Question 1

Write a linear factorization of the function.
-$$f ( x ) = x ^ { 4 } + 6 x ^ { 3 } + 17 x ^ { 2 } + 54 x + 72$$

Accepted Answer

A)  $f ( x ) = ( x + 4 ) ( x + 2 ) \left( x ^ { 2 } + 9 \right)$ 
B)  $f ( x ) = ( x + 4 ) ( x + 2 ) ( x + 3 i ) ( x - 3 i )$ 
C)  $f ( x ) = ( x + 4 ) ( x + 3 ) ( x + 2 i ) ( x - 2 i )$ 
D)  $f ( x ) = ( x + 3 ) ( x + 2 ) ( x + 4 i ) ( x - 4 i )$ 
A)  $f ( x ) = ( x + 4 ) ( x + 2 ) \left( x ^ { 2 } + 9 \right)$ 
B)  $f ( x ) = ( x + 4 ) ( x + 2 ) ( x + 3 i ) ( x - 3 i )$ 
C)  $f ( x ) = ( x + 4 ) ( x + 3 ) ( x + 2 i ) ( x - 2 i )$ 
D)  $f ( x ) = ( x + 3 ) ( x + 2 ) ( x + 4 i ) ( x - 4 i )$

Question 2

Describe how the graph of the given function can be obtained by transforming the graph of the reciprocal function f(x) =
1/x.
-$$f ( x ) = \frac { 1 } { x - 5 }$$

Accepted Answer

A)  Shift the graph of the reciprocal function right 5 units. 
B)  Shift the graph of the reciprocal function down 5 units. 
C)  Shift the graph of the reciprocal function left 5 units. 
D)  Shift the graph of the reciprocal function up 5 units. 
A)  Shift the graph of the reciprocal function right 5 units. 
B)  Shift the graph of the reciprocal function down 5 units. 
C)  Shift the graph of the reciprocal function left 5 units. 
D)  Shift the graph of the reciprocal function up 5 units.

Question 3

Evaluate the limit based on the graph of f shown.
- 
$\lim _ { x \rightarrow \infty } f ( x )$

Accepted Answer

A)  $\infty$ 
B)  $-\infty$ 
C)  0 
D)  3 
A)  $\infty$ 
B)  $-\infty$ 
C)  0 
D)  3

Question 4

Write the word or phrase that best completes each statement or answers the question.
Graph the rational function and analyze it in the following way: find the intercepts, asymptotes, use limits to describe the
behavior at the vertical asymptotes and the end behavior. Find the domain and range. Determine where the function is
continuous and where it is increasing and decreasing. Find any local extrema.
-$$f ( x ) = \frac { x ^ { 3 } - x ^ { 2 } + x - 1 } { x + 3 }$$

Accepted Answer

\[\begin{array} { l } 
\mathrm { x } \te

Question 5

State the power and constant of variation for the function, and then analyze it.
-$$f ( x ) = 4 x ^ { 3 }$$

Accepted Answer

Power is 3; constant of variation is 4;

Question 6

Use limits to describe the behavior of the rational function near the indicated asymptote.
-$$f ( x ) = - \frac { 8 } { x + 2 }$$
Describe the behavior of the function near its vertical asymptote.

