PENN STATE UNIVERSITY
UNIVERSITY PARK
MATH 22
ACTIVITY PACKET
Some activities in this packet will be completed in class and will be turned in. Your
instructor will give further instructions and due dates. If an activity is done in class, you
must be present to receive credit. All activities should be completed during the semester
whether they are collected or not. Some questions similar to exercises in this packet
WILL appear on exams.
Some activities can be completed on the provided activity sheet; most require additional
paper. When you have completed an activity, staple the activity sheet from the packet to
your additional work. Note that neatness counts.
This packet is available on Angel if you wish to print additional copies of activities.
This packet belongs to _________________________________________
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3
SOLVING LITERAL EQUATIONS
Name: _______________________
Section: ___________
Checked by: ________________________
____________________
________________________
Course Objective: Solve literal equations.
Scoring: Your work and answer will be checked on 5 randomly picked questions. Neatness may count for several points.
The process of solving a formula for a given variable is called “solving literal equations.” The rules used are the same ones
used in solving linear equations.
EXAMPLE: Solve the formula A = lw for w.
A = lw
SOLUTION:
A lw
=
l
l
A
=w
l
Instructions: Show neat work to solve each of the following formulas for the indicated variable. You must check your
answers with 2 “neighbors” prior to turning in your work.
1
1. V = Bh , for h
h = ________________
3
(Volume of a pyramid)
2. A = 2! r 2 + 2! rh , for h
h = ________________
(Surface area of a circular cylinder)
3. A = h(b1 + b2 ) , for b1
1
2
(Area of a trapezoid)
b1=________________
5
9
(Celsius to Fahrenheit)
F=_________________
4. C = ( F ! 32) , for F
Score: _______
3
4
For 5-16, solve each equation for x. Attach a separate piece of paper showing your neat work.
5. ax + bx = c
x=________________
6. 1 ! kx = mx
x=________________
7. a ( x + b) = c
x=________________
8. a ( x + b) = b( x ! c)
x=________________
9. x(a ! b) = m( x ! c)
x=________________
1
1
x!a = b
2
3
x=________________
10.
11. 3 x + 5 y = 9
x=________________
12. y = mx + b
x=________________
13. ! x = (5 x + 3)(3k + 1)
x=________________
14. ( x ! 2)(a + 1) = y
x=________________
15. ( x + 1)(a ! 3) = y ! 2
x=________________
16. a 2 (a ! x) = b 2 (b + x) ! 2abx
x=________________
4
5
ABSOLUTE VALUE
Name: _______________________
Section: ___________
Checked by: ________________________
____________________
________________________
Course Objective: Solve various types of equations. Solve absolute value inequalities.
Scoring: Your work and answer will be checked on 5 randomly picked questions. Neatness may count for several points.
Consider the statement: x = ! x
Can this statement be true? _________________________
Fill in the table.
x
x
-x
2
1
0
-1
-2
Opposite and Absolute Value
Opposite: ! x , when read as “opposite of x” (instead of “negative x”) emphasizes an important idea. If we allow x to be any
real number, ! x (opposite of x) can be 0, positive, or negative. If x is 0, then ! x is 0. If x is a positive number, then ! x is a
negative number. If x is a negative number, then ! x is positive.
Absolute value: Geometrically, the absolute value of any number is the distance between the number and 0 on the number
line. For example, the absolute value of 3 is 3, and the absolute value of -2 is 2. Symbolically, |3| = 3; |-2| = 2.
#% x ,
More formally, absolute value is defined as follows: x = "
!x ,
if x < 0
if x $ 0
Solving Absolute Value Equations and Inequalities
First: Isolate the absolute value
Second: Rewrite without the absolute value bars using the following equivalencies:
guts = stuff
is equivalent to
guts < stuff
is equivalent to
guts > stuff
is equivalent to
guts = ! stuff
or
guts = stuff
(you must use both)
! stuff < guts < stuff
guts < ! stuff
or
guts > stuff
(you must use both)
Third: Solve the resulting equations. Be careful of signs.
NOTE: Can the absolute value of something EVER result in a negative value? ______________________
Score:___________
5
6
Exercises:
1. Solve the Absolute Value Equations.
a. x + 2 = 5
b. 3 x ! 7 = 4
c. 3 x ! 7 = 4
d. 3 x ! 7 + 4 = 5
e. 3 x ! 7 + 4 = 1
2. Solve the inequality and give your answer using interval notation.
a. x + 2 < 5
b. 3 x ! 7 > 4
c. 3 x " 7 ! 4
d. 3 x ! 7 + 4 > 5
e. 3 x " 7 + 4 ! 1
f. 3 x " 7 + 4 ! 1
In 3 and 4 solve the inequality for x and specify the answer using interval notation.
3. Assume c > 0 throughout this exercise.
a. a ! x > c
b. a ! x < c
c. a " x ! c
4. Assume b < c and c > 0 throughout this exercise.
a. x ! a + b < c
b. x + a + b < c
c. x + a + b > c
5. If x ! 2 < 3 , find values of m and n such that m < 3 x + 5 < n .
6
7
WORD PROBLEMS – LINEAR EQUATIONS
Name: _______________________
Section: ___________
Checked by: ________________________
________________________
________________________
Course Objective: Translate applications into mathematical models and find logical solutions
Scoring: Your work and answer will be checked on 5 randomly picked questions. Neatness may count for several points.
Throughout your math career you have encountered
Translating from English to Algebra sentence by sentence and following some logical steps can make this process of
approaching usable math rewarding.
Polya’s Problem Solving Steps:
1.
2.
3.
4.
Understand the problem. (Recognize what is asked for.)
Devise a plan. (Respond to what is asked for.)
Carry out the plan. (Develop the result of the response.)
