x positive
Ex g Convert the point-slope equation above to general form.
Parallel and Perpendicular Lines
(Given 2 lines with slopes m1 and m2)
1. The lines are parallel if their slopes are
2. The lines are perpendicular if their slopes:
 have opposite sign
 are reciprocals
To find slope: 1) Isolate y; 2) From y = mx + b, use m, the coefficient of x
*Ex h Determine whether the lines below are parallel, perpendicular, or neither:
y = 2x – 5
2x – 4y = 11
6
R.4 Polynomials and Factoring
Exponent Rules (integer exponents)
1. Product: b m  b n  b mn
2. Power to a power: (bm)n=bmn
3. Product to a power: (ab)n = anbn
n
an
a
4. Quotient to a power:    n
b
b
n
b
5. Quotient: m  b nm
b
6. Zero Exponent: bo = 1
1
7. Negative Exponent: b-n = n
b
 2x 2 y 3
Ex a Simplify: 
5
 x




2
Polynomial Vocabulary
monomial – a number, variable, or product of numbers and variables (which may be
raised to whole-number exponents). We often use the word “term” to have the same
meaning as “monomial” (although there are exceptions).
polynomial – 1 or more terms added or subtracted
coefficient – the number part of a term
degree of a term - for terms with one variable, the exponent of the variable; for terms
with 2 or more variables (rare), the sum of the exponents on the variables
descending order - a polynomial written with highest powers first.
leading term - the highest power term; the first term of a descending order polynomial
degree of a polynomial – the degree of the highest power term only
like terms - terms that have exactly the same variable parts (including exponents)
Ex b Write the polynomial in descending order, fill in the chart for each term and
give the degree of the polynomial: x2 – x + 3x4 + 6
Term
Coefficient
Variable Part
Degree of Term Degree of Polynomial
7
Add/Subtract Polynomials - combine like terms
Ex c Simplify: (x3 + 3x – 6) + (-2x2 + x – 2) – (3x – 4)
Multiply Polynomials
Ex d (5a – 2)(4a2 + 3a – 1)

FOIL - works for _______________________________
Ex e (3x + 2)(4x – 7)

Special Products (should memorize)
 (A + B)(A – B) =

(A + B)2 =

(A – B)2 =
Ex f (6x – 5y)2 =
Division by a Monomial - separate terms and cancel factors in each term
18x 4  3x 2  6x  4
Ex g Simplify:
6x
8
Procedure- Dividing by Polynomials
1. Divide the 2 leading terms. Put quotient piece above bar.
2. Multiply by the divisor
3. Subtract
4. Repeat until terms are used up
5. Write remainder over divisor
Ex h Divide
6x 3  x 2  5

2x  1
2x  1 6x 3 - x 2
5
Synthetic Division - a shortcut (Textbook: Appendix C, page 989)
Procedure:
1. Write coefficients of polynomial inside (upside down division bar)
2. Write the divisor number in front, taking the opposite sign
3. Bring down the first coefficient
4. Multiply the first coefficient by the divisor number, add to next coefficient
5. Repeat until all numbers are used
6. Rewrite the polynomial using the numbers as coefficients, reducing the highest
power by 1 degree.
3x 3  4x 2 - 2x - 1
Ex i Divide
x4
9
Conditions for synthetic division:
1. Divisor must be a binomial (2 terms only)
2. The binomial is linear (no exponents), with coefficient of x = 1
Divisors for problems using synthetic division look like:
Divisors for problems that can't use synthetic division look like:
An example that CANNOT use synthetic division (why?) – use traditional long division
Ex j (long division): x 2  x  1 x 3 
5x  4
10
R.5 Factoring
Factoring: Greatest Common Factors (GCF) – reverse distributive law
Goal: Take out the largest multiplier possible from every term
1. Find GCF (usually found by examining the numbers, then variables in each term).
Write the GCF at the front of the expression.
2. Divide the GCF out of each term, and write the “leftovers” in parentheses
Ex a Factor 10cd2 + 25c3d2
Factoring trinomials
A trinomial with x2 as leading term factors to 2 binomials:
x2 + bx + c = (x +
)(x +
)
Our job: Find the 2 numbers in the binomials (including signs)
1. The 2 binomials will be 2 factors of c (write all factor pairs of c if necessary)
2. The 2 factors will have a sum or difference of b
 If c is positive, 2 factors have same sign and b is a sum. Both signs are
positive if b is positive, both signs are neg. if b is neg.
 If c is negative, 2 factors have different signs and b is a difference
Ex b Factor: x2 – 10x + 9,
x2 – 11x + 24,
x2 + 5x – 14, -x2 + 18x - 72
Factoring by grouping - most common for 4 terms or trinomials with ax2 as leading
term where “a” can’t be factored out:
Ex c Factor x3 – 4x2 – 3x + 12
Ex d Factor: 4x2 + 25x – 21
11
Factoring: Difference of Squares & Sum/Difference of Cubes (should memorize)
1. A2 – B2 =
2. A3 + B3 =
3. A3 – B3 =
Cautions:
A2 – B2  (A – B)2
A3 + B3  (A + B)3
A3 – B3  (A – B)3
Ex e Factor 40c3 – 5d3
Solving – set each factor = 0
Ex f s2 – 2s - 35 = 0
Ex g 4x2 = 20x - 25
12
R.6 Rational Expressions and Equations
P(x)
, where Q(x)  0 (polynomial fraction)
Q(x)
Zeroes in the denominator are to be avoided. (Try 5  0 on your calculator)
Zeroes in the denominator are sometimes called
Rational Expression –
*Ex a Find the values where
x4
is undefined
x  7x 2  10x
3
Reducing/Cancelling Common Factors
Common factors (multiplied) can be cancelled
Common terms (added or subtracted) cannot be cancelled
Ex b Simplify:
x2 - 9
and state the domain. (Contrast with)
5x  15
3x 2  7x  5
3x 2  2x  10
Canceling opposite factors:
x 5
Ex c Simplify:
and state the domain.
5x
Multiplying/Dividing – For division, 1) keep 1st fraction same, 2) change division to
mult., 3) flip second fraction (keep, change, flip). For both operations, factor, cancel
common factors, and gather leftover factors.
x 2  7x  10 x 2  2x  15

Ex d Simplify
3x  6
6x  6
13
Adding & Subtracting Rational Expressions
Case 1: (easy) If same denominators, keep denominator, add numerators
Case 2: If different denominators:
1. Factor
2. Find the LCD by writing each prime factor to the highest power (remember that
the LCD is a Least Common Multiple, and is typically BIGGER than each factor)
3. Write new fractions with same denominator by building each to the LCD
Ex e Simplify:
5
7
 2
4
6x
3x
*Ex f Simplify:
1
2
2x

 2
x  5 2x  10 x  25
Complex Fractions - Method 1
1. Get 4 clearly separated layers to get 2 simple, separate fractions.
2. Convert division to multiplication
x 5
*Ex g Simplify 5
1 1

x 5
14
Solving Fractional Equations
Goal: Use LCD to multiply both sides and get rid of fractions
Caution: Check equation for bad points
Ex h Solve:
x  2 x 1 3


5
6
5
*Ex i Solve:
x
4
18

 2
x 3 x 3 x 9
Proportions – use cross multiplication: If
Ex j Solve:
a c
 , then ad = bc
b d
5
3

x 1 x  3
Applications
1. Distance = rate X time
2. Work = rate X time, but “5 hours to paint a room” is often interpreted as
r = 1/5 room/hour
*Ex k Ian takes 5 hours to rake and bag leaves, and Kyandre takes 3 hours.
Working together, how long does it take to complete the job?
15
*Ex l Two hoses are used to fill a fish pond. Together they take 12 minutes to fill
the pond. Alone, one hose takes 10 minutes longer than the other. How long
does it take each one to fill the pond alone?
*Ex m My husband drives 10 mph faster than me and travels 420 miles in the
same time it takes me to drive 360 miles. How fast is each of us going?
16
7.1 Introduction to Functions
A correspondence connects 2 sets of quantities (usually x and y) to each other.
Domain – set of all possible x values (inputs)
Range – set of all possible y values (outputs)
Some examples of correspondences
1. A set of ordered pairs
Age
4
7
9
12
9
Weight (lb)
42
61
75
92
68
2. A vending machine
A 
B 
C 
D 
E 
Function – a correspondence where each input (x) has exactly one output (y)
 never 2 or more outputs for same input
 OK to have 2 inputs produce same output
 a function is predictable
Ordered Pairs: Deciding if ordered pairs are functions
1. Check to see if 2 or more pairs have same inputs. (If never same inputs, no conflict,
so it’s a function)
2. If same inputs but different outputs, not a function.
Ex a Does the set of pairs {(90,4), (72,2), (94,4), (61,1)} define a function?
17
Ex b Does the set of pairs {(90,4), (72,2), (90,3)} define a function?
Vertical Line Test - A graph is not a function if any vertical line cuts the graph at more
than one point
Ex c Which of the following graphs are functions?
All 2-dimensional inequalities are not functions
Function Notation and Equations
f – name of function
x – domain
f(x) -- range of function
We often replace y with f(x)  y = f(x)
We can also use other symbols in function notation, which are connected to quantities
e.g. C(t), where C is cost, t is time in minutes spent on a phone
Ex d For the function f(x) = x2 – 2x + 3, find
1) f(4)
f(a)
2) f(0)
f(2a)
3) f(-1)
f(a+5)
18
Graphs and function values:
Ex c For the graph find: 1) f(0), and 2) the values of x where f(x) = -1
Ex e Claytons’ profit is based on the formula: P(x) = 0.25x – 3, where
P(x) = profit, x = # of candies sold.
1) Find the profit of selling 40 candies.
2) Find P(0). What is P(0) in real life?
3) Find the break-even point.
4) Graph the function
Equations: Deciding if an equation is a function
Odd Powers Test – If an equation contains only one y term:
1. If the exponent on y is odd (e.g. y, y3, y5…) it is a function
2. If the exponent on y is even (e.g. y2, y4, y6…) it is not a function
Ex f Which of the following equations represent functions?
1) y2 = 2x – 3
2) y = x2 + 4x – 1
3) x = y4
4) x = y3
5) y = x4
6) y = x3
19
7.2 Domain and Range
Domain – set of all possible x values (inputs)
Range – set of all possible y values (outputs)
Domain and Range of Graphs
(empty hole)
Some Restrictions on Specific Domains and Ranges
Ex c Find the domain of f(x) =
x 1
x 3  4x
Ex d A flare is launched from 224 ft. and its height is described by the equation:
h(t) = - 16t2 + 80t + 224
1) Find the value(s) of t when h(t) = 0.
2) Find the domain
20
Piecewise functions - use different formulas for different regions
Ex e A phone company charges $10/month for up to 400 minutes. After 400
minutes, $0.50 is charged for each additional minute. Write a piecewise function.
x for x  0
Ex f Graph f(x) = x  
- x for x  0
- x for x  0

