Question 1

Give the equations of any vertical asymptotes.
-$$h ( x ) = \frac { ( x - 4 ) ( x + 1 ) } { x ^ { 2 } - 9 }$$

Accepted Answer

A)  $x = - 4 , x = 1$ 
B)  $x = 4 , x = - 1$ 
C)  none 
D)  $x = 3 , x = - 3$ 
A)  $x = - 4 , x = 1$ 
B)  $x = 4 , x = - 1$ 
C)  none 
D)  $x = 3 , x = - 3$

Question 2

Use a graphing calculator to approximate the real zeros. Give each zero as a decimal to the nearest tenth.
-$$f ( x ) = x ^ { 4 } - 7 x ^ { 2 } + 12$$

Accepted Answer

A)  $1.7 , - 1.7,2 , - 2$ 
B)  $1.7,2$ 
C)  $3 , - 3,4 , - 4$ 
D)  $- 1,2.3,2.3 , - 2.3$ 
A)  $1.7 , - 1.7,2 , - 2$ 
B)  $1.7,2$ 
C)  $3 , - 3,4 , - 4$ 
D)  $- 1,2.3,2.3 , - 2.3$

Question 3

Sketch the graph of the parabola.
-$$y = \frac { 2 } { 5 } ( x + 1 ) ^ { 2 } - 2$$

Accepted Answer

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

Question 4

Sketch the graph of the rational function.
-$$f ( x ) = \frac { x ^ { 2 } - x - 12 } { x + 4 }$$

Accepted Answer

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

Question 5

Use a graphing calculator to find the coordinates of the turning points of the graph of the polynomial function in the
indicated domain interval. Give answers to the nearest hundredth.
-$$f ( x ) = x ^ { 4 } - 29 x ^ { 2 } + 100 ; [ - 1,0.5 ]$$

Accepted Answer

A)  $( - 3.81 , - 110.25 )$ 
B)  $( 2,174.00 )$ 
C)  $( 3.81,272.02 )$ 
D)  $( 0,100 )$ 
A)  $( - 3.81 , - 110.25 )$ 
B)  $( 2,174.00 )$ 
C)  $( 3.81,272.02 )$ 
D)  $( 0,100 )$

Question 6

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.
-$$f ( x ) = - 6 x ^ { 4 } - 8 x ^ { 3 } - 7 x ^ { 2 } - 8 x + 3$$

Accepted Answer

A) 
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 2 & 2 & 0 \
2 & 0 & 2 \
0 & 2 & 2 \
0 & 0 & 2
\end{array}$ 
B) 
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 1 & 1 & 2
\end{array}$ 
C) 
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 1 & 2 & 1
\end{array}$ 
D) 
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 1 & 3 & 0 \
1 & 1 & 2
\end{array}$ 
A) 
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 2 & 2 & 0 \
2 & 0 & 2 \
0 & 2 & 2 \
0 & 0 & 2
\end{array}$ 
B) 
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 1 & 1 & 2
\end{array}$ 
C) 
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 1 & 2 & 1
\end{array}$ 
D) 
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 1 & 3 & 0 \
1 & 1 & 2
\end{array}$

Question 7

Match the equation to the correct graph.
-$$y = - ( x + 6 ) ^ { 2 } + 4$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D)

Question 8

Solve.
-The data in the table show the number of burglaries reported in a small town for several years on record. $$\begin{array} { c | c } 
\text { Year } & \text { \# of Burglaries } \
\hline 0 & 4 \
2 & 3 \
4 & 2 \
6 & 2 \
7 & 2 \
8 & 3 \
11 & 4 \
13 & 6 \
15 & 8
\end{array}$$ Determine the linear, quadratic, cubic, or quartic function that best fits the data.