Look back. (Check. What does the result tell me? Does it make sense? )
Geometric problem If the length of a side of a square is increased by 3 cm, the perimeter of the new square is 40 cm more
than twice the length of a side of the original square. Find the dimensions of the original square.
Geometric problem A triangle has a perimeter of 30 cm. Two sides of the triangle are both twice as long as the shortest
side. Find the length of the shortest side.
Score:____________
7
8
Uniform-motion problem Chuck travels 80 km in the same time that Mary travels 180 km. Mary travels 50 km/h faster
than Chuck. Find the rate of each person.
Uniform-motion problem Fred and Jan are running in the Blue & White Fun Run. Fred runs at 7 mph, Jan at 5 mph. If they
start at the same time, how long will it be before they are 1/2 mile apart?
Mixture problem A chemist needs a 20% solution of potassium permanganate. She has a 15% solution on hand, as well as a
30% solution. How many liters of the 15% solution should she add to 3 liters of the 30% solution to get her 20% solution?
Mixture problem The manager of a candy store wants to prepare 200kg of a mixture for a Halloween promotion to sell at
$4.84 per kg. How much $4 per kg candy should be combined with $5.20 per kg candy for the required mix?
8
9
RATIONAL EXPRESSIONS
Name: _______________________
Section: ___________
Checked by: ________________________
____________________
________________________
Course Objective: Simplify rational expressions.
Scoring: Your work and answer will be checked on 5 randomly picked questions. Neatness may count for several points.
Throughout Math 022 you will be expected to work with rational expressions in order to solve rational equations and
inequalities and graph rational functions. Getting these expressions in their simplest form is the first step.
1.
What is a rational number? ___________________________________ Give an example _________________
2.
What is a rational expression? ________________________________ Give an example__________________
3.
What is a rational equation? ___________________________________ Give an example _________________
Rational expressions can be reduced (simplified) and added, subtracted, multiplied, and divided.
Reduce to lowest terms.
4.
x 2 ! 5x + 6
x 2 ! 7 x + 10
=
For what values of x is the reduced expression equal the original expression? _______________________
5.
6.
a 2 ! 4b 2
a 3 ! 8b 3
=
x 2 ! x ! 12
8 + 2x ! x 2
=
For what values of x is the reduced expression equal the original expression? _______________________
7.
2 x 3 ! 3x 2 ! 2 x
6 x 4 ! 5x 3 ! 4 x 2
=
For what values of x is the reduced expression equal the original expression? _______________________
Score: _______
9
10
For 8-18, add, subtract, multiply, and divide the rational expressions as indicated. Simplify as much as possible. Work
neatly on your own paper.
8.
9.
10.
11.
12.
1
2
x ! 5x + 6
x!2
2
x +x
!
!
2
2
x ! 4x + 3
=
4
3x
+ 2
=
x +1 x !1
y
x
+
+1 =
x! y y!x
x !1
( x + 1)
3
+
x
( x + 1)
m2 ! n2
2
!
1
=
x +1
n 2 !4mn
=
n 3 ! 16m 2 n n ! m
"
&
xy # &
xy #
!! ( $$ y '
!=
13. $$ x +
x' y" %
x + y !"
%
14.
x 2 + 2ax
x 2 + 4a 2
÷
x 2 ! 4a 2
ax ! 2a 2
=
&1 1#
15. a 2 ' b 2 ÷ $ ' ! =
%b a"
(
16.
)
1 & 1# t 3 & 1#
÷ $1 + ! =
$t ' ! (
t 2 % t " t '1 % t "
1 12
!
a a2
=
17.
6 8
1! + 2
a a
1!
x
!1
=
18. x ! 2
x
1+
2! x
10
11
RATIONAL EQUATIONS
Name: _______________________
Section: ___________
Checked by: ________________________
____________________
________________________
Course Objective: Solve various types of equations. Rational Equations. Literal Equations.
Scoring: Your work and answer will be checked on 5 randomly picked questions. Neatness may count for several points.
Recall, in simplifying rational expressions, care must be taken to not “lose” the denominator. The process of solving rational
equations allows clearing of denominators. HOWEVER, if your answer(s) is/are numbers you must check if the values are
allowed in the original equation. IF an obtained value results in a 0 in the denominator, this value is NOT a solution.
1. Solve for x. y =
y
x!b
(Hint: multiply both sides by m. You can also think of this as writing y as
and “cross
m
1
multiplying”)
2. Solve for x.
x!a y!a
=
b
c
3. Solve for x. y =
3x + 2
5! x
4. Solve for x.
b
a
"
= 0 where a ! b
ax " 1 bx " 1
5. Solve for x.
x!a b!x
=
x!b a ! x
6. Solve for x.
xy 2 ! 5 xy + 4
=2
3x
7. Solve for x.
2a
= a !b
x !1
8. Solve for r.
S=
rl ! a
r !l
Score: _______
11
12
Another way to clear fractions in an equation is to multiply each term by the least common denominator.
As they are written, you cannot use “cross multiplying” on 9-14. Why not?_____________________________________
_________________________________________________________________________________________________
9. Solve for x.
x
+1 = b
a
10. Solve for x.
1 1 1
+ + =3
x y z
11. Solve for x.
3
4
2
!
=
(Check your answer)
2 x + 1 x + 1 2 x 2 + 3x + 1
12. Solve for x.
2( x + 1)
5x ! 1
!3 =
(Check your answer)
x !1
x !1
13. Solve for x.
x + 2p x !2p
4 pq
+
! 2
=0
2q ! x 2q + x 4q ! x 2
14. Solve for x. 1 '
a& a# b& b#
$1 ' ! ' $1 ' ! = 0
b % x" a% x"
Mixed Practice.