Accepted Answer

A)  $\lim _ { x \rightarrow 2 ^ { - } } f ( x ) = \infty , \lim _ { x \rightarrow 2 ^ { + } } f ( x ) = \infty$ 
B)  $\lim _ { x \rightarrow 2 ^ { - } } f ( x ) = \infty , \lim _ { x \rightarrow 2 ^ { + } } f ( x ) = - \infty$ 
C)  $\lim _ { x \rightarrow 2 ^ { - } } f ( x ) = \infty , \lim _ { x \rightarrow 2 ^ { + } } f ( x ) = - \infty$. 
D)  $\lim _ { x \rightarrow 2 ^ { - } } f ( x ) = 0 , \quad \lim _ { x \rightarrow 2 ^ { + } } f ( x ) = 0$ 
A)  $\lim _ { x \rightarrow 2 ^ { - } } f ( x ) = \infty , \lim _ { x \rightarrow 2 ^ { + } } f ( x ) = \infty$ 
B)  $\lim _ { x \rightarrow 2 ^ { - } } f ( x ) = \infty , \lim _ { x \rightarrow 2 ^ { + } } f ( x ) = - \infty$ 
C)  $\lim _ { x \rightarrow 2 ^ { - } } f ( x ) = \infty , \lim _ { x \rightarrow 2 ^ { + } } f ( x ) = - \infty$. 
D)  $\lim _ { x \rightarrow 2 ^ { - } } f ( x ) = 0 , \quad \lim _ { x \rightarrow 2 ^ { + } } f ( x ) = 0$

Question 7

Match the equation to one of the curves (for x ≥ 0).
-$$f ( x ) = - \frac { 3 } { 7 } x ^ { 4 }$$

Accepted Answer

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

Question 8

Match the graph of the rational function with its equation.
-

Accepted Answer

A)  $f ( x ) = \frac { - 1 } { x ^ { 2 } + x - 30 }$ 
B)  $f ( x ) = \frac { x - 1 } { x ^ { 2 } + x - 30 }$ 
C)  $f ( x ) = \frac { 1 } { x ^ { 2 } + x - 30 }$ 
D)  $f ( x ) = \frac { x + 1 } { x ^ { 2 } + x - 30 }$ 
A)  $f ( x ) = \frac { - 1 } { x ^ { 2 } + x - 30 }$ 
B)  $f ( x ) = \frac { x - 1 } { x ^ { 2 } + x - 30 }$ 
C)  $f ( x ) = \frac { 1 } { x ^ { 2 } + x - 30 }$ 
D)  $f ( x ) = \frac { x + 1 } { x ^ { 2 } + x - 30 }$

Question 9

Find the requested function.
-Find the polynomial function with leading coefficient 7 ; degree 4 ; and $0 , - 2,3$, and $\frac { 7 } { 6 }$ as zeros.

Accepted Answer

A)  $f ( x ) = 7 x ( x + 2 ) ( x - 3 ) ( 6 x - 7 )$ 
B)  $f ( x ) = 7 x ( x + 2 ) ( x - 3 ) ( x - 7 )$ 
C)  $f ( x ) = \frac { 7 } { 6 } x ( x + 2 ) ( x + 3 ) ( 6 x - 7 )$ 
D)  $f ( x ) = \frac { 7 } { 6 } x ( x + 2 ) ( x - 3 ) ( 6 x - 7 )$ 
A)  $f ( x ) = 7 x ( x + 2 ) ( x - 3 ) ( 6 x - 7 )$ 
B)  $f ( x ) = 7 x ( x + 2 ) ( x - 3 ) ( x - 7 )$ 
C)  $f ( x ) = \frac { 7 } { 6 } x ( x + 2 ) ( x + 3 ) ( 6 x - 7 )$ 
D)  $f ( x ) = \frac { 7 } { 6 } x ( x + 2 ) ( x - 3 ) ( 6 x - 7 )$

Question 10

List the x- and y-intercepts, and graph the function.
-$$f ( x ) = \frac { x + 3 } { 2 x ^ { 2 } - 5 x - 3 }$$

Accepted Answer

A)  No x-intercepts, $y$-intercept: $( 0 , - 1 )$;
  
B)  $x$-intercept: $( - 3,0 ) , y$-intercept: $( 0 , - 1 )$;
  
C)  x-intercept: $( - 3,0 ) , y$-intercept: $( 0,1 )$;
  
D)  No $x$-intercepts , $y$-intercept: $( 0 , - 1 )$;
  
A)  No x-intercepts, $y$-intercept: $( 0 , - 1 )$;
  
B)  $x$-intercept: $( - 3,0 ) , y$-intercept: $( 0 , - 1 )$;
  
