Question 1

Give the equations of any horizontal asymptotes.
-$$h ( x ) = \frac { 3 x ^ { 2 } - 4 x - 2 } { 8 x ^ { 2 } - 7 x + 9 }$$

Accepted Answer

A)  $y = \frac { 4 } { 7 }$ 
B)  $y = 0$ 
C)  $y = \frac { 3 } { 8 }$ 
D)  none 
A)  $y = \frac { 4 } { 7 }$ 
B)  $y = 0$ 
C)  $y = \frac { 3 } { 8 }$ 
D)  none

Question 2

Give the equation of any oblique asymptotes.
-$$f ( x ) = \frac { 9 x ^ { 2 } } { 9 x ^ { 2 } + 3 }$$

Accepted Answer

A)  none 
B)  $y = x + 1$ 
C)  $y = x + 9$ 
D)  $x = y + 9$ 
A)  none 
B)  $y = x + 1$ 
C)  $y = x + 9$ 
D)  $x = y + 9$

Question 3

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative
real zeros for the function.
-$$f ( x ) = 7 x ^ { 8 } + 3 x ^ { 6 } + 2 x ^ { 4 } + 5 x ^ { 2 } + 2$$

Accepted Answer

A) 
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 4 & 4 & 0
\end{array}$ 
B) 
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 4 & 0 & 4
\end{array}$ 
C) 
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 0 & 4 & 4
\end{array}$ 
D) 
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 0 & 0 & 8
\end{array}$ 
A) 
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 4 & 4 & 0
\end{array}$ 
B) 
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 4 & 0 & 4
\end{array}$ 
C) 
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 0 & 4 & 4
\end{array}$ 
D) 
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 0 & 0 & 8
\end{array}$

Question 4

Answer the question
-$$\text { How can the graph of } f ( x ) = \frac { 1 } { ( x - 9 ) ^ { 2 } } + 15 \text { be obtained from the graph of } y = \frac { 1 } { x ^ { 2 } } ?$$

Accepted Answer

A)  By making a horizontal shift of 15 units to the right and a vertical shift of 9 units down 
B)  By making a horizontal shift of 15 units to the left and a vertical shift of 9 units up 
C)  By making a horizontal shift of 9 units to the right and a vertical shift of 15 units up 
D)  By making a horizontal shift of 9 units to the left and a vertical shift of 15 units down 
A)  By making a horizontal shift of 15 units to the right and a vertical shift of 9 units down 
B)  By making a horizontal shift of 15 units to the left and a vertical shift of 9 units up 
C)  By making a horizontal shift of 9 units to the right and a vertical shift of 15 units up 
D)  By making a horizontal shift of 9 units to the left and a vertical shift of 15 units down

Question 5

Give the equations of any horizontal asymptotes.
-$$f ( x ) = \frac { 5 x ^ { 2 } + 7 } { 5 x ^ { 2 } - 7 }$$

Accepted Answer

A)  none 
B)  $y = 1$ 
C)  $y = - 7$ 
D)  $\mathrm { y } = 7$ 
A)  none 
B)  $y = 1$ 
C)  $y = - 7$ 
D)  $\mathrm { y } = 7$

Question 6

Translate the given formula to an English phrase using the word "varies".
-Suppose $\mathrm { z }$ varies directly as $\mathrm { x }$ and inversely as the square of $\mathrm { y }$. If $\mathrm { x }$ is doubled and $\mathrm { y }$ is tripled, what happens to $\mathrm { z }$ ?

Accepted Answer

A)  $\mathrm { z }$ is six times as large as before 
B)  $\mathrm { z }$ is multiplied by $\frac { 1 } { 18 }$ 
C)  $\mathrm { z }$ is multiplied by $\frac { 2 } { 9 }$ 
D)  $\mathrm { z }$ is one-sixth as large as before 
A)  $\mathrm { z }$ is six times as large as before 
B)  $\mathrm { z }$ is multiplied by $\frac { 1 } { 18 }$ 
C)  $\mathrm { z }$ is multiplied by $\frac { 2 } { 9 }$ 
D)  $\mathrm { z }$ is one-sixth as large as before

Question 7

Use synthetic division to divide f(x) by x - k for the given value of k. Then express f(x) in the form f(x) = (x - k)q(x) + r for
the given value of k.
-$$f ( x ) = - 6 x ^ { 3 } + 2 x ^ { 2 } + 5 x - 10 ; \quad k = 2$$