15. Solve for x. ax + b 2 = bx ! a 2
j
16. Solve for j. A = I (1 ! )
n
17. Solve for y.
x1 x
a
2
+
y1 y
b2
=1
Sometimes a word problem results in a rational equation that must be solved.
18. What number must be added to both the numerator and the denominator of the fraction 3/7 to yield a fraction that is equal
to 2/3?
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GRAPHS OF FUNDAMENTAL FUNCTIONS
Name:_______________________
Section: ___________
Checked by: _________________________
_____________________
_________________________
Course Objective: Graph fundamental relations and functions
Note: Keep this sheet as a reference for this and future math courses
Scoring: Your work and answer will be checked on 5 randomly picked questions. Neatness may count for several points.
Throughout algebra of functions and calculus it is expected that students know how to sketch graphs of certain functions.
With the idea of the graph firmly in mind, students are prepared to answer questions about domain & range and other
properties of each function. In this activity we will explore these basic functions through their graphs.
Directions:
1. Plot points.
2. Consider values that x can’t be and what happens as x gets close to these values from each side.
3. Consider what happens as x gets very large positive and very large negative.
y = x2
y= x
Domain:
Range:
Domain:
Range:
Score:____________
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14
y = x3
y=3 x
Domain:
Range:
Domain:
Range:
y =| x |
y = r 2 ! x 2 (graph for r = 1 )
Domain:
Range:
Domain:
Range:
14
15
y=
1
x
y=
1
x2
Domain:
Range:
Domain:
Range:
LATER:
y = a x , for a > 1
y = log a x, for a > 1
Domain:
Range:
Domain:
Range:
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16
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17
QUADRATIC FUNCTIONS
Name:_______________________
Section: ___________
Checked by: _________________________
__________________________
__________________________
Course Objective: Analyze fundamental relations and functions (Quadratic functions)
Scoring: Your work and answer will be checked on 5 randomly picked questions. Neatness may count for several points.
Quadratic functions and their graphs appear throughout algebra and calculus. Practically, graphs of quadratics have one highest
point (maximum) or one lowest point (minimum) and can be used to model real physical or economic situations that can then
be maximized or minimized (optimization problems).
Recall: Linear functions are polynomial functions of degree _________.
A quadratic function is any function of the form f ( x) = ax 2 + bx + c , where a, b, and c are real numbers and a
! 0.
Quadratic functions are polynomial functions of degree _________.
A quadratic equation is an equation of the form ax 2 + bx + c = 0 . In earlier work we solved for x in quadratic equations. In
this study we will often deal with just the right hand side of f ( x) = ax 2 + bx + c . Note that ax 2 + bx + c is an expression. It
is NOT an equation; you can change its form, but cannot solve for anything.
RECALL FROM PREVIOUS STUDIES:
Graph:
f ( x) = !2( x + 1) 2 ! 2 using transformations of the basic function y = x 2 .
A quadratic function written in the form f ( x) = a ( x ! h) 2 + k , can be easily graphed. We will use a method of completing the
square, similar to (but not exactly like) the method learned to solve quadratic equations.
Score: _______
17
18
2
2
Example: Rewrite g ( x) = x ! 2 x + 3 in the form f ( x) = a ( x ! h) + k .
1. Write the function spaced as shown. g ( x) = ( x 2 ! 2 x + _____ ! _____ ) + 3
2.
3.
4.
Complete the square by taking ½ of 2 and squaring that number, 12 = 1 Add and subtract this number in the
parentheses to get g ( x) = x 2 + 2 x + 1 ! 1 + 3 .
Now remove the –1 from the parentheses and combine it with the +3.
(
)
g ( x) = _________________________ (fill in the blank).
Write the portion in parentheses in factored form as a binomial squared; keep the stuff outside the parentheses what
and where it is.
g ( x) = _________________________ (fill in the blank).
5. Graph this quadratic function and fill in the information asked about the graph.
Vertex: ____________
Circle whichever is correct:
Opens up
Opens down
*Axis of Symmetry____________
y-intercept: ______________
Domain: ___________
Sketch the graph.
x-intercept(s): ____________
Range: __________
Interval increasing: _________ Interval decreasing: _________
*The axis of symmetry is the vertical line through the vertex
Example: Rewrite f ( x) = 2 x 2 + 8 x + 3 in the form f ( x) = a ( x ! h) 2 + k .
1. Factor a 2 from the first two terms _______________________________
(BE CAREFUL!)
2. Complete the square by taking 1 2 of 4 and squaring that number. Add and subtract this number in the parentheses to
3.
(
)
get g ( x) = 2 x 2 + 4 x + _____ ! _____ + 3 .
Remove the subtracted number from the parentheses by multiplying it by 2.
g ( x) = _________________________ (fill in the blank).
4. Write the completed square in factored form; don’t forget the a value in front, and
be sure to combine the numbers outside of the parentheses.
Your quadratic function should be of the form
f ( x ) = a ( x ! h) 2 + k .
g ( x) = _________________________ (fill in the blank).
Graph this quadratic function and fill in the information asked about the graph.
Vertex: ____________
Circle whichever is correct:
Opens up
Opens down
Axis of Symmetry____________
y-intercept: ______________
Domain: ___________
Sketch the graph.
x-intercept(s): ____________
Range: __________
Interval increasing: _________ Interval decreasing: _________
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19
CONCLUSIONS:
A quadratic function written in the form
f ( x ) = a ( x ! h) 2 + k
a. has vertex __________.
b.
has axis of symmetry ___________.
c.
opens up if ___________.
d.
opens down if __________.
e.
has domain ___________. (true for all quadratic functions)
f.
if a > 0, the range is ____________.
g.
If a < 0, the range is ____________. k is called the maximum value of the function.
k is called the minimum value of the function.