C)  x-intercept: $( - 3,0 ) , y$-intercept: $( 0,1 )$;
  
D)  No $x$-intercepts , $y$-intercept: $( 0 , - 1 )$;

Question 11

State how many complex and real zeros the function has.
-$$f ( x ) = x ^ { 3 } + 5 x ^ { 2 } + x + 5$$

Accepted Answer

A)  3 complex zeros; 1 real 
B)  3 complex zeros; all 3 real 
C)  3 complex zeros; 2 real 
D)  3 complex zeros; none real 
A)  3 complex zeros; 1 real 
B)  3 complex zeros; all 3 real 
C)  3 complex zeros; 2 real 
D)  3 complex zeros; none real

Question 12

Find the zeros of the polynomial function and state the multiplicity of each.
-$$f ( x ) = 2 x ( x - 7 ) ( x + 9 )$$

Accepted Answer

A)  $- 9$, multiplicity 1; 7 , multiplicity 1 
B)  $- 9$, multiplicity $1 ; 7$, multiplicity 2 
C)  $- 9$, multiplicity $1 ; 0$, multiplicity $1 ; 7$, multiplicity 1 
D)  $- 9$, multiplicity $1 ; 9$, multiplicity $1 ; 7$, multiplicity 1 
A)  $- 9$, multiplicity 1; 7 , multiplicity 1 
B)  $- 9$, multiplicity $1 ; 7$, multiplicity 2 
C)  $- 9$, multiplicity $1 ; 0$, multiplicity $1 ; 7$, multiplicity 1 
D)  $- 9$, multiplicity $1 ; 9$, multiplicity $1 ; 7$, multiplicity 1

Question 13

Write the statement as a power function equation. Use k as the constant of variation.
-The surface area of a sphere $\mathrm { S }$ varies directly as the square of its radius $\mathrm { r }$.

Accepted Answer

A)  $S = \frac { k } { r ^ { 2 } }$ 
B)  $S = \frac { r ^ { 2 } } { k }$ 
C)  $S = k ^ { 2 } r$ 
D)  $\mathrm { S } = \mathrm { kr } ^ { 2 }$ 
A)  $S = \frac { k } { r ^ { 2 } }$ 
B)  $S = \frac { r ^ { 2 } } { k }$ 
C)  $S = k ^ { 2 } r$ 
D)  $\mathrm { S } = \mathrm { kr } ^ { 2 }$

Question 14

Use a graphing calculator to approximate the real zeros of the function defined by f(x). Express decimal approximations to
the nearest hundredth.
-$f ( x ) = - \sqrt { 11 } x ^ { 3 } + \sqrt { 10 } x + \sqrt { 7 }$

Accepted Answer

A)  $1.26$ 
B)  $1.47 , - 1.47$ 
C)  $1.44$ 
D)  No real zeros 
A)  $1.26$ 
B)  $1.47 , - 1.47$ 
C)  $1.44$ 
D)  No real zeros

Question 15

Describe how the graph of the given function can be obtained by transforming the graph of the reciprocal function f(x) =
1/x.
-$$f ( x ) = \frac { 3 x - 4 } { x + 4 }$$

Accepted Answer

A)  Shift the graph of the reciprocal function left 4 units, reflect across the $x$-axis, stretch vertically by a factor of 16 , and then shift 3 units down. 
B)  Shift the graph of the reciprocal function left 4 units, reflect across the x-axis, and then shift 3 units up. 
C)  Shift the graph of the reciprocal function left 4 units, reflect across the $x$-axis, stretch vertically by a factor of 16 , and then shift 3 units up. 
D)  Shift the graph of the reciprocal function left 4 units, reflect across the y-axis, stretch vertically by a factor of 16 , and then shift 3 units up. 
A)  Shift the graph of the reciprocal function left 4 units, reflect across the $x$-axis, stretch vertically by a factor of 16 , and then shift 3 units down. 
B)  Shift the graph of the reciprocal function left 4 units, reflect across the x-axis, and then shift 3 units up. 
C)  Shift the graph of the reciprocal function left 4 units, reflect across the $x$-axis, stretch vertically by a factor of 16 , and then shift 3 units up. 
D)  Shift the graph of the reciprocal function left 4 units, reflect across the y-axis, stretch vertically by a factor of 16 , and then shift 3 units up.