Ex g Graph f(x) = x 2 for 0  x  2
3 for x  2

21
7.3 Graphs of Functions
Linear Functions
Linear functions look like f(x) = ax + b
Ex a Write x + 2y = 6 as a linear function and graph it.
Some special linear functions:
Constant Function: f(x) = k, where k is a fixed number
Ex b My garbage bill:
Identity Function: f(x) = x
Ex c Matching grant:
Ex d A facility rents for $200, and charges $15/meal, represented by the linear
function
C(x) = 200 + 15x, where C(x) is the total cost and x is the # of guests
Calculate some ordered pairs and graph the function.
Domain and Range of a Linear Function
Ex d Find the domain and range of:
1) g(x) = - 3x + 2
2) h(x) = 1
22
Graphing Non-Linear Equations
Non-linear equations – don't look like f(x) = ax + b
Some examples:
Polynomial
Absolute value
f(x) = x3 + 1
f(x) = |x|
Rational
1
f(x) =
x
They tend to be unpredictable. Graph by plotting points
Ex e Graph f(x) = x3 + 1
x
f(x)
Ex f Graph f(x) =
x
f(x)
1
x
23
Supplement 2.5 Transformations of Curves
Basic graphs to memorize
1. f(x) = x
2. f(x) = x2
5. f(x) = |x|
6. f(x) = 1/x
3. f(x) = x3
7. f(x) =
x
Vertical Translation
f(x) + k shifts f(x) up k units
f(x) – k shifts f(x) down k units
Ex a f(x) = x3 + 3
f(x) = x3 - 2
Horizontal Translation
f(x - h) shifts f(x) right by k units
f(x + h) shifts f(x) left by k units
Ex b f(x) = (x – 2)4
f(x) = (x + 1)4
4. f(x) = x4
8. x = y2
24
x-axis Reflection
- f(x) reflects f(x) across the x-axis (above/below)
Ex c f(x) =  x
Ex d f(x) = - |x|
y-axis Reflection
f(-x) reflects f(x) across the y-axis (left/right)
Ex e f(x) =  x
Ex f f(x) = |  x|
Vertical Stretching
For c > 1, c  f(x) stretches f(x) vertically
For 0 < c < 1, c  f(x) shrinks f(x) vertically
Ex g f(x) = 2 x
Ex h f(x) = 2 x
g(x) =
x
g(x) = x
h(x) =
1
x
2
h(x) =
1
x
2
25
Horizontal Stretching (somewhat counter intuitive)
For c > 1, f(cx) shrinks f(x) horizontally (narrower)
For 0 < c < 1, f(cx) stretches f(x) horizontally (wider)
Ex i For f(x) below, graph f(2x) and f(½x)
Ex j
f(x) =
4x
g(x) =
x
h(x) =
1
x
4
Successive Transformations
1. Do shrinking, stretching, and reflections first
2. Do translations (vertical and horizontal) last
2x + 3
Ex i
f(x) =
Ex j
f(x) = - ½ (x + 1)3 + 2
26
7.4 The Algebra of Functions
Addition, subtraction, multiplication, and division all work as expected
Ex a For f(x) = x2 + 6x + 8 and g(x) = x + 2
(f+g)(x) =
(f – g)(x)=
(f  g)(x) =
(f/g)(x) =
Ex b For f(x) and g(x) above, find (f – g)(3)
Domains and Graphs
1. For (f+g), (f – g), and (f  g): Remove bad points from domains of f and g. Remaining
points are domain.
2. For (f/g): Remove bad points from domains of f and g. Remove points where
denominator, g(x) = 0. Remaining points are domain
Ex c For f(x) =
2
x 1
and g(x) =
find the domains of (f+g)(x), (f – g)(x), (f  g)(x),
x 3
x
and (f/g)(x)
Ex c For f(x) = 2  x and g(x) = 2 x  3
1. Sketch the graphs
2. Find the domains of (f+g)(x),
(f – g)(x), (f  g)(x), and (f/g)(x)
27
7.5 Applications and Variation
Formulas- Solving for a Specified Variable
Solving for a Specified Variable (procedure)
1. Get rid of denominators (multiply by LCD or cross multiply)
2. Get all terms with desired variable on one side, all other terms on other side
3. Factor out the desired variable
4. Divide by "junk"
Ex a Solve the equation
d
5