A quadratic function written in the form f ( x) = ax 2 + bx + c
a.
has y-intercept _____________.
b.
opens up if ________________ .
c.
opens down if ______________.
d.
The x-coordinate of the vertex is
e.
The axis of symmetry is __x =_________________
!b
!b
, and the y-coordinate is f ( ) .
2a
2a
Intercepts (Review):
1.
To find the x-intercept(s) set _____________ and solve for __________
Use methods of Section 1.5
2. To find the y-intercept set _______________ and solve for ___________.
Example: Solve the inequality x 2 ! x ! 2 > 0 by graphing.
1.
2.
Graph f ( x) = x 2 ! x ! 2 by methods of this section. Make sure you identify the x-intercepts.
Note where the function is above the x-axis. Use interval notation to describe x values for which the function is
greater than 0 (above the x-axis).
Exercise: Sketch a graph of each quadratic function by completing the square to get the function in graphing form OR by
gathering information about intercepts and vertex.
1. f ( x) = ! x 2 ! x + 12
2. g ( x) = 3 x 2 ! 6 x + 2
3. h( x) = 2 x 2 ! 12 x + 19
19
20
20
21
WORD PROBLEMS
Name: _______________________
Section: ___________
Checked by: ________________________
________________________
________________________
Course Objective: Translate applications into mathematical models and find logical solutions
Scoring: Your work and answer will be checked on 5 randomly picked questions. Neatness may count for several points.
Throughout your math career you have encountered
Translating from English to Algebra sentence by sentence and following some logical steps can make this process of
approaching usable math rewarding.
Polya’s Problem Solving Steps:
5.
6.
7.
8.
Understand the problem. (Recognize what is asked for.)
Devise a plan. (Respond to what is asked for.)
Carry out the plan. (Develop the result of the response.)
Look back. (Check. What does the result tell me? Does it make sense? )
Exercises:
1. The product of two numbers is 16. Express the sum of the squares of the two numbers as a function of a single variable.
2. A piece of wire !y inches long is bent into a circle.
a. Express the area of the circle as a function of y.
b. If the original piece of wire were bent into a square instead of a circle, how would you express the area of the
square in terms of y?
3. A point P(x,y) lies on the curve of
y = x as shown. Express the area of the shaded triangle as a function of x.
Score:________
21
22
4. The point P lies in the first quadrant on the graph of the line y = 7 ! 3 x . From the point P, perpendiculars are drawn to
both the x-axis and the y-axis. What is the largest possible area for the rectangle thus formed?
5. Find the point on the curve y = x that is nearest to the point (3, 0).
6. What number exceeds its square by the greatest amount?
22
23
POLYNOMIAL INEQUALITIES
Name:_______________________
Section: ___________
Checked by: _____________________________
____________________________
_____________________________
Course Objective: Solve absolute value, polynomial, and rational inequalities
Scoring: Your work and answer will be checked on 5 randomly picked questions. Neatness may count for several points.
In calculus, a method of solving factorable polynomial inequalities may be used 3 times in one exercise as follows:
1. Solve a polynomial inequality to determine where a graph is above/below the x-axis
2. Find the derivative of the polynomial function (requires easy calculus)
3. Solve the resulting polynomial inequality to determine where the graph is increasing or decreasing.
4. Find the next derivative (more easy calculus)
5. Solve the resulting polynomial inequality to determine how the graph curves
Example:
a) Use the graph to solve x 2 ! 2 x ! 3 < 0 ;
b) Use the graph to solve x 2 " 2 x " 3 ! 0
20
SOLUTION:
a)The graph of y = x 2 ! 2 x ! 3 is below the x-axis ( y < 0 ) only
15
10
for x-values strictly between -1 and 3, (!1,3) .
5
_4
_2
2
4
6
b) The graph of y = x 2 ! 2 x ! 3 is above the x-axis( y ! 0 ) for
x-values less than or equal to -1,or for x-values greater
than or equal to 3. (#!,#1] " [3, !)
To develop an algebraic method of solving inequalities, we look at factors and how they work together.
Recall: Let P and Q be factors of a polynomial.
then PQ < 0 if P < 0 or Q < 0 but not both;
and PQ > 0 if P < 0 and Q < 0 or if P > 0 and Q > 0
It is important to determine x-values at which P and Q may change signs (from positive to negative or from negative to
positive).
We will call these values key numbers.
METHOD: to solve x 2 " 2 x " 3 ! 0
1. Factor the polynomial, ( x " 3)( x + 1) ! 0 .
2. Find the key numbers. These are the numbers for which each factor equals zero. Here: x = 3 and x = !1 .
3. List factors in a column. Below and to the right, draw a number line and label key numbers.
(x + 1)
( x ! 3)
___________|_____________________|_________________
-1
3
4. Test each factor in the three regions determined.
(x + 1)
-+
+
(x – 3)
--+
___________|_____________________|_________________
+
-1
-3
+
5. “Multiply” signs and place resultant sign below number line.
6. Conclude: x 2 " 2 x " 3 ! 0 for x-values in (#!,#1] " [3, !) (intervals for which the resultant sign is +, i.e. > 0)
7. Ideas extend for more than two factors.
Score:_____________
23
24
Exercises:
Solve the polynomial inequalities algebraically (using a number line, also known as a sign chart).
Remember to always get 0 alone on one side of the inequality sign.
Write your answer in interval notation.
1. 5 x 2 + 11x " 12 ! 0
2. 14 ! 13 x > 12 x 2
3. 2 x 2 + 1 ! 0
4. 4m 3 + 7 m 2 ! 2m > 0
5. 1 + x 2 ! 0
6. ( x ! 2) 5 (3 ! x)( x + 1) 2 < 0
Notice patterns of negatives and positives. How does a negative in front of the x affect this pattern?