Accepted Answer

A)  $f ( x ) = 0.28 x + 1.7$ 
B)  $f ( x ) = - 0.0007 x ^ { 3 } + 0.086 x ^ { 2 } - 0.79 x + 4.07$ 
C)  $f ( x ) = - 0.00004 x ^ { 4 } + 0.0006 x ^ { 3 } + 0.07 x ^ { 2 } - 0.76 x + 4.05$ 
D)  $f ( x ) = 0.07 x ^ { 2 } - 0.71 x + 3.99$ 
A)  $f ( x ) = 0.28 x + 1.7$ 
B)  $f ( x ) = - 0.0007 x ^ { 3 } + 0.086 x ^ { 2 } - 0.79 x + 4.07$ 
C)  $f ( x ) = - 0.00004 x ^ { 4 } + 0.0006 x ^ { 3 } + 0.07 x ^ { 2 } - 0.76 x + 4.05$ 
D)  $f ( x ) = 0.07 x ^ { 2 } - 0.71 x + 3.99$

Question 9

Translate the given formula to an English phrase using the word "varies".
-If y varies directly as the square of x, and x is doubled, how is y changed?

Accepted Answer

A)  y is one-quarter as large as before 
B)  y is four times as large as before 
C)  y is twice as large as before 
D)  y is half as large as before 
A)  y is one-quarter as large as before 
B)  y is four times as large as before 
C)  y is twice as large as before 
D)  y is half as large as before

Question 10

Find the zeros of the polynomial function and state the multiplicity of each.
-$$f ( x ) = \left( x ^ { 2 } + x - 12 \right) ^ { 7 } ( x - 4 + \sqrt { 7 } ) ^ { 2 }$$

Accepted Answer

A)  $- 4 , - 3$ (multiplicity 7), $- 4 - \sqrt { 7 }$ (multiplicity 2) 
B)  $4 , - 3$ (multiplicity 7 ), $4 - \sqrt { 7 }$ (multiplicity 2) 
C)  $- 4,3$ (multiplicity 7 ), $4 + \sqrt { 7 }$ (multiplicity 2) 
D)  $- 4,3$ (multiplicity 7), $4 - \sqrt { 7 }$ (multiplicity 2) 
A)  $- 4 , - 3$ (multiplicity 7), $- 4 - \sqrt { 7 }$ (multiplicity 2) 
B)  $4 , - 3$ (multiplicity 7 ), $4 - \sqrt { 7 }$ (multiplicity 2) 
C)  $- 4,3$ (multiplicity 7 ), $4 + \sqrt { 7 }$ (multiplicity 2) 
D)  $- 4,3$ (multiplicity 7), $4 - \sqrt { 7 }$ (multiplicity 2)

Question 11

Find the equation that the given graph represents.
-

Accepted Answer

A)  $P ( x ) = x ^ { 5 } - 4 x ^ { 3 } + 12 x ^ { 2 } + 10$ 
B)  $P ( x ) = - x ^ { 5 } - 10 x ^ { 2 } - 10$ 
C)  $P ( x ) = x ^ { 4 } + x ^ { 3 } - 10 x ^ { 2 } + 10$ 
D)  $P ( x ) = - x ^ { 4 } + x ^ { 3 } - 12 x ^ { 2 } + 10$ 
A)  $P ( x ) = x ^ { 5 } - 4 x ^ { 3 } + 12 x ^ { 2 } + 10$ 
B)  $P ( x ) = - x ^ { 5 } - 10 x ^ { 2 } - 10$ 
C)  $P ( x ) = x ^ { 4 } + x ^ { 3 } - 10 x ^ { 2 } + 10$ 
D)  $P ( x ) = - x ^ { 4 } + x ^ { 3 } - 12 x ^ { 2 } + 10$

Question 12

Solve the problem
-If an object is propelled upward from a height of 96 feet at an initial velocity of 64 feet per second, then its height after $t$ seconds is given by the equation $h ( t ) = - 16 t ^ { 2 } + 64 t + 96$, where height is in feet. After how many seconds will the object reach a height of 160 feet?

Accepted Answer

A)  $2 \mathrm { sec }$ 
B)  $1 \mathrm { sec }$ 
C)  $8 \mathrm { sec }$ 
D)  $4 \mathrm { sec }$ 
A)  $2 \mathrm { sec }$ 
B)  $1 \mathrm { sec }$ 
C)  $8 \mathrm { sec }$ 
D)  $4 \mathrm { sec }$

Question 13

Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary.
-$$f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 5 x + 39$$