Question 16

Choose the one alternative that best completes the statement or answers the question.
Solve the equation.
-$$\frac { 5 } { x + 7 } + \frac { 11 } { x + 8 } = \frac { 9 } { x ^ { 2 } + 15 y + 56 }$$

Accepted Answer

A)  $x = - \frac { 27 } { 4 }$ 
B)  $x = - \frac { 38 } { 5 }$ 
C)  No solution 
D)  $x = - \frac { 73 } { 11 }$ 
A)  $x = - \frac { 27 } { 4 }$ 
B)  $x = - \frac { 38 } { 5 }$ 
C)  No solution 
D)  $x = - \frac { 73 } { 11 }$

Question 17

List the x- and y-intercepts, and graph the function.
-$$f ( x ) = \frac { x - 2 } { x + 3 }$$

Accepted Answer

A)  $x$-intercept: $( 2,0 ) ; y$-intercept: $\left( 0 , \frac { 2 } { 3 } \right)$;
  
B)  $x$-intercept: $( 2,0 ) ; y$-intercept: $\left( 0 , - \frac { 2 } { 3 } \right)$
  
C)  $x$-intercept: $( 2,0 ) ; y$-intercept: $\left( 0 , \frac { 2 } { 3 } \right)$;
  
D)  x-intercept: $( 2,0 ) ; y$-intercept: $\left( 0 , - \frac { 2 } { 3 } \right)$;
  
A)  $x$-intercept: $( 2,0 ) ; y$-intercept: $\left( 0 , \frac { 2 } { 3 } \right)$;
  
B)  $x$-intercept: $( 2,0 ) ; y$-intercept: $\left( 0 , - \frac { 2 } { 3 } \right)$
  
C)  $x$-intercept: $( 2,0 ) ; y$-intercept: $\left( 0 , \frac { 2 } { 3 } \right)$;
  
D)  x-intercept: $( 2,0 ) ; y$-intercept: $\left( 0 , - \frac { 2 } { 3 } \right)$;

Question 18

For the given function, find all asymptotes of the type indicated (if there are any)
-$f ( x ) = \frac { 4 x ^ { 2 } + 2 } { 4 x ^ { 2 } - 2 }$, horizontal

Accepted Answer

A)  $y = - 2$ 
B)  $y = 2$ 
C)  $y = 1$ 
D)  None 
A)  $y = - 2$ 
B)  $y = 2$ 
C)  $y = 1$ 
D)  None

Question 19

If the following is a polynomial function, then state its degree and leading coefficient. If it is not, then state this fact.
-$$f ( x ) = 5 x ^ { - 7 } + 17$$

Accepted Answer

A)  Degree: -7; leading coefficient: 5 
B)  Degree: 5 ; leading coefficient: $- 7$ 
C)  Degree: 7; leading coefficient: 5 
D)  Not a polynomial function 
A)  Degree: -7; leading coefficient: 5 
B)  Degree: 5 ; leading coefficient: $- 7$ 
C)  Degree: 7; leading coefficient: 5 
D)  Not a polynomial function

Question 20

Describe the end behavior of the polynomial function by finding  
-$$f ( x ) = 7 x ^ { 4 } - 7 x ^ { 2 } + 2$$

Accepted Answer

A)  $-\infty , -\infty$ 
B)  $-\infty , \infty$ 
C)  $\infty , -\infty$ 
D)  $\infty , \infty$ 
A)  $-\infty , -\infty$ 
B)  $-\infty , \infty$ 
C)  $\infty , -\infty$ 
D)  $\infty , \infty$

Write a linear factorization of the function. - $f ( x ) = x ^ { 4 } + 6 x ^ { 3 } + 17 x ^ { 2 } + 54 x + 72$