7. ( x " 2) 5 (3 " x)( x + 1) 2 ! 0
8. (2 x ! 1)(3 x ! 1) 3 ( x ! 1) 4 < 0
Use patterns instead of testing particular values.
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25
RATIONAL INEQUALITIES
Name:_____________________________________
Section: ___________
Checked by: ________________________________
________________________________
________________________________
Course Objective: Solve absolute value, polynomial, and rational inequalities
Scoring: Your work and answer will be checked on 5 randomly picked questions. Neatness may count for several points.
Like polynomial inequalities, rational inequalities appear in calculus, and the methods illustrated below may be used multiple
times in one problem.
Here again you need to recall how factors work together:
Let P and Q be factors of a rational expression
P
< 0 if P < 0 or Q < 0 but not both;
then
Q
P
> 0 if P < 0 and Q < 0 or if P > 0 and Q > 0
Q
It is important to determine x-values at which P and Q change signs (from positive to negative or from negative to positive).
We will call these values key numbers.
and
Method for solving Rational Inequalities using a sign chart
1.
2.
3.
4.
5.
6.
7.
8.
9.
Get 0 on one side of the inequality sign
Find a common denominator (NOTE: you CANNOT get rid of your denominator)
Factor & simplify if possible
Find “key” numbers… “key numbers” are numbers that will make the numerator or the denominator equal 0.
Plot all “key numbers” on a number line. NOTE: circle any key numbers that are associated with factors in the
denominator, since these can never = 0.
List all factors from the numerator and all factors from the denominator above and to the left of the number line.
“Test” each factor in each region…use patterns you may have determined from the polynomial inequalities
worksheet.
“Multiply” signs to get either + or – for each region.
Conclude:
If “expression > 0”, (or greater or equal to 0), look for regions with a positive sign; these are the intervals of your
answer.
If “expression < 0”, (or less than or equal to 0), look for regions with a negative sign; these are the intervals of your
answer.
Be sure not to include a value that will make the denominator 0.
Example:
25
26
1
!1
x +1
+
+
-!x
(x +1)
-+
+
1
"1 ! 0
___________|_____________________|_________________
x +1
--1
+
0
-1
x +1
"
!0
x +1 x +1
Solution: (#!,#1) " [0, !)
1 " ( x + 1)
!0
x +1
Note: -1 is not included since x = !1 would make the denominator 0.
1" x "1
!0
x +1
"x
!0
x +1
Exercises:
Solve the rational inequalities algebraically (using a number line, also known as a sign chart).
Remember to always get 0 alone on one side of the inequality sign.
Write your answer in interval notation.
1.
2.
2x " 3
x 2 +1
!0
(2 x ! 1)( x + 7)
(3 x + 5)( x ! 3) 2
3.
k !1
>1
k +2
4.
5
12
!
p +1 p +1
5.
3
!4
<
2r ! 1 r
6.
1" x2
x 2 + 5x + 6
>0
!0
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27
Determine the domain of each function. The domain is the set of all real-number values for the variable for which the
expression is defined.
7. y =
8. y =
x!2
x+3
1
2
x ! 4x ! 5
GRAPHING RATIONAL FUNCTIONS
Name: ________________________________
Section: ___________
Checked by:_____________________________
_____________________________
_____________________________
Course Objective: Graph fundamental relations and functions using analysis
Scoring: Your work and answer will be checked on 5 randomly picked questions. Neatness may count for several points.
If P(x) and Q(x) are polynomials, then a function of the form
P( x)
f ( x) =
is called a rational function.
Q( x)
Domain – consists of all real numbers for which Q( x) ! 0 . That means that you cannot use numbers that will make the
denominator 0.
x!3
f ( x) =
Example:
x !1
x cannot be ______ because the denominator will equal 0.
Therefore, the domain in interval notation is __________________.
Example:
f ( x) =
2x ! 3
x2 ! 4
Factor the denominator completely. x 2 ! 4 = ___________________
x cannot be ______ or ______ because the denominator will equal 0.
Therefore, the domain in interval notation is ______________________
Vertical Asymptotes – occur when the rational function is in lowest terms and correspond to each root of Q(x) = 0. A graph
never crosses a vertical asymptote.
x!3
f ( x) =
Example:
x !1
The vertical asymptote is the line _______________.
Example:
f ( x) =
2x ! 3
x2 ! 4
The vertical asymptotes are the lines ______________________.
Holes - IF the rational function is not in lowest terms, then a hole occurs instead of a vertical asymptote.
( x + 1)( x + 3)
f ( x) =
Example:
x( x + 1)
________ is a common factor and at x = ______ a hole occurs and NOT a vertical asymptote.
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28
A vertical asymptote occurs at _________
Horizontal Asymptotes – occur when the degree of the numerator is less than or equal to the degree of the denominator. A
graph may cross a horizontal asymptote.
If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote is the line
______________ or the ______-axis.
Example:
f ( x) =
x
2
x !4
,
f ( x) =
1
,
x
f ( x) =
3
2
x + 2x +1
All of the above functions will have a horizontal asymptote of ____________ because the degree of the numerator is
less than the degree of the denominator.
Score:___________
If the degree of the numerator = the degree of the denominator, then the horizontal asymptote is determined by the ratio of the
leading coefficients.
2x + 1
f ( x) =
Example:
x!3
The ratio of the leading coefficients is ______
and the horizontal asymptote is the line _____________.
f ( x) =
Example:
! x 2 + 7x ! 9
x 2 ! 6x + 9
The ratio of the leading coefficients is _________
and the horizontal asymptote is the line _____________.
Find any point(s) where the graph crosses a horizontal asymptote as follows:
determine the horizontal asymptote y = a
-
P( x)
=a
Q( x)
solve for x.
set
Other asymptotic behavior - Determine other asymptotic behavior by dividing (polynomial long division) f (x) by g (x) .