Accepted Answer

A)  $f ( x ) = ( x - 2 ) \left( - 6 x ^ { 2 } + 10 x + 25 \right) - 40$ 
B)  $f ( x ) = ( x + 2 ) \left( - 6 x ^ { 2 } + 10 x + 25 \right) - 40$ 
C)  $f ( x ) = ( x - 2 ) \left( - 6 x ^ { 2 } - 10 x + 15 \right) - 40$ 
D)  $f ( x ) = ( x - 2 ) \left( - 6 x ^ { 2 } - 10 x - 15 \right) - 40$ 
A)  $f ( x ) = ( x - 2 ) \left( - 6 x ^ { 2 } + 10 x + 25 \right) - 40$ 
B)  $f ( x ) = ( x + 2 ) \left( - 6 x ^ { 2 } + 10 x + 25 \right) - 40$ 
C)  $f ( x ) = ( x - 2 ) \left( - 6 x ^ { 2 } - 10 x + 15 \right) - 40$ 
D)  $f ( x ) = ( x - 2 ) \left( - 6 x ^ { 2 } - 10 x - 15 \right) - 40$

Question 8

Give the equations of any vertical asymptotes.
-$$f ( x ) = \frac { 5 x + 2 } { 3 x - 8 }$$

Accepted Answer

A)  $x = - \frac { 3 } { 8 }$ 
B)  $x = \frac { 5 } { 3 }$ 
C)  $x = \frac { 8 } { 3 }$ 
D)  $x = \frac { 3 } { 8 }$ 
A)  $x = - \frac { 3 } { 8 }$ 
B)  $x = \frac { 5 } { 3 }$ 
C)  $x = \frac { 8 } { 3 }$ 
D)  $x = \frac { 3 } { 8 }$

Question 9

Round your answer to two decimal places.
-If $x$ varies inversely as $y ^ { 2 }$, and $x = 6$ when $y = 12$, find $x$ when $y = 3$

Accepted Answer

A)  4 
B)  54 
C)  96 
D)  144 
A)  4 
B)  54 
C)  96 
D)  144

Question 10

Sketch the graph of the parabola.
-$$y = - 5 x ^ { 2 }$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 11

Find all rational zeros and factor f(x).
-$$f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - x - 4$$

Accepted Answer

A)  $1 , - 1 , - 5 ; f ( x ) = ( x - 1 ) ( x + 1 ) ( x + 5 )$ 
B)  $1 , - 1 , - 4 ; f ( x ) = ( x - 1 ) ( x + 1 ) ( x + 4 )$ 
C)  $1 , - 2,6 ; f ( x ) = ( x - 1 ) ( x + 2 ) ( x - 6 )$ 
D)  $1 , - 1,4 ; f ( x ) = ( x - 1 ) ( x + 1 ) ( x - 4 )$ 
A)  $1 , - 1 , - 5 ; f ( x ) = ( x - 1 ) ( x + 1 ) ( x + 5 )$ 
B)  $1 , - 1 , - 4 ; f ( x ) = ( x - 1 ) ( x + 1 ) ( x + 4 )$ 
C)  $1 , - 2,6 ; f ( x ) = ( x - 1 ) ( x + 2 ) ( x - 6 )$ 
D)  $1 , - 1,4 ; f ( x ) = ( x - 1 ) ( x + 1 ) ( x - 4 )$

Question 12

Find the equation of the axis of symmetry of the parabola.
-$$y = 3 x ^ { 2 } + 30 x + 73$$

Accepted Answer

A)  $x = - 2$ 
B)  $x = 5$ 
C)  $x = 2$ 
D)  $x = - 5$ 
A)  $x = - 2$ 
B)  $x = 5$ 
C)  $x = 2$ 
D)  $x = - 5$

Question 13

Solve the inequality. Write the solution set in interval notation.
-$$x ^ { 2 } + 3 x - 18 < 0$$

Accepted Answer

A)  $( - 6,3 )$ 
B)  $( \infty , - 6 ) \cup ( 3 , \infty )$ 
C)  $( \infty , - 3 ) \cup ( 6 , \infty )$ 
D)  $( - 3,6 )$ 
A)  $( - 6,3 )$ 
B)  $( \infty , - 6 ) \cup ( 3 , \infty )$ 
C)  $( \infty , - 3 ) \cup ( 6 , \infty )$ 
D)  $( - 3,6 )$

Question 14

Find the equation that the given graph represents.
-$$f ( x ) = - \frac { 1 } { 2 } ( x + 3 ) ^ { 3 } + 4$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 15