for r.
r r 2
Ex b Solve the equation
1
1 1
 
for r1 (formula for parallel resistors)
R eq r1 r2
28
Direct relationship - 2 quantities go up together
Let s(t) = salary , t = # of hours worked
Inverse relationship - one goes up as the other goes down
Let P(x) = inheritance payment, x = # of “kids”
Goal: Get one of 2 formulas, and find a number value for k, to write final formula
 y = kx
(direct) OR
k
 y=
(inverse)
x
Direct Variation
y = kx
y = kxn
“y varies as x” or “y varies directly as x”
“y is directly proportional to x”
“y varies directly as the nth power of x”
(first) = k  (second)
“first varies as second”
“first is directly proportional to second”
Typical procedure (not all parts are always asked for)
1. Write a generic equation, using "direct" or "inverse" to connect the quantities in the
correct relationship. Use k, which represents "varies" or "proportional"
2. Calculate a number for k, using given data.
3. Write a specific equation, exchanging the number for k in the previous equation.
4. Use the specific equation to find the new data point, plugging in the new numbers
and solving.
Ex a Distance, d, varies directly as the square of time. A rock falls 64 feet in 2
seconds.
1) Find the proportionality constant, k, and write an equation for d(t)
2) How far does a rock fall in 6 seconds?
29
Inverse Variation
k
y=
“y varies inversely as x”
x
“y is inversely proportional to x”
(first) =
k
(second)
“first varies inversely as second”
“first is inversely proportional to second”
Ex b The volume of a gas is inversely proportional to air pressure. If the volume
is 80 cubic inches at 15 psi, what is the volume at 25 psi?
Joint/Combined Variation
Ex c Drag force on a boat, f, varies jointly as the wet surface area, A, and the
square of velocity, v. Write a generic equation connecting these quantities.
Ex d Body Mass Index (B, in this example) varies directly as weight (w) and
inversely as the square of height(h).
1. If B = 23.0 at when h = 70” and w = 160 lb, calculate the proportionality
constant, k, and write a formula for B.
2. What is the BMI of a 60”, 100 lb. girl?
30
8.1 Solving Systems of Equations -- Graphing Method
A system of equations has 2 (or more) equations with 2 variables (or more
We are interested in finding points that are solutions of both equations (the system
solution)
Ex a Is (0, 0) a solution of the system below? Is (1, 3) a solution?
y = 3x
y=-x+4
3 possible outcomes (3 kinds of systems)
1. The 2 lines intersect at one point
y = 3x
y=-x+4
solution(s)
-
system is consistent (has
solution)
equations are independent
2. The 2 lines never intersect
y=x–1
y=x+3
solution(s)
-
system is
-
equations are
3. The 2 lines lie on top of each other
y=x
3x – 3y = 0
solution(s)
-
system is
-
equations are
The most common (and expected) outcome is
31
Ex b Solve by graphing:
y=½x–3
y = - 2x + 2
Ex c Solve by graphing:
3x + 3y = 6
y=-x+2
32
8.2 Solving Systems – Substitution or Elimination
Graphing gives a "big picture" view of the solution, but it has limitations:
- depends on drawing accuracy
- can’t “eyeball” answers with fractional coordinates (what does 5/11, -2/7 look like?)
Substitution and Elimination are more precise
Procedure - Solving by Substitution - best when one variable is easily isolated
1. Isolate 1 variable in 1 equation
2. Substitute that variable into the other equation. You should now have an equation
with variable.
3. Solve that equation for the variable
4. Solve for the other variable. Write your solution as an ordered pair
5. Check both equations (optional)
Ex a Solve the system:
2x + 2y = 7
x – 4y = 1
Ex b Solve the system
x + 2y = 9
3x + 5y = 20
Ex c The sum of 2 numbers is 70. They differ by 11. Find the numbers
33
Procedure for Solving by Elimination - use when substitution is hard or coefficients are
easy to “match”
1. Write both equations in general form (Ax + By = C) and stack with like types in a
column.
2. Choose which variable to eliminate
3. Multiply one or both equations so the coefficients have opposite sign same amount
4. Add 2 equations. One variable should drop out
5. Solve for the remaining variable
6. Solve for the other variable. Write as an ordered pair
7. Check solution in both equations (optional)
Ex d Solve the system:
2x + 3y = 21
5x – 2y = -14
Ex e Solve the system:
3x + 4y = 2.5
5x – 4y = 25.5
Ex f Nadia buys 3 candy bars and 4 fruit roll-ups for $2.84.Peter buys 3 candy
bars and 1 fruit roll-up for $1.79. What is the cost each candy bar/fruit roll-up?
34
8.3 Solving Applications – Systems of 2 Equations
(also see Khan Academy applications from 8.2, Ex b and Ex d)
Procedure for solving
1. Choose variables to represent the desired quantities (choose sensible ones!)
2. Write a system of equations (usually 2) from the information given
3. Solve the system
4. Answer the question
5. Check
Money Value
Ex a Zoe bought burgers for $3 and sodas for $0.50. If the # of sodas was 3 less
than twice the # of burgers, and the total cost was $34.50, how many of each was
bought?
Interest
Ex b An investor splits $10,000 between a “safe” fund at 5% and a “risky” fund at
20% interest. If $1100 in interest was earned, how much was in each fund?
Mixture
Ex c Cashews cost $7/lb and almost cost $4/lb. How many pounds of are used to
make 18 lb. of a mix costing $5/lb.
35
Distance/Rate/Time
Wind/Current
Let
s = speed of plane in still air (the speed produced by the plane alone)
w = wind speed
s + w = speed with the wind
s – w = speed against the wind
Ex d A plane flies 300 miles in 2 hours with the wind. The return trip takes 3
hours. Find the plane speed and the wind speed
36
8.4 Systems of Equations in 3 Variables
An example of an equation in 3 variables:
This describes a plane, not a line.
A solution to this is an ordered ________________
All solutions lie on the plane.
Some examples of solutions to this plane:
A system of 3 equations describes 3 planes. The intersection may produce:
1. One solution
2. No solution
3. Infinitely many solutions
We usually hope for the first case. Goal: Solve for x, y, and z
Solve by Substitution
Ex a Solve the system:
x - 3y + 4z = 15
5y – 2z = - 16
3z = 9
Solve by Elimination (with back substitution)
1. Choose which variable to eliminate - use an equation with coefficient = 1 if possible
2. Multiply that equation and add to another row
3. Replace the added to row with the new coefficients
Ex b 2x + 3y – 4z = -10
2y + z = 8
4x – 5y + 3z = 2
37
Ex c 4x + 3y – 5z = - 29
3x – 7y – z = - 19
2x + 5y + 2z = - 10
38
8.5 Applications
Ex a Chidren’s tickets to a museum cost $4, adult tickets cost $9, and senior
tickets cost $7. A group of 27 people pay $171 total. There are twice as many
children as seniors. How many of each ticket type is bought?
Ex b The SAT has 3 parts: math, critical reading, and writing. The average
composite score in 2007 was 1511. The average math score was 13 points higher
than the average reading score, and the average writing score was 8 points less
than the average reading score. What was the average score for each category?
39
9.1 Inequalities and Domain
Solving Inequalities Graphically
Ex a (One variable) Solve: 2x – 2 > - x + 4
Ex b Graph the functions f(x) = 2x – 2 and g(x) = - x + 4 on the same graph.
1) Find the value(s) of x where f(x) = g(x)
2) Find the value(s) of x where f(x) > g(x)
Domain Restrictions - function is undefined for:
1. 0 in denominator (bad points)
2. Negative #’s inside radicals
Ex c Find the domain of f(x) = 4  2x
Applications
Ex d College A charges $500 per unit. College B charges a flat rate of $5000 up
to 15 units, then an additional $1000/unit above 15 units. When is College A
cheaper?
40
9.2 Intersections, Unions, and Compound Inequalities
1. "and" -- Statements using "and" are the intersection of 2 sets -- conjunctions.
Equivalent statements:
A and B
A  B (A intersect B)
How many cards can be described as being “red and a king”?
2. "or" -- Statements using "or" are the union of 2 sets -- called disjunctions
Equivalent statements:
A or B
A  B (A union B)
How many cards can be described as being “red or a king"?
Ex a For A = {0, 1, 2, 3, 4, 5} and B = {1, 3, 5, 7, 9}, find (A  B) and (A  B)
Ex b Solve, express in interval notation, and write as 2 statements using AND:
- 3 < 2x + 3 < 5
Ex c Graph and express in interval notation: x > 4 or x < 2
41
Ex d Graph and express in interval notation: 2x > - 4 or x < 0
Ex e Graph and express in interval notation: x > 4 and x < - 3
Interval Notation and Domains
Ex f Write the domain of g(x) =
3
in interval notation.
(x  2)(x  5)
Ex g Write the domain of f(x) = x  3 in interval notation.
42
9.3 Absolute Value Equations and Inequalities
Definition of Absolute Value:
a if a  0
|a| = 
- a if a  0
|a| is also defined as the distance between 0 and a on a number line - useful
definition.
Property of absolute value equations
If k is positive, then |x| = k is has 2 possible solutions:
1)
2)
Caution: Don’t confuse this with x = |k|
Ex a Solve |x| = 3
Ex b Solve |2x + 7| = 5
Ex c Solve |2x + 7| = - 5
Ex d Solve 2x + 7 = |5|
43
Property of absolute value inequalities – LESS THAN case
If k is positive |x| < k is equivalent to:
1)
AND
2)
Note: Distance less than – "close in"
Ex e Solve and graph |4x – 3| < 1
Ex f Solve |3x| + 5 < 3
Property of absolute value inequalities – GREATER THAN case
If k is positive, |x| > k is equivalent to:
1)
OR
2)
Note: Distance greater than – "far out"
Ex g Solve: |4x + 2| > 12
44
9.4 Inequalities in 2 Variables
Inequalities in one variable (e.g. x < -5) had solutions graphed as a shaded portion of a
number line. For 2-variable inequalities, the solution is the shaded portion of a plane.
Graphing 1 linear inequality - Procedure
1. Graph the inequality as if it were an equation
a. Use solid line for > or <
b. Use dotted line for > or <
2. Decide where to shade using a test point
Ex a Graph the solution of 3x + 5y > 15
Solving (Graphing) a System of 2 Linear Inequalities - Procedure
1. Graph the 1st line and shade its solution
2. Graph the 2nd line and shade its solution
3. The overlapping region is the final solution
Note: The double shaded quarter is opposite the "empty" quarter.
Ex b Graph the solution of the system:
y < 3x
y > - 3x + 6
Alternate way to shade:
If y is isolated on the left: 1) shade above for > or >
2) shade below for < or <
45
Ex c Graph the solution of the system:
y>-1
2x + y < 4
x>-2
46
10.1 Radical Expressions and Functions
square root - the reverse/inverse process of square
Ex a Solve for x:
x2 = 16
4
x= 
25
x = 16
“Generic” Square Root(s): if c2 = a, c =+ a
Note 2 roots:
Principal Square Root: c = a
Note 1 root – sign is
Squaring a radical “undoes” the radical; non-radical parts may change
Ex b ( 291 )2
(-2 3 )2
Types of Roots
1. Rational Roots – If "a" is a perfect square, then
a is rational
4
25
2. Irrational Roots – If "a" is positive but not a perfect square,
55

To get a ballpark idea of a number:
3. Non-real roots – If "a" is negative,
9
a is non-real
Cube Roots
If c3 = a, the signs of “c” and “a” are the same
For x3 = 8
x= 3 8
"The cube root of 8"
3
3
For x = - 8 x =  8
“The cube root of – 8”
Ex c
3
125x 6 y 3
Higher roots of negative numbers
 If n is odd, n  a is negative