Example:
f ( x) =
2x 2 + 1
x!3
Divide 2 x 2 + 1 by x ! 3 . The resulting quotient is _________________
Consider this quotient as a function y = ______________
As x gets large (negative or positive), the graph of f (x) acts very much like the line you got by dividing.
x-intercepts – Let y = 0 and solve. A fraction will = 0 if its numerator = 0.
y-intercept - Let x = 0 and solve.
x!2
Example: f ( x) =
x+3
The x-intercept is _________ because the solution of _________ is __________.
The y-intercept is ________ because letting x = 0 gives ___________ = __________.
Putting it all together.
( x ! 1)( x + 2)
3 x( x ! 4)
The domain is _______________________
Example:
f ( x) =
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The vertical asymptotes are ____________________
The horizontal asymptote is ___________________
The graph crosses its horizontal asymptote at the point ___________________
The x-intercepts are ___________________
There is no y-intercept because________________________.
Sketch the graph.
GRAPHING RATIONAL FUNCTIONS-STEPS
Course Objective: Graph fundamental relations and functions using analysis
.
METHOD:
P( x)
, where P (x) and Q(x) are polynomial functions
Q( x)
(1) Determine domain. (Recall that the denominator cannot equal zero)
Given a Rational Function y =
(2) Find vertical asymptote(s) by setting the denominator equal to zero and solving. (If any of the factors cancel, the result is
a hole, not a vertical asymptote.)
State vertical asymptotes as x = ____
(Vertical asymptotes are lines)
(3) Find x- and y-intercepts
(4) Find horizontal asymptotes (if any)
Horizontal asymptotes may be determined by “comparing powers” in numerator and denominator.
If degree of P > degree of Q, then there is no horizontal asymptote.
If degree of P < degree of Q, y = 0 is the horizontal asymptote.
If degree of P = degree of Q, then the horizontal asymptote is
a
y= n ,
bn
where an is the leading coefficient of f,
and bn is the leading coefficient of g.
State horizontal asymptotes as y = ____
(Horizontal asymptotes are lines)
(5) Determine other asymptotic behavior by dividing (polynomial long division) P (x) by Q(x) .
(6) Consider graph’s behavior near x-intercepts (If a factor has an even exponent, the graph “touches” the x-axis; if a factor
has an odd exponent, the graph passes through the x-axis.)
(7) Find any point(s) where the graph crosses a horizontal asymptote as follows:
determine the horizontal asymptote y = a
-
P( x)
=a
Q( x)
solve for x.
set
(8) Use the information to sketch a graph.
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30
NOTE:
Any graph of the form y =
ax + b
cx + d
has “basic” graph (draw)
Any graph of the form
y=
a
for n>2 even, has “basic” graph (draw)
( x + b)n
Any graph of the form
y=
a
for n>1 odd, has “basic” graph (draw)
( x + b)n
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31
POLYNOMIAL & RATIONAL GRAPHING REVIEW
Name: _______________________
Section: ___________
Checked by: ________________________
________________________
________________________
Course Objective: Graph fundamental relations and functions using analysis.
Scoring: Your work and answer will be checked on 5 randomly picked questions. Neatness may count for several points.
Given the function
f (x) = 5x(2x " 3) 2 (x + 4) 3
a) The domain of
f (x) is ______________
!
b) The x-intercept(s) is/are ______________
c) Describe the behavior of the graph at each x-intercept
(i.e. the graph touches the x-axis when x = _____
the graph crosses the x-axis when x = _____ )
_______________________________________________
_______________________________________________
_______________________________________________
_______________________________________________
d) What is the degree of
f (x) ?
e) What is the leading coefficient of
_____________
f (x) ? _____________
f) Complete the following statements.
As x ! "#, y ! _________
As x ! +", y ! _________
g) Draw a sketch of y = f (x) .
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32
Score:_____________
Given the function f ( x) =
a) The domain of
( x ! 3)(2 x + 5)
(4 x + 3)( x ! 2)
f (x) is
__________________
b) The vertical asymptote(s) is/are
__________________
c) The horizontal asymptote is
__________________
d) Does the graph cross its horizontal asymptote? _____________
If it does, what is the point of intersection? _____________
e) The x-intercept(s) is/are
__________________
f) The y-intercept is
__________________
Sketch the graph of
y = f (x) . Label important points and asymptotes.
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33
EXTREME FACTORING
Name:_______________________
Section: ___________
Checked by: _________________________
____________________
_________________________
Course Objective: Use advanced factoring techniques to solve equations and simplify expressions
Scoring: Your work and answer will be checked on 5 randomly picked questions. Neatness may count for several points.
Most of the following exercises come directly from intermediate steps in working calculus problems. The goal in calculus is to
either simplify the expression to make the next step easier or to find for which values the expression is equal zero or is
undefined. Factoring provides an efficient means of simplifying the expression.
Exercises: Factor each of the following expressions. Remember, these are expressions NOT equations. They cannot be
“solved”.
1. x !1 ! 10 x
!2
+ 21 =
Are there any values for which the above expression is not defined?
Hint: multiply through by x 2 and factor the resulting quadratic.
For what values is the expression equal zero?
2. x
4
3
! 5x
2
3
! 36 =
Are there any values for which the above expression is not defined?
2
Hint: substitute: p = x 3 . Why does this work?
For what values is the expression equal zero?
3. 3 x( x + 2) 2 + ( x + 2) 3 =
Are there any values for which the above expression is not defined?
Hint: Factor out ( x + 2) 2 .
For what values is the expression equal zero?
4. 4( x + 2) 2 + (8 x + 4)( x + 2) =
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34
Are there any values for which the above expression is not defined?