Accepted Answer

A)  $- 3,3 + 2 i , 3 - 2 i$ 
B)  $- 3,3 + 4 i , 3 - 4 i$ 
C)  $- 3,1 + 2 i , 1 - 2 i$ 
D)  $- 3,1 + 2 \sqrt { 13 } i , 1 - 2 \sqrt { 13 } i$ 
A)  $- 3,3 + 2 i , 3 - 2 i$ 
B)  $- 3,3 + 4 i , 3 - 4 i$ 
C)  $- 3,1 + 2 i , 1 - 2 i$ 
D)  $- 3,1 + 2 \sqrt { 13 } i , 1 - 2 \sqrt { 13 } i$

Question 14

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions.
-Zeros of 3 and $i , P ( 1 ) = - 4$

Accepted Answer

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

Question 15

Solve the problem
-The polynomial $G ( x ) = - 0.006 x ^ { 4 } + 0.140 x ^ { 3 } - 0.53 x ^ { 2 } + 1.79 x$ measures the concentration of a dye in the bloodstream $x$ seconds after it is injected. Does the concentration increase between 12 and 13 seconds?

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 16

Factor f(x) into linear factors given that k is a zero of f(x).
-$$f ( x ) = 4 x ^ { 3 } + 24 x ^ { 2 } + 17 x - 15 ; k = \frac { 1 } { 2 }$$

Accepted Answer

A)  $( 2 x + 1 ) ( 2 x + 3 ) ( x - 5 )$ 
B)  $( 2 x + 1 ) ( 2 x + 1 ) ( x - 15 )$ 
C)  $( 2 x - 1 ) ( 2 x + 3 ) ( x + 5 )$ 
D)  $( 2 x - 1 ) ( 2 x - 3 ) ( x - 5 )$ 
A)  $( 2 x + 1 ) ( 2 x + 3 ) ( x - 5 )$ 
B)  $( 2 x + 1 ) ( 2 x + 1 ) ( x - 15 )$ 
C)  $( 2 x - 1 ) ( 2 x + 3 ) ( x + 5 )$ 
D)  $( 2 x - 1 ) ( 2 x - 3 ) ( x - 5 )$

Question 17

Use the factor theorem to decide whether or not the second polynomial is a factor of the first.
-$3 x ^ { 2 } - 22 x + 40 ; x - 4$

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 18

Match the equation to the correct graph.
-$$y = - ( x + 5 ) ^ { 2 } + 5$$

Accepted Answer

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

Question 19

Give the equation of any oblique asymptotes.
-$$f ( x ) = \frac { x ^ { 2 } - 8 x + 4 } { x + 2 }$$

Accepted Answer

A)  none 
B)  $x = y + 8$ 
C)  $y = x - 10$ 
D)  $y = x + 12$ 
A)  none 
B)  $x = y + 8$ 
C)  $y = x - 10$ 
D)  $y = x + 12$

Question 20

Answer the question
-$$\text { How can the graph of } f ( x ) = - \frac { 1 } { x ^ { 2 } } - 4 \text { be obtained from the graph of } y = \frac { 1 } { x ^ { 2 } }$$

Accepted Answer

A)  By making a horizontal shift of 4 units to the left and a vertical shift of 1 unit down 
B)  By reflecting across the x-axis and making a vertical shift of 4 units down 
C)  By making a horizontal shift of 1 unit to the right and a vertical shift of 4 units up 
D)  By reflecting across the y-axis and making a horizontal shift of 4 units to the left 
A)  By making a horizontal shift of 4 units to the left and a vertical shift of 1 unit down 
B)  By reflecting across the x-axis and making a vertical shift of 4 units down 
C)  By making a horizontal shift of 1 unit to the right and a vertical shift of 4 units up 
D)  By reflecting across the y-axis and making a horizontal shift of 4 units to the left

Give the equations of any vertical asymptotes. - $h ( x ) = \frac { ( x - 4 ) ( x + 1 ) } { x ^ { 2 } - 9 }$

Use a graphing calculator to approximate the real zeros. Give each zero as a decimal to the nearest tenth. - $f ( x ) = x ^ { 4 } - 7 x ^ { 2 } + 12$

Sketch the graph of the parabola. - $y = \frac { 2 } { 5 } ( x + 1 ) ^ { 2 } - 2$ $Sketch the graph of the parabola. - y = \frac { 2 } { 5 } ( x + 1 ) ^ { 2 } - 2$