If n is even
n
 a is non-real
5
 32
4
 16
a is irrational
47
Higher roots – It's helpful to know some perfect squares, cubes, 4th powers, etc.
x
1
2
3
4
5
6
7
8
9
10
11 12 13 14 15
x2
1
4
9
16 25 36 49 64 81 100 121 144 169 196 225
x3
1
8
27 64 125 216
x4
1
16 81 256 625
x5
1
32 243
Roots of Expressions with Variables – absolute values
Consider
So
( 3) 2 =
9 = 3. If x = -3,
x 2  x , the outcome has the opposite sign
x 2  x is only true if x is positive. If x is negative
x 2 = ______
We need |x| when the original expression has ____________ power, but the final
has ______________ power
Ex d
x6
x8
x 2  6x  9
Domains of radicals
2x  8
Ex e Find the domain of f(x) =
Ex e Find the domain of g(x) =
3
x2
48
10.2 Rational Numbers (Fractions) as Exponents
By definition:
a ½= a
a 1/n = n a
Ex a 25½
Ex b (y6)1/3
Ex c d(4d4)1/4
Ex d
4
5a 4b12
Other Fractional Exponents
 
m
a m/n = n a m  n a
a m/n = (am)1/n = (a 1/n)m
Ex e – 163/4
Ex f (32)2/5
Negative Fractional Exponents
Recall: 3-2 =
Ex e 49-1/2
Neg. exponent moves factor from top to bottom or bottom to top
Ex f 27-2/3
Laws of Exponents (same as before)
Recall:
Product rule: a m  a n  a mn
am
Quotient rule: n  a m-n
a
Power to a power rule: (a m )n  a mn
Product to a power rule: (ab)m = ambm
y 2/3  y 8/3
Ex g Simplify:
y 7/3
 x 2/3 y 2
Ex h Simplify:  1/12
 z




6
49
10.3 Multiplying Radical Expressions
Multiplying: Product Rule (also applies for higher roots)
a b  ab and ab  a b for any real numbers “a” and “b”
Ex a Find the products:
3 7
3689  3689
Caution: The Product Rule does NOT work for addition
Simplifying: Simplified radicals have no perfect square factor inside the radical symbol
(or perfect nth power factor for higher roots)
2 Processes: 1) Separate (perfect squares from “leftovers) into multiplied pieces
2) Convert perf. square (radical to non-radical -- “roof” AND changing #)
Goal: Look for perfect squares. If you can’t “see” them, break down to prime factors
Recall perf. squares: 4, 9, 16, 25, 36, 47…. and x2, x4, x6, x8,…
Ex b Simplify:
72
50
Ex c Simplify: 7 320
Ex d Simplify:
243
Ex e Simplify: 3 500x 3
Ex f Simplify:
75x 11y 8 z 2
7 27
4 25
3 29
50
Higher Roots/Powers (may have “leftover” powers)
Ex g
4
16
Ex h
6
64x 8
Multiplying
Ex g 53 2x 2  33 4x 4
Ex h
6x 3 y  8x 2 y 7
51
10.4 Dividing Radical Expressions
Dividing: Quotient Rule
a
a
a
a
and
for real numbers “a” and “b”,


b
b
b
b
b  0
60x 2 y
48x
Ex a Simplify:
3
Ex b Simplify:
3
10y 2 z
54y 5
Simplified radicals have:
1. No perfect squares inside radical sign
2. No fractions inside radical sign
3. No radicals in the denominator
1
1


2
2
Ex c Simplify:
Procedure for Rationalizing the Denominator (monomial)
1. Simplify the denominator radical as much as possible
2. Examine the what remains inside the radical and determine "what's missing" to
make a whole (non-radical)
3. Multiply top and bottom by the missing part
7
Ex d Simplify:
15
7x 3
18y
Ex e Simplify:
Higher roots - “fill the pie”
1
Ex f Simplify: 5 3
y
Ex g Simplify:
4
7
32ab 6
52
10.5 Expressions with Several Radical Terms
Completely simplified radicals should have:
1. No parentheses (distribute or multiply out all terms)
2. No perfect square factors inside radical
3. No fractions inside radical or radicals in the denominator
4. Products of radicals should be written under one "roof"
5. Sums of like radicals should be combined
Additing & Subtracting:
Don't confuse with multiplication:
ab  a  b
Radicals are similar to variable like terms – only like types can be added/subtracted
Ex a
3 3 3 5 3 7 2
Ex b
50  7 18
Ex c 5p 12p  27p 3
Multiplying – distributive law
Ex d
5

20  10

Multiplying – FOIL
Ex e

2 5 3 2 2  3

Ex f

x y

x y


Conjugates – a pair of binomials that have 1) 1st terms exactly the same and 2) 2nd
terms that are the same, except for opposite signs
Ex g Find the conjugate of 3 2  5 , and also of  x  3
53
Rationalizing a binomial denominator: multiply denom (and numerator) by conjugate
Ex h Rationalize
Ex i Rationalize
12
2 5
5y
2 y 5
Writing a radical in lowest terms
You CAN'T cancel terms (things added/subtracted) in the top & bottom
You CAN cancel factors (things multiplied) in top & bottom
Method 1: Factor top and bottom, then cancel
Method 2: Separate numerator terms, putting each over its own denominator, then
cancel
Ex j Simplify
5 3  10
5
Ex k Simplify
15  72
24
54
10.6 Solving Radical Equations
Principle of Powers: If a = b, then an = bn for any exponent n.
Goal: Isolate the variable
Recall: Squaring is the reverse of square root, so squaring a radical "undoes" it
Ex a Solve:
x2  3
Ex b Solve:
y  3  2
What happened?
Solving Procedure:
1. Isolate the radical. If there are 2 radicals, get one on each side
2. Square both sides. Combine like terms
3. If there is still a radical, isolate it. Repeat steps 1 & 2.
4. Solve for potential solutions
5. Check all potential solutions – this is mandatory!
Ex c Solve:
Ex c1 Solve:
5x 2  8  2x
2x  x4
55
Observation on solutions:
Linear (first degree) equations typically have 1 solution
Quadratic (2nd degree) equations typically have 2 solutions
Radical equations (1/2 power) typically have solutions that are sometimes good
Ex d Solve: 3  5x  6  12
Ex e Solve:
2x  2x  9  9
Ex f Solve:  3 y  4 3 y  5
56
10.7 Distance & Midpoint Formulas; Applications
Pythagorean theorem
c2 = a2 + b2 or c  a 2  b 2
Ex a A canal has dimensions as shown. Find x, the length of the canal side.
45 – 45 – 90 Triangles
The sides of a 45 – 45 – 90 triangle have the following relationship:
Ex b A baseball diamond is 90 ft on one side.
1) How far is it from 2nd base to home?
2) How far is it from the pitcher’s mound to home?
30 – 60 - 90 Triangles
The sides of a 30 – 60 – 90 triangle have the following relationship:
Ex c A 16-ft ladder leans against a building and makes a 60o angle. How high up
the building does the ladder reach?
57
Distance Formula:
d=
(x 2  x 1 ) 2  (y 2  y 1 ) 2
Ex d Find the distance between the points (3, -4) and (6, 0)
Midpoint – the point halfway between 2 points on a line segment
Midpoint coordinates:
 x1  x 2 y1  y 2 
,


2 
 2
Note: the x coordinate is the
of x1 and x 2
Ex e Find the midpoint of (4,3) and (2, -4)
Caution:
There are 3 similar formulas which are often confused with each other:
midpt:
distance
slope
58
10.8 Complex Numbers
The number i is defined as i =
i 2 = -1
- 1 (a non-real number)
Complex numbers have the form
a + bi
e.g. -3 + 2i or 7 - 4i
There are pure real, pure imaginary numbers, and numbers with a mix of each.
Imaginary numbers refers to any number with an imaginary part (either pure imaginary
or a mix of real & imaginary parts) : a + bi , where b  0
Complex numbers include all 3 types (pure real, pure imaginary, mixed)
Adding and Subtracting – add/subtract real and imaginary parts separately, similar to
like terms
Ex a (5 – 2i) – (3 – 8i)
Ex b ( 7  i 3 )  ( 5  5i 3 )
Multiplying - Use distributive law or FOIL
Ex c -4i (2 + 5i)
Ex d ( 3 - i )(2 + 7i)
Caution: Product rule
Wrong way:
Right way:
-3  -5
-3  -5
a  b  ab only applies if
a and b are real numbers:
59
Conjugates and Dividing
 Imaginary numbers are not allowed in the denominator for simplified expression
 For monomials in denominator, multiply by i
 For binomials, rationalize by multiplying top and bottom by the conjugate, similar
to the way radicals in the denominator are eliminated.
Ex e Simplify
1  4i
3i
Ex f Simplify
2  3i
7i
Powers of i (cycle)
i
i2
i3
i4
Ex g Simplify
i 100
i 501
i 7321
i -13
60
11.1 Equations – Square Root & Complete the Square
Recall: Quadratic equations look like ax2 + bx + c = 0 (2 solutions maximum)
Ex a Solve x2 – 5x = - 4
4 Common Methods of Solving Quadratic Equations
Method
Advantages
Disadvantages
1. Factoring
fast, simple
Doesn't solve every equation
2. Square Root
fast, simple
Doesn't solve every equation
3. Complete the Square Solves every equation Requires thinking
4. Quadratic Formula
Solves every equation Tedious, requires many steps
Square Root Method – works when b = 0 (only have x2 and constant terms, no x term)
Square Root Property
If a2 = k, then a = + k or a = – k
Procedure: Isolate the square before taking the square root.
Ex b Solve: x2 = 23
Ex c Solve: 3(x – 3)2 = 48
Ex d Solve: (2x + 1)2 = - 3
Ex e Solve: (x + 5)2 = 0
Remember: Quadratics may have
1. 2 solutions  x2 = positive number (or (stuff)2 = positive #)
2. no solution  x2 = negative number
3. one solution  x2 = 0
61
Completing the Square
Any quadratic equation can be “packaged” into a perfect square.
x2 + 10x + 31 = 0
Procedure for Completing the Square
1. Get equation in the form x2 + bx = c (x2 and x terms on left, number on right)
2. Find the needed number to complete the square
2
3.
4.
5.
6.
b
nn =  
2
Add it to both sides.
Factor the perfect square. Write it as a square that looks like (x + b/2)2
Square root both sides
Solve for x
Ex f Solve: x2 – 8x – 2 = 0
Exg Solve: 4x2 – 40x - 300 = 0 (similar to Ex f, with Khan video)
Ex h Solve 3k2 + k + 2 = 0 (This is nasty)
62
11.2 Quadratic Formula
Recall:
Quadratic equations have the standard form ax2 + bx + c = 0
Quadratic equations have 2, 1, or 0 real solutions
Ex a Find a, b, and c for the equations:
3x2 – 2 = 5x
x2 – 17 = 0
Quadratic Formula
The equation ax2 + bx + c = 0 has solutions:
 b  b 2  4ac
 b  b 2  4ac
 b  b 2  4ac
or x =