Find the zeros of the polynomial function and state the multiplicity of each.
-$$f ( x ) = \left( x ^ { 2 } + 13 x + 40 \right) ^ { 2 }$$

Accepted Answer

A)  8 (multiplicity 2 ), 5 (multiplicity 2 ) 
B)  $- \sqrt { 5 }$ (multiplicity 4) 
C)  $\sqrt { 5 }$ (multiplicity $2 ), - \sqrt { 5 }$ (multiplicity 2 ) 
D)  $- 5$ (multiplicity 2), $- 8$ (multiplicity 2 ) 
A)  8 (multiplicity 2 ), 5 (multiplicity 2 ) 
B)  $- \sqrt { 5 }$ (multiplicity 4) 
C)  $\sqrt { 5 }$ (multiplicity $2 ), - \sqrt { 5 }$ (multiplicity 2 ) 
D)  $- 5$ (multiplicity 2), $- 8$ (multiplicity 2 )

Question 16

Give the domain and range for the rational function. Use interval notation.
-$$f ( x ) = \frac { 1 } { ( x - 7 ) ^ { 2 } } + 5$$

Accepted Answer

A)  Domain: $( - \infty , 5 ) \cup ( 5 , \infty )$; Range: $( 0 , \infty )$ 
B)  Domain: $( - \infty , 7 ) \cup ( 7 , \infty )$; Range: $( - \infty , 0 ) \cup ( 0 , \infty )$ 
C)  Domain: $( - \infty , - 7 ) \cup ( - 7 , \infty )$; Range: $( - \infty , 5 )$ 
D)  Domain: $( - \infty , 7 ) \cup ( 7 , \infty )$; Range: $( 5 , \infty )$ 
A)  Domain: $( - \infty , 5 ) \cup ( 5 , \infty )$; Range: $( 0 , \infty )$ 
B)  Domain: $( - \infty , 7 ) \cup ( 7 , \infty )$; Range: $( - \infty , 0 ) \cup ( 0 , \infty )$ 
C)  Domain: $( - \infty , - 7 ) \cup ( - 7 , \infty )$; Range: $( - \infty , 5 )$ 
D)  Domain: $( - \infty , 7 ) \cup ( 7 , \infty )$; Range: $( 5 , \infty )$

Question 17

Solve the problem
-The number of mosquitoes $\mathrm { M } ( \mathrm { x } )$, in millions, in a certain area depends on the June rainfall $\mathrm { x }$, in inches: $\mathrm { M } ( \mathrm { x } ) = 2 \mathrm { x } - \mathrm { x } ^ { 2 }$. What rainfall produces the maximum number of mosquitoes?

Accepted Answer

A)  0 in. 
B)  2 in. 
C)  1 in. 
D)  4 in. 
A)  0 in. 
B)  2 in. 
C)  1 in. 
D)  4 in.

Question 18

Use synthetic division to divide f(x) by x - k for the given value of k. Then express f(x) in the form f(x) = (x - k)q(x) + r for
the given value of k.
-$$f ( x ) = 3 x ^ { 5 } + 4 x ^ { 4 } + 2 x ^ { 2 } - 1 ; \quad k = - 2$$

Accepted Answer

A)  $f ( x ) = ( x + 2 ) \left( 3 x ^ { 4 } - 2 x ^ { 3 } + 4 x ^ { 2 } + 6 \right) - 13$ 
B)  $f ( x ) = ( x + 2 ) \left( 3 x ^ { 4 } + 2 x ^ { 3 } + 4 x ^ { 2 } + 8 x \right) - 15$ 
C)  $f ( x ) = ( x + 2 ) \left( 3 x ^ { 4 } - 2 x ^ { 3 } + 4 x ^ { 2 } - 6 x + 12 \right) - 25$ 
D)  $f ( x ) = ( x + 2 ) \left( 3 x ^ { 4 } - 2 x ^ { 3 } + 6 x ^ { 2 } - 12 \right) + 23$ 
A)  $f ( x ) = ( x + 2 ) \left( 3 x ^ { 4 } - 2 x ^ { 3 } + 4 x ^ { 2 } + 6 \right) - 13$ 
B)  $f ( x ) = ( x + 2 ) \left( 3 x ^ { 4 } + 2 x ^ { 3 } + 4 x ^ { 2 } + 8 x \right) - 15$ 
C)  $f ( x ) = ( x + 2 ) \left( 3 x ^ { 4 } - 2 x ^ { 3 } + 4 x ^ { 2 } - 6 x + 12 \right) - 25$ 
D)  $f ( x ) = ( x + 2 ) \left( 3 x ^ { 4 } - 2 x ^ { 3 } + 6 x ^ { 2 } - 12 \right) + 23$