Sketch the graph of the rational function. - $f ( x ) = \frac { x ^ { 2 } - x - 12 } { x + 4 }$ $Sketch the graph of the rational function. - f ( x ) = \frac { x ^ { 2 } - x - 12 } { x + 4 }$

Use a graphing calculator to find the coordinates of the turning points of the graph of the polynomial function in the indicated domain interval. Give answers to the nearest hundredth. - $f ( x ) = x ^ { 4 } - 29 x ^ { 2 } + 100 ; [ - 1,0.5 ]$

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. - $f ( x ) = - 6 x ^ { 4 } - 8 x ^ { 3 } - 7 x ^ { 2 } - 8 x + 3$

Match the equation to the correct graph. - $y = - ( x + 6 ) ^ { 2 } + 4$

Solve. -The data in the table show the number of burglaries reported in a small town for several years on record. Year \# of Burglaries 0 4 2 3 4 2 6 2 7 2 8 3 11 4 13 6 15 8 Determine the linear, quadratic, cubic, or quartic function that best fits the data.

Translate the given formula to an English phrase using the word "varies". -If y varies directly as the square of x, and x is doubled, how is y changed?

Find the zeros of the polynomial function and state the multiplicity of each. - $f ( x ) = \left( x ^ { 2 } + x - 12 \right) ^ { 7 } ( x - 4 + \sqrt { 7 } ) ^ { 2 }$

Find the equation that the given graph represents. -

Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary. - $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 5 x + 39$

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions. -Zeros of 3 and $i , P ( 1 ) = - 4$

Solve the problem -The polynomial $G ( x ) = - 0.006 x ^ { 4 } + 0.140 x ^ { 3 } - 0.53 x ^ { 2 } + 1.79 x$ measures the concentration of a dye in the bloodstream $x$ seconds after it is injected. Does the concentration increase between 12 and 13 seconds?

Factor f(x) into linear factors given that k is a zero of f(x). - $f ( x ) = 4 x ^ { 3 } + 24 x ^ { 2 } + 17 x - 15 ; k = \frac { 1 } { 2 }$

Use the factor theorem to decide whether or not the second polynomial is a factor of the first. - $3 x ^ { 2 } - 22 x + 40 ; x - 4$

Match the equation to the correct graph. - $y = - ( x + 5 ) ^ { 2 } + 5$

Give the equation of any oblique asymptotes. - $f ( x ) = \frac { x ^ { 2 } - 8 x + 4 } { x + 2 }$

Answer the question - $\text { How can the graph of } f ( x ) = - \frac { 1 } { x ^ { 2 } } - 4 \text { be obtained from the graph of } y = \frac { 1 } { x ^ { 2 } }$

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 vertical asymptotes. - h(x)=(x−4)(x+1)x2−9h ( x ) = \frac { ( x - 4 ) ( x + 1 ) } { x ^ { 2 } - 9 }h(x)=x2−9(x−4)(x+1)​

Use a graphing calculator to approximate the real zeros. Give each zero as a decimal to the nearest tenth. - f(x)=x4−7x2+12f ( x ) = x ^ { 4 } - 7 x ^ { 2 } + 12f(x)=x4−7x2+12

Sketch the graph of the parabola. - y=25(x+1)2−2y = \frac { 2 } { 5 } ( x + 1 ) ^ { 2 } - 2y=52​(x+1)2−2

Sketch the graph of the rational function. - f(x)=x2−x−12x+4f ( x ) = \frac { x ^ { 2 } - x - 12 } { x + 4 }f(x)=x+4x2−x−12​

Use a graphing calculator to find the coordinates of the turning points of the graph of the polynomial function in the indicated domain interval. Give answers to the nearest hundredth. - f(x)=x4−29x2+100;[−1,0.5]f ( x ) = x ^ { 4 } - 29 x ^ { 2 } + 100 ; [ - 1,0.5 ]f(x)=x4−29x2+100;[−1,0.5]

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)=−6x4−8x3−7x2−8x+3f ( x ) = - 6 x ^ { 4 } - 8 x ^ { 3 } - 7 x ^ { 2 } - 8 x + 3f(x)=−6x4−8x3−7x2−8x+3

Match the equation to the correct graph. - y=−(x+6)2+4y = - ( x + 6 ) ^ { 2 } + 4y=−(x+6)2+4

Solve. -The data in the table show the number of burglaries reported in a small town for several years on record. Year \# of Burglaries 0 4 2 3 4 2 6 2 7 2 8 3 11 4 13 6 15 8 Determine the linear, quadratic, cubic, or quartic function that best fits the data.