2a
2a
2a
(pos and neg solutions written separately)
(2 solutions written together)
x=
Ex b Solve: -7q2 + 2q + 9 = 0
Ex c Solve x2 + 10x + 25 = 0 by the quadratic formula
This is easier to solve by factoring, but there’s an important observation to note.
Ex d Solve x2 – 2x + 7 = 0 for complex solutions
63
Compound Interest
A = P(1 + r)t
Application of Quadratic Formula and Fractional Equations
Based on the compound interest formula A = P(1 + r)t, if you solve for P, the equation
looks like:
A
P=
(1  r) t
Problem:
Gloria wants to get a BA degree in 3 years. After paying for her first year up front, she
has $11,000 left to pay for the last 2 years. Each year costs $6000, and payment is due
at the beginning of each year. At what interest rate must she invest to earn enough
interest to cover the costs? The formula for investments at 2 annual payments, with
interest compounded annually is:
A
A

P=
; P = principal invested, A = amount paid out, r = interest rate
1  r (1  r) 2
Solution:
From our data, P = 11,000 and A = 6000. Plugging into the formula:
6000 6000
11,000 =

1  r (1  r) 2
To get rid of fractions, use the LCD: (1 + r)2
(11000)(1 + r)2 = 6000(1 + r) + 6000
(11000)(1 + 2r + r2) = 6000 + 6000r + 6000
11000 + 22000r + 11000r2 = 12000 + 6000r
11000r2 + 16000 r – 1000 = 0
1000(11r2 + 16r – 1) = 0
 16  16 2  4(11)( 1)
By the quadratic formula, r =
= 0.06 or -1.5
2(11)
So the interest rate needed is .06 = 6%.
Business majors' note:
When calculating interest we commonly ask the question, "If I invest P dollars for t years
at r% interest, how much will I have at the end?"
Sometimes the question needs to be asked in reverse. For example, "If I want to have
an income of A dollars each year paid out over t years, how much Principal is invested
originally at r% interest?" This kind of calculation is called Net Present Value (NPV).
64
11.3 Studying Solutions of the Quadratic Equations
Recall: The equation ax2 + bx + c = 0 has solutions:
 b  b 2  4ac
2a
The inside of the square root is sometimes called the discriminant.
 If b2 – 4ac > 0 
root(s)
x1,2 =

If b2 – 4ac = 0 
root(s)

If b2 – 4ac < 0 
root(s)
For each of the following, determine the number and type of solutions (without solving)
Ex a - x2 + 3x – 6 = 0
Ex b
5x2 – 6x = 0
Ex c 41x2 – 31x – 52 = 0
Ex d x2 – 8x + 16 = 0
Intercepts of a Graph of a Quadratic Function
If we set f(x) = 0, this is equivalent to y = 0, which gives the x intercepts. We can see
how many times the graph crosses the x axis in each of the graphs below:
f(x) = - x2 + 3x – 6
no intercepts
f(x) = 0 has imaginary
solutions at x =
 3  i 15 3  i 15

2
2
f(x) = 5x2 – 6x
2 intercepts
f(x) = 0 has real solutions at
x = 0, 6/5
f(x) = x2 – 8x + 16
one intercept
f(x) = 0 has one real
solution at x = -4
65
Writing Equations from Solutions
Ex d Write an equation if the solutions are x = -5 or x = 2/3
Ex e Write an equation if the solutions are x = i 3 or
x=-i 3
66
11.4 Applications of Quadratic Equations
Distance/Rate/Time d = 45, t = d/r, r= d/t (variations of the same equation)
Ex a I drive 330 miles from Modesto to Carlsbad. My husband drives 11 mph
faster and saves 1 hour. How fast is each of us traveling?
Distance
Rate
Time
Solving (isolating) a specified variable
Ex b The Pendulum Formula is T  2
l
g
Solve for l.
1
Ex c The Displacement formula: s  v o t  at 2
2
Solve for t
Ex d From the formula from Ex c, s = displacement (distance), vo= initial velocity,
a = acceleration due to gravity = 32 ft/sec2, and t = time
If a bridge is 64 ft above water, how long does it take for a jumper to hit the water
(assume vo = 0?
67
11.5 Equations Reducible to Quadratics
Sometimes we can substitute “u” for an expression – this allows the equation to be
rewritten as a quadratic equation using “u.” You may be able to “intuitively” factor
without using “u” – do so if you can.
Ex a Solve: x4 – 19x2 + 48 = 0
Ex b Solve: (x2 + 2)2 – 6(x2 + 2) + 8 = 0
Ex c Solve: x - 2 x - 15 = 0
Ex d Solve: x-2 + 3x-1 + 2 = 0
68
11.6 Quadratic Functions and Graphs (graph by shifting, stretching, reflecting)
Recall some features of linear functions (1st degree):
1. They can be written as y = mx + b
2. They have the shape of ________________________
3. Important landmarks are m (slope) and b (y-intercept)
Some features of quadratic equations (2nd degree):
1. They can be written as y = ax2 + bx + c
2. They have the shape of ___________________________
3. Important landmarks include:
-Direction of opening
-Vertex (shifting) and axis of symmetry
-Width (of parabola)
Stretching and Reflection - Graph f(x) = ax2
Recall the “basic” parabola, f(x) = x2. Note what happens when we multiply by large,
small, and negative coefficients:
x
-2
-1
0
1
2
y = f(x) = x2
4
1
0
1
4
g(x) = 3x2
h(x) = ½ x2
p(x) = - x2
Ex a Graph the functions f(x), g(x), h(x) and p(x) above:
Note: When “a” is positive, the parabola is face ___________
When “a” is negative, the parabola is face ___________
All parabolas have either a “peak” called the _______________ or a “valley” called the
_______________
Shifting - Graph f(x) = a(x – h)2 + k
69
Ex b Graph f(x) = (x + 2)2
x
f(x)
-2
-1
0
1
2
Ex c Graph y = -2(x – 2)2 + 5
x
f(x)
-2
-1
0
1
2
Setting y = f(x), we have y = a(x – h)2 + k. Subtracting k,
(y – k) = a(x – h)2. The vertex of this parabola is located at (h, k)
Note: When asked to find the maximum or minimum, you are really being asked to find
the _____________________
70
11.7 More About Graphing Quadratic Functions - Equations in standard form.
Most parabolas are written as f(x) = ax2 + bx + c (standard form) However, from 11.6,
we used equations that were “packaged up” to look like f(x) = (x – h)2 + k to show
horizontal and vertical shift (vertex form) How do we get vertex form?
Complete the Square – Procedure -- for the equation f(x) = ax2 + bx + c
1. Factor out “a” from every term – pull to the “front” as a multiplier
2. Find the needed number (nn) to complete the square
2
b
nn   
2
3. Add and subtract nn from the polynomial. Group the added term with ax2 + bx
(making a perfect square), group the “leftovers” (c and the subtracted term).
2
b