Question 19

Find the equation that the given graph represents.
-

Accepted Answer

A)  $P ( x ) = x ^ { 6 } - 2 x ^ { 2 } - 6$ 
B)  $P ( x ) = x ^ { 4 } - 3 x ^ { 2 } + 6$ 
C)  $P ( x ) = - x ^ { 5 } - 3 x ^ { 2 } + 6$ 
D)  $P ( x ) = x ^ { 5 } + 2 x ^ { 3 } + 6$ 
A)  $P ( x ) = x ^ { 6 } - 2 x ^ { 2 } - 6$ 
B)  $P ( x ) = x ^ { 4 } - 3 x ^ { 2 } + 6$ 
C)  $P ( x ) = - x ^ { 5 } - 3 x ^ { 2 } + 6$ 
D)  $P ( x ) = x ^ { 5 } + 2 x ^ { 3 } + 6$

Question 20

Give the equations of any vertical asymptotes.
-$$h ( x ) = \frac { 4 x - 7 } { x ^ { 2 } + 3 x - 40 }$$

Accepted Answer

A)  $y = 5 , y = - 8$ 
B)  $y = 4$ 
C)  $x = 5 , x = - 8$ 
D)  $x = - 5 , x = 8$ 
A)  $y = 5 , y = - 8$ 
B)  $y = 4$ 
C)  $x = 5 , x = - 8$ 
D)  $x = - 5 , x = 8$

Give the equations of any horizontal asymptotes. - $h ( x ) = \frac { 3 x ^ { 2 } - 4 x - 2 } { 8 x ^ { 2 } - 7 x + 9 }$

Give the equation of any oblique asymptotes. - $f ( x ) = \frac { 9 x ^ { 2 } } { 9 x ^ { 2 } + 3 }$

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - $f ( x ) = 7 x ^ { 8 } + 3 x ^ { 6 } + 2 x ^ { 4 } + 5 x ^ { 2 } + 2$

Answer the question - $\text { How can the graph of } f ( x ) = \frac { 1 } { ( x - 9 ) ^ { 2 } } + 15 \text { be obtained from the graph of } y = \frac { 1 } { x ^ { 2 } } ?$

Give the equations of any horizontal asymptotes. - $f ( x ) = \frac { 5 x ^ { 2 } + 7 } { 5 x ^ { 2 } - 7 }$

Translate the given formula to an English phrase using the word "varies". -Suppose $\mathrm { z }$ varies directly as $\mathrm { x }$ and inversely as the square of $\mathrm { y }$ . If $\mathrm { x }$ is doubled and $\mathrm { y }$ is tripled, what happens to $\mathrm { z }$ ?

Use synthetic division to divide f(x) by x - k for the given value of k. Then express f(x) in the form f(x) = (x - k)q(x) + r for the given value of k. - $f ( x ) = - 6 x ^ { 3 } + 2 x ^ { 2 } + 5 x - 10 ; \quad k = 2$

Give the equations of any vertical asymptotes. - $f ( x ) = \frac { 5 x + 2 } { 3 x - 8 }$

Round your answer to two decimal places. -If $x$ varies inversely as $y ^ { 2 }$ , and $x = 6$ when $y = 12$ , find $x$ when $y = 3$

Sketch the graph of the parabola. - $y = - 5 x ^ { 2 }$ $Sketch the graph of the parabola. - y = - 5 x ^ { 2 }$

Find all rational zeros and factor f(x). - $f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - x - 4$

Find the equation of the axis of symmetry of the parabola. - $y = 3 x ^ { 2 } + 30 x + 73$

Solve the inequality. Write the solution set in interval notation. - $x ^ { 2 } + 3 x - 18 < 0$

Find the equation that the given graph represents. - $f ( x ) = - \frac { 1 } { 2 } ( x + 3 ) ^ { 3 } + 4$ $Find the equation that the given graph represents. - f ( x ) = - \frac { 1 } { 2 } ( x + 3 ) ^ { 3 } + 4$

Find the zeros of the polynomial function and state the multiplicity of each. - $f ( x ) = \left( x ^ { 2 } + 13 x + 40 \right) ^ { 2 }$

Give the domain and range for the rational function. Use interval notation. - $f ( x ) = \frac { 1 } { ( x - 7 ) ^ { 2 } } + 5$