Hint: Factor out ( x + 2) 2 .
For what values is the expression equal zero?
5.
( x ! 1) 2 ! 2 x( x ! 1)
( x ! 1) 4
=
Are there any values for which the above expression is not defined?
Hint: Factor out of the numerator ( x ! 1) . Reduce.
For what values is the expression equal zero?
Score:__________
6.
! ( x ! 1) 3 + 3( x + 1)( x ! 1) 2
( x ! 1) 6
=
Are there any values for which the above expression is not defined?
Hint: Factor out of the numerator ( x ! 1) 2 . Reduce.
For what values is the expression equal zero?
7. 2 x ! x =
Are there any values for which the above expression is not defined?
1
Hint: Factor out x 2 .
For what values is the expression equal zero?
8. x
5
! 5x
3
2
3
=
Are there any values for which the above expression is not defined?
2
Hint: Factor out x 3 .
For what values is the expression equal zero?
9.
5
3
x
2
3
! 103 x
! 13
=
Are there any values for which the above expression is not defined?
Hint: Factor out
5
3
x
! 13
and write the expression as a fraction.
For what values is the expression equal zero?
10.
10
9
x
! 13
+ 109 x
!43
=
Are there any values for which the above expression is not defined?
Hint: Factor out
10
9
x
!43
and write the expression as a fraction.
For what values is the expression equal zero?
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35
2 2
11.
2
2
2(1 + x ) ! 8 x (1 + x )
(1 + x 2 ) 4
=
Are there any values for which the above expression is not defined?
Hint: Factor out of the numerator 2(1 + x 2 ) 2 . Reduce.
For what values is the expression equal zero?
12.
( x 2 + 1)
1
2
! x 2 ( x 2 + 1)
! 12
=
( x 2 + 1)
Are there any values for which the above expression is not defined?
!1
Hint: Factor out of the numerator (1 + x 2 ) 2 . Reduce.
For what values is the expression equal zero?
LOGARITHM PROPERTIES
Name:_______________________
Section: ___________
Checked by: _________________________
_________________________
_________________________
Course Objective: Use logarithm properties
Scoring: Your work and answer will be checked on 5 randomly picked questions. Neatness may count for several points.
Properties of Logarithms
True or False?
________ 1.
log(x + y) = log x + log y
________ 2.
1
AB
log A + log B " logC = log
2
C
!
________ 3.
!
________ 4.
!
log x
= log x " log y
log y
ln e =
1
2
________ 5.
ln x 3 = ln 3x
!
________
6.
ln x 3 = 3ln x
!
________
7.
(log x) k = k log x
!
________
8.
ln2x 3 = 3ln2x
________
9.
!
(log x)(log y) = log x + log y
!
_______
10.
log a c = b means a b = c .
!
_______
11.
log 5 24 is between 51 and 5 2 .
!
!
!
!
35
!
36
_______ 12.
log 5 24 is between 1 and 2.
_______ 13.
log 5 24 is closer to 1 than to 2.
_______
14. The domain of
!
g(x) = ln x is the set of all real numbers.
_______
15. The range of
!
g(x) = ln x is the set of all real numbers.
_______ 16. The!function
g(x) = ln x is one-to-one.
!
!
Score:____________
A. Simplify the expression by using the definition and properties of logarithms.
1. log(70) ! log(7) = _________________________
2. log 9 (25) ! log 9 (75) =______________________
3. log 7 ( 7 ) = _____________________________
4. log 3 (108) + log 3 (3 / 4) =___________________
5. ! 12 + ln e = ____________________________
6. 2 log 2 (5) ! 3 log 5 (3 5 ) _______________________
7. log 2 (54) ! log 2 (24) ! log 2 (9) = ________________
B. Write the expression as a single logarithm with a coefficient of 1.
8. log(4) + 3[log(1 + x) ! 12 log(1 ! x)] = ___________________________________
9. 4 ln(3) ! 6 ln( x 2 + 1) + 12 [ln( x + 1) ! 2 ln(5)] = ____________________________
C. Write the quantity using sums and differences of simpler logarithmic expressions.
Express the answer so that logarithms of products, quotients and powers do not appear.
10. log( ( x + 1)( x + 2) = ______________________________________________
& x ' 3 4x + 1 #
! = ________________________________________________
11, ln$
$
2x - 1 !"
%
36
37
& x+4 #
! = __________________________________________________
12. log$ 3
$
!
x
%
"
&
4 ' x2
13. ln$
3
$
2
% ( x ' 1)( x + 2)
#
! =_____________________________________________
!
"
EXPONENTIAL and LOGARITHMIC EQUATIONS
Name: _______________________
Section: ___________
Checked by: ________________________
____________________
________________________
Course Objective: Solve exponential and logarithmic equations.
Scoring: Your work and answer will be checked on 5 randomly picked questions. Neatness may count for several points.
Exponential and logarithmic equations appear in calculus for engineering, science, and business. Applications of exponential
and logarithmic functions are abundant in these fields.
Method: Look at the equation and determine which of the categories it falls into. Use the method shown to solve it.
I. If the equation involves a single logarithmic expression or a single exponential expression, use
y = log a ( x) if and only if a y = x .
In other words, if the original equation involves a log, change it to its exponential form;
if the original equation involves an exponential expression, change it to its log form.
II. Use previous properties when applicable.
a. if a M = a N then M = N
b. if log a ( M ) = log a ( N ) then M = N (check your answer)
III. If an equation has several logs with the same base, combine them using properties, then use method II. b.
Combining properties:
a. log a ( M ) + log a ( N ) = log a ( M ! N )
b. log a ( M ) ! log a ( N ) = log a (
M
)
N
IV. If an equation has only exponential expressions with different bases on each side, take log or ln of each side and use the
power property. (Often the logarithmic base to use will be designated in the problem.)
aM = bN
ln(a M ) = ln(b N )
M ln(a ) = N ln(b)
Continue solving for the designated variable.