4. Factor the perfect square. Write it as  x   ; put the “leftovers” after the square.
2

5. If needed, multiply “a” by 1) the perfect square, and 2) the “leftovers”
6. The function should now look like f(x) = f(x) = (x – h)2 + k
Ex a For f(x) = x2 – 6x + 7, convert to vertex form, graph, and find the minimum
function value.
a) b) c) d) e) f) g) h) i) j) k) l)
m)n) o) p) q) r) s) t) u) v) w) x)
y) z) aa)bb)cc)dd)ee)ff) gg)hh)ii) jj)
kk)ll) mm)
nn)oo)pp)qq)rr) ss)tt) uu)vv)
ww)
xx)yy)zz)aaa)
bbb)
ccc)
ddd)
eee)
fff)ggg)
hhh)
iii)jjj)kkk)
lll)mmm)
nnn)
ooo)
ppp)
qqq)
rrr)sss)ttt)
uuu)
vvv)
www)
xxx)
yyy)
zzz)
aaaa)
bbbb)
cccc)
dddd)
eeee)
ffff)
gggg)
hhhh)
iiii)jjjj)kkkk)
llll)mmmm)
nnnn)
oooo)
pppp)
qqqq)
rrrr)
ssss)
tttt)uuuu)
vvvv)
wwww)
xxxx)
yyyy)
zzzz)
aaaaa)
bbbbb)
ccccc)
ddddd)
eeeee)
fffff)
ggggg)
hhhhh)
iiiii)
jjjjj)
kkkkk)
lllll)
mmmmm)
nnnnn)
ooooo)
ppppp)
qqqqq)
rrrrr)
sssss)
ttttt)
uuuuu)
vvvvv)
wwwww)
xxxxx)
yyyyy)
zzzzz)
aaaaaa)
bbbbbb)
cccccc)
dddddd)
eeeeee)
ffffff)
gggggg)
hhhhhh)
iiiiii)
jjjjjj)
kkkkkk)
llllll)
mmmmmm)
nnnnnn)
Ex b Convert y = -3x2 + 24x + 27 to vertex form by completing the square. Then
graph the equation by plotting the vertex, axis of symmetry, and at least one
other point. (Before you begin, what is the direction of opening?)
71
Vertex Formula - an alternate way to graph a parabola
1. Find the direction of opening using the sign of "a"
2. Find the vertex coordinates (x, f(x)) using the formula
b
a) Find x: x =
2a
b) Find f(x) by plugging the value of x you found above
4ac  b 2
(or if you prefer, use formula, f(x) =
)
4a
3. Graph another point to find width
Ex b2 Find the vertex of f(x) = -3x2 + x – 2 and graph the function
Ex c Find the vertex of f(x) = ½ x2 – 3x + 2 and graph the function
Intercepts – good for extra points on parabola
f(x)-intercept (or y-intercept) - let x = 0
x-intercept(s) – let f(x) = 0
There is always an f(x) intercept, but not always x-intercept(s)
Ex d Find the intercepts of f(x) = -x2 + 4x + 5. Then find the vertex, axis of
symmetry and graph
72
11.8 Problem Solving and Quadratic Functions
Maximum and Minimum -- Think:
Vertex formula:
x=
b
2a
f(x) =
(sometimes f(x) is not even needed)
Ex a
A profit function is represented by the equation:
P(x) = - x2 + 120x – 2700, where x is the number of items sold. Find
a) The # of items to sell to maximize the profit.
b) The maximum profit.
Ex b A pig pen is built against a barn, with fencing on the other 3 sides. If 80 ft.
of fencing is available, what dimensions produce the maximum area? Let x = the
width of the pen.
73
Modeling: Fitting an Equation to Real Data
Some models we have learned already:
 Linear, rising: f(x) = mx + b, m > 0
 Linear, falling: f(x) = mx + b, m < 0
 Quadratic, face up: f(x) = ax2 + bx + c, a > 0
 Quadratic, face down : f(x) = ax2 + bx + c, a < 0
Some models we will learn:
 Exponential
 Logarithmic
What formula (generic form) would you fit to the following data?
Calculating Quadratic Equation Constants from Data
Need at least ______ points to determine a line
Need at least ______ points to determine a parabola.
Ex d Find the equation of the parabola that passes thru (-3, -30), (3, 0), and (6,6).
74
11.9 Polynomial & Rational Inequalities
A quadratic inequality looks like x2 + x – 6 > 0.
Procedure for Solving Rational Inequalities:
1. Get a single expression on left side, zero on other side.
2. Factor (if needed).
3. Set each numerator & denominator factor equal to zero.
4. Use solutions as break point to divide the number line into regions.
5. Test a point in each region.
6. Shade each “true” region.
7. Determine the shape of the end points.
8. Write the solution in interval notation.
Ex a Solve the inequality: x2 + x – 6 > 0
Ex b Solve the inequality: -x2 – 3x + 28 > 0
Ex c Solve:
x 2
0
x 1
Ex d Solve:
2x  1
1
x4
75
Square (or any even-powered) factors mess up the pattern:
Ex e Solve (x – 3)2(x + 1) < 0
76
12.1 Composite and Inverse Functions
Composite Function Notation: f  g(x) = f(g(x))
NOT multiplication, but “plug in”
Ex a For f(x) = x2 + x and g(x) = 4 - x, find
1) f  g(x)
2) g  f(x )
Note: Composition is not necessarily commutative (order matters!)
We can plug specific values into composite functions – easier than performing the
composition of the entire functions:
Ex b For f(x) =
1) f  g(1)
2) g  f(3)
x  3 and g(x) =
4
find
5 - x2
84
Decomposition - Separating composite functions into smaller functions
Assign one function to a “cluster” – where would you put parentheses?
Ex c Express the following functions as the composition of functions.
3
1) h(x) = (2x + 3)3
2) h(x) = 5x  1
3) h(x) =
+ 4 (modified)
x 5
It’s possible to decompose in more than one way
Inverses and One-to-One Functions
Function
1. Never 2 outputs for same input
2. Passes vertical line test – no vertical line cuts graph twice
One-to-One – does not apply if it’s not a function
1. Never has 2 inputs produce same output
2. Passes horizontal line test – no horizontal line cuts graph twice.
Which are functions? Which are one-to-one?:
One-to-one functions have the properties
 Each f(x) has only one x associated with it, and each x has only one f(x)
 If f(x1) = f(x2), then x1 = x2 (if 2 outputs are the same, the inputs must be same)
 They are invertible (they have an inverse)
 f  f -1(x)  x, and f -1  f(x)  x , where f-1(x) is the inverse of f(x)
85
Finding formulas for the inverse of a function
We use the notation f-1(x) = “the inverse of f(x)” or “f-inverse of x” (NOT an exponent)
Procedure for finding f-1(x)
1. Let y = f(x)
2. Exchange x and y (this creates the inverse relationship)
3. Isolate y
4. Replace y with f-1(x)
Ex d Find the inverse of f(x) =
4
x7
(for x  -7)
Graphs of Functions and Their Inverses
Ex e Graph f(x) = x2 for x > 0, and g(x) =
limited domains)
x
for x > 0 (both functions have
What is significant about the line y = x in this context?
86
Inverse Functions and Composition
We can use composition to answer the question, “Are f(x) and g(x) inverses?”
Recall: f  f -1(x)  x, and f -1  f(x)  x
Ex e Are f(x) = 4x and g(x) =
x
4
inverses? Graph these 2 functions on the
same grid.
Ex f Determine if f(x) = 7x – 3 and g(x) =
x3
are inverses.
7
87
12.2 Exponential Functions
f(x) = ax (a > 0, a  1 is the exponential function with base “a”)
Ex a
Graph f(x) = 2x
xx x f(x f(x)
-2
-1
0
1
2
Ex b
Graph f(x) = (½)x
xx x f(x f(x)
-2
-1
0
1
2
Some observations for f(x) = ax:
 When a > 1, the function, f(x) ________________________