Translate the given formula to an English phrase using the word "varies". -If y varies directly as the square of x, and x is doubled, how is y changed?

Find the zeros of the polynomial function and state the multiplicity of each. - f(x)=(x2+x−12)7(x−4+7)2f ( x ) = \left( x ^ { 2 } + x - 12 \right) ^ { 7 } ( x - 4 + \sqrt { 7 } ) ^ { 2 }f(x)=(x2+x−12)7(x−4+7​)2

Find the equation that the given graph represents. -

Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary. - f(x)=x3−3x2−5x+39f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 5 x + 39f(x)=x3−3x2−5x+39

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions. -Zeros of 3 and i,P(1)=−4i , P ( 1 ) = - 4i,P(1)=−4

Factor f(x) into linear factors given that k is a zero of f(x). - f(x)=4x3+24x2+17x−15;k=12f ( x ) = 4 x ^ { 3 } + 24 x ^ { 2 } + 17 x - 15 ; k = \frac { 1 } { 2 }f(x)=4x3+24x2+17x−15;k=21​

Use the factor theorem to decide whether or not the second polynomial is a factor of the first. - 3x2−22x+40;x−43 x ^ { 2 } - 22 x + 40 ; x - 43x2−22x+40;x−4

Match the equation to the correct graph. - y=−(x+5)2+5y = - ( x + 5 ) ^ { 2 } + 5y=−(x+5)2+5

Give the equation of any oblique asymptotes. - f(x)=x2−8x+4x+2f ( x ) = \frac { x ^ { 2 } - 8 x + 4 } { x + 2 }f(x)=x+2x2−8x+4​

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

Give the equations of any vertical asymptotes. - $h ( x ) = \frac { ( x - 4 ) ( x + 1 ) } { x ^ { 2 } - 9 }$

Use a graphing calculator to approximate the real zeros. Give each zero as a decimal to the nearest tenth. - $f ( x ) = x ^ { 4 } - 7 x ^ { 2 } + 12$

Sketch the graph of the parabola. - $y = \frac { 2 } { 5 } ( x + 1 ) ^ { 2 } - 2$ $Sketch the graph of the parabola. - y = \frac { 2 } { 5 } ( x + 1 ) ^ { 2 } - 2$

Sketch the graph of the rational function. - $f ( x ) = \frac { x ^ { 2 } - x - 12 } { x + 4 }$ $Sketch the graph of the rational function. - f ( x ) = \frac { x ^ { 2 } - x - 12 } { x + 4 }$

Use a graphing calculator to find the coordinates of the turning points of the graph of the polynomial function in the indicated domain interval. Give answers to the nearest hundredth. - $f ( x ) = x ^ { 4 } - 29 x ^ { 2 } + 100 ; [ - 1,0.5 ]$

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. - $f ( x ) = - 6 x ^ { 4 } - 8 x ^ { 3 } - 7 x ^ { 2 } - 8 x + 3$

Match the equation to the correct graph. - $y = - ( x + 6 ) ^ { 2 } + 4$

Find the zeros of the polynomial function and state the multiplicity of each. - $f ( x ) = \left( x ^ { 2 } + x - 12 \right) ^ { 7 } ( x - 4 + \sqrt { 7 } ) ^ { 2 }$

Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary. - $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 5 x + 39$

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions. -Zeros of 3 and $i , P ( 1 ) = - 4$

Factor f(x) into linear factors given that k is a zero of f(x). - $f ( x ) = 4 x ^ { 3 } + 24 x ^ { 2 } + 17 x - 15 ; k = \frac { 1 } { 2 }$

Use the factor theorem to decide whether or not the second polynomial is a factor of the first. - $3 x ^ { 2 } - 22 x + 40 ; x - 4$

Match the equation to the correct graph. - $y = - ( x + 5 ) ^ { 2 } + 5$

Give the equation of any oblique asymptotes. - $f ( x ) = \frac { x ^ { 2 } - 8 x + 4 } { x + 2 }$