37
38
V. If it looks like a quadratic, deal with it like a quadratic.
EXAMPLES: State the numeral of the method and solve the equations for x.
1. 7 !4 x = 31+3 x Answer in terms of ln.
Method:________
2. log( x + 1) = 2 log( x ! 1)
Method:________
Score:_____________
3. log 2 (2 x 2 ! 4) = 5
Method:________
4. log 2 (log 3 ( x)) = 0
Method:________
5. 2 3 x = 7
Method:________
6. ln x + ln( x + 1) = ln(2 x + 6)
Method:________
7. e1!5 x = 3 e Answer in terms of ln. Use properties to get “nicest” form
Method:________
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39
8. log 2 x + log 2 (3 x + 10) ! 3 = 0
Method:________
9. a) e 2 x ! 2e x + 1 = 0
Method:________
b) e 2 x ! 2e x ! 3 = 0
Method:________
EXPONENTIAL/LOGARITHMIC INVERSE
Name: ______________________________
Section: ___________
Checked by: __________________________
____________________________
____________________________
Course Objective(s): Find inverse functions with appropriate domain restrictions. Analyze and graph exponential &
logarithmic functions.
Scoring: Your work and answer will be checked on 5 randomly picked questions. Neatness may count for several points.
Given
!
f (x) = ln(x + 1)
1.
The x-intercept of
f (x) is __________.
2.
The y-intercept of
f (x) is __________.
3.
The domain of
4.
The range of
5.
The asymptote of
6.
Find
f
!1
f (x) is ____________.
f (x) is ___________.
f (x) is __________. This is a __________ asymptote.
( x) .
f "1 (x) = ________________
7.
The x-intercept of
f
8.
The y-intercept of
f
9.
The domain of
!1
( x) is ___________
!
10. The range of
f
f
!1
!1
!1
( x) is __________.
( x) is ____________.
( x) is ___________.
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40
!1
f
11. The asymptote of
12. Sketch the graphs of
( x) is __________. This is a __________ asymptote.
f (x) and f
!1
( x) , labeling asymptotes and intercepts.
Score:___________
Given
f ( x) = e 2 x +1
13. The x-intercept of
f (x) is __________.
14. The y-intercept of
f (x) is __________.
15. The domain of
16. The range of
f (x) is ____________.
f (x) is ___________.
f (x) is __________. This is a __________ asymptote.
17. The asymptote of
18. Find
f
!1
( x) .
f "1 (x) = ________________
19. The x-intercept of
f
20. The y-intercept of
f
!1
( x) is ___________
!
21. The domain of
22. The range of
f
f
!1
!1
23. The asymptote of
!1
( x) is __________.
( x) is ____________.
( x) is ___________.
f
!1
24. Sketch the graphs of
( x) is __________. This is a __________ asymptote.
f (x) and f
!1
( x) , labeling asymptotes and intercepts.
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41
GENERAL EXPONENTIAL AND LOGARITHM REVIEW
Name: _______________________
Section: ___________
Checked by: ___________________________
___________________________
____________________________
Course Objective: Review for Final Exam.
Scoring: Your work and answer will be checked on 5 randomly picked questions. Neatness may count for several points.
1.
y = bx , b > 1
Domain:
Range:
y-intercept:
x-intercept:
Asymptote(s):
Graph:
2.
y = bx , 0 < b < 1
Domain:
Range:
y-intercept:
x-intercept:
Asymptote(s):
Graph:
3.
y = log b x , b > 1
Domain:
Range:
y-intercept:
x-intercept:
Asymptote(s):
Graph:
4.
y = 3 " x +1 " !
Domain:
Range:
(Exact values for all answers)
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42
y-intercept:
x-intercept:
Asymptote(s):
Graph:
y = ! log 3 ( x ! 2) + 1
5.
Domain:
Range:
y-intercept:
x-intercept:
Asymptote(s):
Graph:
Score: ____________
f ( x) = log 3 (2 ! x) , find f
6.
7.
8.
!1
Write as a single logarithm with a coefficient of 1. Assume all variables represent positive real numbers.
a)
! 23 log 5 5m 2 + 12 log 5 25m 2
b)
2(log 5 3 ! 3 log 5 4) + log 5 8
Solve for x.
a)
e 2 x ! 5e x = 6
b)
log 4 x + log 4 ( x ! 3) = 1
c)
! 2e 3 x !1 = 9
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9.
Express log10 2 in terms of base e logarithms.
GENERAL GRAPHING REVIEW
Name: _______________________
Section: ___________
Checked by: ___________________________
___________________________
____________________________
Course Objective: Review for Final Exam.
Scoring: Your work and answer will be checked on 5 randomly picked questions. Neatness may count for several points.
Graph. Find asymptotes, x- and y-intercepts. Note domain and range.
1.
t ( x) = ! x + 4 ! 2
2.
t ( x) = 3 ! ( x + 2) 2
3.
t ( x) = x ! 2 + 3
4.
t ( x) = !
5.
t ( x) = !3 x + 2 ! 1
6.
t ( x) = e ! x + 5
7.
t ( x) = ! log 2 ( x ! 3)
8.
t ( x) = ln( x + 1) ! 1
9.
p ( x) = x( x ! 3) 2 ( x + 1) 3 ( x ! 5)
1
!2
x+3
10.
p ( x) = x 3 ! 10 x 2 + 25 x
11.
r ( x) =
x!2
x2 !1
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44
2
12.
r ( x) =
2x + 7x ! 4
x2 + x ! 2
Score: _____________
44