When 0 < a < 1, f(x) ________________________

The graph always passes through the y-intercept _________

The graph never goes below the line: ______________________

Does the graph pass the horizontal line test?
Equations with y and x interchanged
Ex c
Graph x = 2y , then plot y = 2x on the same grid
xx x f(x f(x)=y
-2
-2 -2
-1
-1
0
0
1
1
2
2
88
Application - Compound Interest (annual compounding)
A = P(1 + r)t
Ex d A loan of $20,000 is taken out. If the interest rate is 20%, how much is
owed after:
1) 1 year?
2) 5 years?
Transformations - still apply (shifting, flipping, stretching)
Ex e Graph the following functions:
y = 2x + 3
y = 2x+3
y = - 2x
y = 2-x
89
12.3 Logarithms and Logarithmic Functions
Sometimes we "create" operations to provide an inverse for an operation we already
have (when we need to isolate x)
Classic example:
When x = y3 
For an exponent equation, how do we isolate x?
2x = 5, x = ? What is the power needed to change 2power into 5?
Try 21 = 2
22 = 4
23 = 8
We create a new operation, the logarithm, to be the inverse of y = 2 x
By definition: x  log 2 y , where x is the “power needed”
Logarithmic Functions
f(x)  log b x is the log function with base b. It has a graph.
Ex a Graph f(x) = 2x and g(x)  log2 x on the same grid
x
0
1
2
3
-1
-2
f(x) = 2x
x
g(x)  log2 x
90
Equivalent formulas:
*****
m = log a x  am = x *****
(very important relationship!)
Conversion of log form to exponent form
Ex b Convert to exponent form:
log 3 81  4
log 3 1 = 0
log 81 9 = 1/2
log 5
1
 2
25
Conversion of exponent form to log form
Ex c Convert to log form
10y = 1000
10z = 29.8
2x = 128
Principle of Exponential Equality - used to solve log and exponent equations
bm = bn is equivalent to m = n
Solve y = log 3
1
27
Solve y = log10 0.001
Solve log2 x  3
Solve log 32 x 
2
5
Solve log 5 x  0
(for b real, b  -1, 0, 1)
91
12.4 Properties of Logarithmic Functions
Properties of Logarithms
1. 1. log a a  1
Equivalent Properties of Exponents
1. 1. a1 = a
2. 2. loga 1  0
2. 2. ao = 1
3. 3. alogak= k
4. 4. loga ak  k
5.
10log10 7 =
Ex a log3 3 x =
More Properties of Logs
5. 5. logaM  N  logaM  loga N
M
6. 6. log a  log aM  log a N
N
7. 7. logaMp = p logaM
Equivalent Properties of Exponents
5. aMaN = aM+N
aM
6. N  aM-N
a
Caution: log(x + y)  log(x) + log(y)
(When no base written, base 10 assumed)
log(xy)  (log x)(log y)
Ex b Expand: log 2 (x 4 y 5 )
Ex c Expand: ln
x 3 y2
z5
Ex d Express as one log: 5 ln x 
1
ln y - 7 ln z
2
Ex e Given log2 3  1.585 and log2 5  2.322 , find:
1) log2 15
2) log2 81
3) log 2
3
5
92
12.5 Common and Natural Logarithms
Scientific calculators typically perform log calculations in 2 bases: base 10 and base e
Common Logarithms – base 10
log x = log10 x (by definition, when no base is shown, assume base 10)
Use a calculator to compute the following:
Ex a log 110 =
Optional check: 10
log 0.425 =
= 110?
10
= 0.425
Ex b Solve for x: 10x = 5.6
Ex c log(-3.5) =
Natural Logarithms – base e
ln x = log e x (by definition)
Ex d ln 5 =
Ex e Solve: ex = 0.8
Ex f Solve: ln x = 0.75
log (0) =
The natural number e  2.718
ln 1.25 =
93
Graphs of log functions
Ex g Graph f(x) = ex and g(x) = ln x on the same grid
x
f(x) = ex
x
g(x) = ln x
-2
-1
0
1
2
The common log has a similar shape, but it “flattens out” faster, just as 10 x rises faster
Transformations (shift, flip, stretch) still apply
Note: Landmark point is (1, 0), not (0,0)
Ex h Graph f(x) = ln(x - 3)
and
g(x) = 2 – ln(x)
Changing Logarithmic Bases (Simplifying with base other than 10 or e)
Ex i Solve for x: 5x = 476
Change of base property** (easy but important property)
logbr
log ar 
(Choose
as the new base)
logb a
Ex j Find: log 5 62
log2 15
94
12.6 Solving Exponential and Logarithmic Equations
Equal Exponent Property (for b>0, b  1, -1, m and n real numbers)
bn = bm if and only if n = m
Logarithmic Property of Equality Property (for b>0, b  1, -1, m and n real numbers)
logb x = logb y
if and only if x = y
Caution: The argument (inside) of a log expression must be > 0 (=0 not allowed). All
solutions of log equations must be checked for this.
Ex a Solve 7x-1 = 22x–1
Solving Log Equations
Type A - Mix of log and non-log terms
1. Combine all log terms into one log term. Get all non-log terms on other side.
2. You now have logax = m. Convert to exponent form: am = x
3. Solve.
4. Check for false solutions (not in domain of log function)
Ex b Solve: log4(2x – 4) = 2
95
Ex c Solve: log(x – 15) + log(x) = 2
Type B - All terms have logs
1. Get everything on left hand side into 1 log term.
2. Get everything on right hand side into 1 log term.
3. Set expressions inside the logs equal to each other
4. Solve
Ex d Solve: ln x + ln(x – 4) = ln(3x)
Ex e Solve: log 2 x 2 - log 2 (x  4)  3 (This problem is more challenging than a
typical Math 90 exam problem, but similar problems will appear in Math 121, and
you have the all the tools to complete it).
96
12.7 Applications of Exponential and Logarithmic Functions
Loudness (decibels):

, where I = intensity of the sound, and  = reference intensity
L  log

Ex a If an amplifier makes a sound 500 times greater than the original sound,
what is the decibel change?
Richter Numbers

R  log
where Io is the reference intensity

Ex b A tsunami had an estimated 8.0 Richter number. The actual Richter
number was 9.1. How many times greater was the actual wave than the
estimated wave?
Applications
Basic exponent formula, when the base is given:
P(t)
 at
P(t) = Poat , or
Pο
where P(t) = amount, Po = original amount, a = base, t = time
Exponential function - when no base is given, the base is assumed to be “e”
P(t)
 e kt
P(t) = Poekt . or
Pο
where k is the percent/rate of increase or decrease
The function increases for_________________ and decreases for _____________
97
Compound Interest - A type of growth (increase)
1. Compounded Annually
A = P(1 + r)t , where A = current amount, P = Principal (original amount),
r = interest rate (% converted to decimal), t = time
2. Compounded More Than Annually
nt
r

A  P1   , where n = number of compoundings per year
n

3. Compounded Continuously
A = Pert, where e is the natural number  2.718
Ex c How much is repaid on a loan of $20,000 at 20% interest over 5 years,
compounded annually? Monthly? Daily? One million times/year?
n
1
12
365
106
Notice what happens to (1 + 1/n)n as n becomes very large.
This number “naturally” appears, and is called the natural number, e.
98
Ex d The population of a city grows exponentially at 5% annually. If the original
population is 10,000
1) Write an equation for P(t)
2) Find the population after 10 years and 30 years.
Exponential Decay/Depreciation (decrease) - uses a base < 1, or negative exponent
Ex d Assuming 15% depreciation, if a car originally costs $20,000, how much is it
worth after 1 year? After 3 years? After 10 years?
Half Life - The amount of time it takes for a value to decrease to half the original value.
P(t) 1
=
2
Pο
a) Write an equation, showing k, and calculate the value of k.
Ex e Carbon-14 has a half-life of 5650 years, i.e.
b) If 6% of the original amount of C-14 remains, how many years have passed?
99
13.1 Parabolas and Circles
Quadratics often come in “stretched out” (general form) and “useful” (vertex) form
Parabolas (vertical)
General form: y = ax2 + bx + c
Vertex form: y = a(x – h)2 + k
or
(y – k) = a(x – h)2
Parabolas (horizontal)
General: x = ay2 + by + c
Vertex: x = a(y - k)2 + k
or (x – h) = a(y – k)2
Graphing Procedure (from Chapter 11)
1. Decide direction of opening
2. Find vertex
3. Plot another point to find width
Ex a Graph x = - y2 – 2y + 3
Which way does it face?
Bonus material in videos (skip if you like): Vertex form can be written with the
coefficient
1
“a” =
, so the vertex forms look like
4p
4p(y – k) = (x – h)2 (vertical parabola) or
4p(x – h) = (y – k)2 (horizontal parabola)
The value “p” is significant – it’s the distance from the vertex to the focus of the
parabola.
Ex 1-bonus: For the equation x2 + 8x = 4y – 8, write in standard form. Find
the vertex, focus and graph
(see video at
https://www.youtube.com/watch?v=CKepZr52G6Y&ab_channel=Mathispower4u
100
Circles
Circle (definition) - the set of points (x, y) whose distance from the center, r, is constant.
Choosing the origin as our center: x2 + y2 = r2 or r =
x2  y2
Distance from points (x,y) to origin:
General form: x2 + y2 + Dx + Ey + F = 0
Useful (standard) form of circle with center at (h,k): (x – h)2 + (y – k)2 = r2
Ex b Graph the equation (x – 2)2 + (y + 3)2 = 16
Ex c Find the equation of a circle with r =
general form.
3 and center at (-1, 3) in standard and
Ex d Find the standard form equation of a circle with center (-1, 4), containing the
point (2, -2)
101
Ex d Find the center, radius, and graph of x2 + y2 + 8x – 12y + 43 = 0
102
13.2 Ellipses
An ellipse has 2 foci (F1 and F2).
Ellipse (definition) - The set of points (x, y) whose sum of distances from F1 and F2 is
constant.
General form: Ax2 + By2 + Cx + Dy + E = 0
x2 y2
Standard form of ellipse (center at origin): 2  2  1
a
b
Horizontal Ellipse a > b
Vertical Ellipse b > a

Has vertices at (a, 0), (-a, 0)

Has vertices at (0, b), (0, -b)

Has semivertices at (0, b), (0, -b)

Has semivertices at (a, 0), (-a, 0)
Ex a Graph
x 2 y2

 1 and give the coordinates of the vertices and semivertices.
9 25
Ex b Graph 3x2 + 4y2 = 36, and give the coordinates of the vertices and
semivertices.
103
Standard form of ellipse - center at (h, k):
(x - h) 2 (y - k) 2

1
a2
b2
(y  1) 2 (x  2) 2

 1 . Give the coordinates of the center, vertices,
Ex c Graph
4
9
and semivertices.
Ex d Graph 25x 2  100x  4y 2  0 . Give the coordinates of the center, vertices,
and semivertices.
Vocabulary and Information:
vertices – end points of the ellipse in the “long” direction
semivertices – endpoints of the ellipse in the “short” direction (also called endpoints of
minor axis)
major axis - the distance across the ellipse in the “long” direction (2a or 2b)
minor axis – the distance across the ellipse in the “short” direction (2a or 2b)
Bonus material:
 It’s possible to find the distance from the center to the focus (c), where
c2 = a2 – b2 or c2 = b2 – a2 (whichever difference is positive)
 It’s also possible to find the equation of an ellipse, given the center (h,k) and the
values a and b (or given c, using it to calculate a or b). You enter the values of h,
k, a, and b into standard form, then get rid of the fractions for general form.
Ex 1-bonus Find the equation of the ellipse with center at (5,4), focus at (1,4),
and vertex at (0,4) ---see video at:
https://www.youtube.com/watch?v=RWaEIJOlHlw&feature=youtu.be&ab_channel=M
athispower4u
104
13.3 Hyperbolas
Hyperbola - Set of points whose difference of distances from 2 foci, F1 and F2 is
constant.
General Form: Ax2 – By2 + Cx + Dy + E = 0 (coefficients of x2 and y2 have opp. sign)
Standard Form Horizontal Hyperbola
(center @ origin)
x2 y2
Equation: 2  2  1
a
b
Standard Form Vertical Hyperbola (center
@ origin)
y2 x2
Equation: 2  2  1
b
a