Use synthetic division to divide f(x) by x - k for the given value of k. Then express f(x) in the form f(x) = (x - k)q(x) + r for the given value of k. - $f ( x ) = 3 x ^ { 5 } + 4 x ^ { 4 } + 2 x ^ { 2 } - 1 ; \quad k = - 2$

Find the equation that the given graph represents. -

Give the equations of any vertical asymptotes. - $h ( x ) = \frac { 4 x - 7 } { x ^ { 2 } + 3 x - 40 }$

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Trigonometric Identities and Equations

Applications of Trigonometry

Systems and Matrices

Analytic Geometry

Further Topics in Algebra

Filters

Exam 4: Polynomial and Rational Functions

Give the equations of any horizontal asymptotes. - h(x)=3x2−4x−28x2−7x+9h ( x ) = \frac { 3 x ^ { 2 } - 4 x - 2 } { 8 x ^ { 2 } - 7 x + 9 }h(x)=8x2−7x+93x2−4x−2​

Give the equation of any oblique asymptotes. - f(x)=9x29x2+3f ( x ) = \frac { 9 x ^ { 2 } } { 9 x ^ { 2 } + 3 }f(x)=9x2+39x2​

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - f(x)=7x8+3x6+2x4+5x2+2f ( x ) = 7 x ^ { 8 } + 3 x ^ { 6 } + 2 x ^ { 4 } + 5 x ^ { 2 } + 2f(x)=7x8+3x6+2x4+5x2+2

Give the equations of any horizontal asymptotes. - f(x)=5x2+75x2−7f ( x ) = \frac { 5 x ^ { 2 } + 7 } { 5 x ^ { 2 } - 7 }f(x)=5x2−75x2+7​

Translate the given formula to an English phrase using the word "varies". -Suppose z\mathrm { z }z varies directly as x\mathrm { x }x and inversely as the square of y\mathrm { y }y . If x\mathrm { x }x is doubled and y\mathrm { y }y is tripled, what happens to z\mathrm { z }z ?

Use synthetic division to divide f(x) by x - k for the given value of k. Then express f(x) in the form f(x) = (x - k)q(x) + r for the given value of k. - f(x)=−6x3+2x2+5x−10;k=2f ( x ) = - 6 x ^ { 3 } + 2 x ^ { 2 } + 5 x - 10 ; \quad k = 2f(x)=−6x3+2x2+5x−10;k=2

Give the equations of any vertical asymptotes. - f(x)=5x+23x−8f ( x ) = \frac { 5 x + 2 } { 3 x - 8 }f(x)=3x−85x+2​

Round your answer to two decimal places. -If xxx varies inversely as y2y ^ { 2 }y2 , and x=6x = 6x=6 when y=12y = 12y=12 , find xxx when y=3y = 3y=3

Sketch the graph of the parabola. - y=−5x2y = - 5 x ^ { 2 }y=−5x2

Find all rational zeros and factor f(x). - f(x)=x3+4x2−x−4f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - x - 4f(x)=x3+4x2−x−4

Find the equation of the axis of symmetry of the parabola. - y=3x2+30x+73y = 3 x ^ { 2 } + 30 x + 73y=3x2+30x+73

Solve the inequality. Write the solution set in interval notation. - x2+3x−18<0x ^ { 2 } + 3 x - 18 < 0x2+3x−18<0

Find the equation that the given graph represents. - f(x)=−12(x+3)3+4f ( x ) = - \frac { 1 } { 2 } ( x + 3 ) ^ { 3 } + 4f(x)=−21​(x+3)3+4

Find the zeros of the polynomial function and state the multiplicity of each. - f(x)=(x2+13x+40)2f ( x ) = \left( x ^ { 2 } + 13 x + 40 \right) ^ { 2 }f(x)=(x2+13x+40)2

Give the domain and range for the rational function. Use interval notation. - f(x)=1(x−7)2+5f ( x ) = \frac { 1 } { ( x - 7 ) ^ { 2 } } + 5f(x)=(x−7)21​+5

Use synthetic division to divide f(x) by x - k for the given value of k. Then express f(x) in the form f(x) = (x - k)q(x) + r for the given value of k. - f(x)=3x5+4x4+2x2−1;k=−2f ( x ) = 3 x ^ { 5 } + 4 x ^ { 4 } + 2 x ^ { 2 } - 1 ; \quad k = - 2f(x)=3x5+4x4+2x2−1;k=−2