Has vertices at (a, 0), (-a, 0)
 Has vertices at (0, b), (0, -b)
“a” = distance from center to edge in x
 “a” = distance from center to edge in x
direction
direction
 “b” = distance from center to edge in y
 “b” = distance from center to edge in y
direction
direction
b
Asymptotes for both equations: y   x
a
2
2
y
x

 1 , showing asymptotes, and labeling vertices
Ex a Graph
4
9
Ex b Graph 5x2 – 4y2 = 20, showing asymptotes, and labeling vertices
105
Quick trick for telling if a hyperbola is vertical or horizontal:
x2 y2

1
e.g.
9 16
Hyperbolas with center at (h, k)
Horizontal
(x - h) 2 (y - k) 2
or

1
a2
b2
Ex c Graph
Vertical
(y - k) 2 (x - h) 2

1
b2
a2
(x  1) 2 (y  1) 2

1
16
4
Non-standard hyperbola form (special cases)
xy = c , where c is a constant
Ex d xy = - 4 How would you rewrite this equation?
x
y
-2
-1
0
1
2
Classifying Graphs
106
Shape
Standard Form
("useful" form)
y = mx + b
Landmarks
General Form
("pretty" form)
Ax + By = C
3x – y = 5
Ex: y = 3x + 5
(y – k) = a(x – h)2
Ax2 + Cx + Dy + E = 0
Ex: 4(y – 3) = x2
(x – h) = a(y – k)2
x2 – 4y + 12 = 0
By2 + Cx + Dy + E = 0
Ex: 8(x – 3) = (y – 2)2
(x – h)2 + (y – k)2 = r2
y2 – 4y – 8x + 28 = 0
x2 + y2 + Cx + Dy + E = 0
Ex:
(x+ 4)2 + (y – 6)2 = 25
(x - h) 2 (y - k) 2

1
a2
b2
x2 + y2 + 8x – 12y + 43 = 0
Ex:
(x - 2) 2 y 2

1
4
25
(x - h) 2 (y - k) 2

1
a2
b2
x 2 (y  1) 2
Ex:

1
9
4
(y - k) 2 (x - h) 2

1
b2
a2
Ex:
(y  4) 2 (x - 1) 2

1
4
4
Ax2 + By2 + Cx + Dy + E = 0
(coefficients of x2 & y2 same
sign)
25x2 + 4y2 – 100x + 8y – 96 =
0
Ax2 - By2 + Cx + Dy + E = 0
(coefficients of x2 & y2
opposite signs)
4x2 – 9y2 – 18y – 45 = 0
By2 – Ax2 + Cx + Dy + E = 0
(coefficients of x2 & y2
opposite signs)
x2 – y2 – 2x – 8y – 11 = 0
107
Identifying Quadratic (and Linear) Equations
For each equation, tell what shape it is (line, circle, parabola, ellipse, hyperbola).
(y  3) 2 x 2
1.

1
4
9
2. x2 – 6x + 4y2 + 8y – 4 = 0
3. x2 – 6x – 4y2 + 8y – 4 = 0
4. x2 – 8x + y2 + 2y + 5 = 0
5. x – y = 9
6. x2 – y = 9
7. x2 – y2 = 9
8. x2 + y2 = 9
9. x2 + 9y2 = 9
(y  1) 2 (x - 2) 2

1
16
9
11. 4x2 + 8y2 = 32
10.
12. x = 16y2 – 32
108
14.1 Sequences and Series
A sequence is a string of numbers that follow a pattern - can be finite or infinite
1, 4, 9, 16, 25...
2, 4, 8, 16, 32
Each term has a “rank”
General Term (formula to generate each term) an
Ex a For a n 
1
, write the first 4 terms
n1
Ex b Write the general term an of the sequences:
2, 5, 8, 11…
-2, 4, -8, 16…
4, 9, 16, 25…
(we will learn systematic ways of finding the general term in 14.2 and 14.3
Sums and Series - A series is a sequence where the terms are added
S  = sum of an infinite number of terms
Sn = the sum of the first n terms
Surprisingly, S  sometimes approaches a finite number:
Ex c For the series 2 + 5 + 8 + … find S3 and S5
Sigma (Summation) Notation
5
An example:
 2n  1
n1
109
5
Ex d Write the terms and find the sum of
 n2  1
n 0
Ex e Write in sigma notation: -3 + 6 – 9 + 12 – 15
110
14.2 Arithmetic Sequences and Series
An arithmetic sequence has a common difference between terms (number added to
each term to get to the next term). We call this difference "d".
-5, -3, -1, …
8, 5, 2, …
Formula for the general term: an
=
a1 + (n – 1)d
Ex a Find the common difference and the general term of the sequence
2, 8, 14, 20…
Sums of Arithmetic Sequences - The Karl Gauss Problem (1784)
Ex b Find the partial sum 1 + 2 + 3 + …+ 100
Formula for the sum of an arithmetic sequence: Sn =
n(a 1  a n )
2
Ex c Find sum of the first 50 terms of the sequence 4 + 1 + (-2) + …
111
Ex d Find the partial sum: 2 + 7 + 12 + … + 257
Ex e Find the number of seats in an auditorium with 52 rows if there are 24 seats
in the first row, 28 seats in the second row, 32 seats in the third row, and so on.
Ex f Ed earns $10K in his first year, and his salary increases $1K each year.
1) What is his salary in his 40th year?
2) What is the total of his earnings over 40 years?
112
14.3 Geometric Sequences and Series
A geometric sequence has a common ratio between terms (number multiplied by each
term to get the next)
Ex a Find the ratio of the terms in the sequence 2, -6, 18, -54…
A formula to find ratio: r =
Formula for the general term of a geometric sequence: a n  a1r n1
Ex b Find the 10th term of the sequence 2, -6, 18…
Ex c Find the general term of the sequence 2, 1, ½ , ¼, ….
Sum of Geometric Sequences
Ex d Find the sum: S = 2 + 6 + 18 + …+ 4374
Formulas for the sum of a geometric sequence:
a (1 - r n ) a - a r n a - a r
Sn  1
 1 1  1 n
1- r
1- r
1- r
Ex e Find the sum of the first 10 terms of the series 1 + 2 + 4 + 8….
113
Applications
Ex f At a conference, each person shakes hands with one person at each session.
If 10 sessions are attended, and there are no repeat handshakes, how many
people’s germs will a person receive by the last hand shake?
Ex g A company offers to pay $0.10 the first day, $0.20 the second, $0.40 the
third, etc. What will be the total salary for the month (30 days)?
A more realistic application (compare to 14.2, Ex f:
Ann earns 10K the first year and receives a 5% raise each year.
1) How much does she earn in the 40th year?
2) How much total salary does she earn?
114
Sum of an Infinite Sequence
Some sequences add up to a fixed number as n approaches infinity. Even though there
are an infinite number of terms, they add up to a finite number
a
Formula for the sum of an infinite sequence: S  1
1- r
Ex h Find the sum 5 + 2 + 4/5 + 8/25 + ….
Repeating Decimals (how to convert to fractions)
A terminating decimal can be converted by a fraction by dividing the number by 10 n,
where n is the number of decimal places.
Ex i Convert 0.3636… to a fraction.
115
14.4 Binomial Theorem
Taking a binomial to a power:
(x + y)0 =
(x + y)1 =
(x + y)2 =
(x + y)3 =
(x + y)4 =
Patterns:
1. For each polynomial
powers of x _________________________
powers of y __________________________
2. Coefficients (numbers)____________________________________
How do we get the coefficients?
Pascal's Triangle (most efficient for small powers)Pascal's Triangle
0
 
0
1  1
   
 0  1
 2  2  2
     
 0  1   2 
3 3 3 3
       
 0  1   2   3 
 4  4  4  4  4
         
 0  1   2   3   4 
Ex a Expand (x + 3)4
116
Foundational Tools for Binomial Theorem
Factorials
1!
2!
3!
4!
n!
Note: 0! =
“Choose” Notation
n
n!
  =
 r  (n - r)! r!
5
Ex b  
 2
10 
Ex c  
1 
n
Binomial Coefficients   = the binomial coefficient
r 
Binomial Theorem
n
(x + y)n =
n
  i x
i0
 
n -i
n 
y i =   x n y 0 +
0
 n  n-1 1
  x y +
1 
 n  n-2 2
  x y + … +
 2
Ex d (same as Ex a) Expand (x + 3)4 using binomial coefficients
n 0 n
  x y
n
117
Finding a Specific Term
Let p = term asked for
n
p = r + 1, or r = p – 1, where r is the lower number (power of y) of  
r 
Ex e Find the 4th term of (x + y)11
p = 4, r =