Describe how the graph of the given function can be obtained by transforming the graph of the reciprocal function f(x) = 1/x. - $f ( x ) = \frac { 1 } { x - 5 }$

Evaluate the limit based on the graph of f shown. - $Evaluate the limit based on the graph of f shown. - \lim _ { x \rightarrow \infty } f ( x )$ $\lim _ { x \rightarrow \infty } f ( x )$

State the power and constant of variation for the function, and then analyze it. - $f ( x ) = 4 x ^ { 3 }$

Use limits to describe the behavior of the rational function near the indicated asymptote. - $f ( x ) = - \frac { 8 } { x + 2 }$ Describe the behavior of the function near its vertical asymptote.

Match the equation to one of the curves (for x ≥ 0). - $f ( x ) = - \frac { 3 } { 7 } x ^ { 4 }$ $Match the equation to one of the curves (for x ≥ 0). - f ( x ) = - \frac { 3 } { 7 } x ^ { 4 }$

Match the graph of the rational function with its equation. -

Find the requested function. -Find the polynomial function with leading coefficient 7 ; degree 4 ; and $0 , - 2,3$ , and $\frac { 7 } { 6 }$ as zeros.

List the x- and y-intercepts, and graph the function. - $f ( x ) = \frac { x + 3 } { 2 x ^ { 2 } - 5 x - 3 }$ $List the x- and y-intercepts, and graph the function. - f ( x ) = \frac { x + 3 } { 2 x ^ { 2 } - 5 x - 3 }$

State how many complex and real zeros the function has. - $f ( x ) = x ^ { 3 } + 5 x ^ { 2 } + x + 5$

Find the zeros of the polynomial function and state the multiplicity of each. - $f ( x ) = 2 x ( x - 7 ) ( x + 9 )$

Write the statement as a power function equation. Use k as the constant of variation. -The surface area of a sphere $\mathrm { S }$ varies directly as the square of its radius $\mathrm { r }$ .

Use a graphing calculator to approximate the real zeros of the function defined by f(x). Express decimal approximations to the nearest hundredth. - $f ( x ) = - \sqrt { 11 } x ^ { 3 } + \sqrt { 10 } x + \sqrt { 7 }$

Describe how the graph of the given function can be obtained by transforming the graph of the reciprocal function f(x) = 1/x. - $f ( x ) = \frac { 3 x - 4 } { x + 4 }$

Choose the one alternative that best completes the statement or answers the question. Solve the equation. - $\frac { 5 } { x + 7 } + \frac { 11 } { x + 8 } = \frac { 9 } { x ^ { 2 } + 15 y + 56 }$

List the x- and y-intercepts, and graph the function. - $f ( x ) = \frac { x - 2 } { x + 3 }$ $List the x- and y-intercepts, and graph the function. - f ( x ) = \frac { x - 2 } { x + 3 }$

For the given function, find all asymptotes of the type indicated (if there are any) - $f ( x ) = \frac { 4 x ^ { 2 } + 2 } { 4 x ^ { 2 } - 2 }$ , horizontal

If the following is a polynomial function, then state its degree and leading coefficient. If it is not, then state this fact. - $f ( x ) = 5 x ^ { - 7 } + 17$

Describe the end behavior of the polynomial function by finding $Describe the end behavior of the polynomial function by finding - f ( x ) = 7 x ^ { 4 } - 7 x ^ { 2 } + 2$ - $f ( x ) = 7 x ^ { 4 } - 7 x ^ { 2 } + 2$

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An Introduction to Calculus: Limits, Derivatives, and Integrals

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Exam 2: Polynomial, Power, and Rational Functions

Write a linear factorization of the function. - f(x)=x4+6x3+17x2+54x+72f ( x ) = x ^ { 4 } + 6 x ^ { 3 } + 17 x ^ { 2 } + 54 x + 72f(x)=x4+6x3+17x2+54x+72

Describe how the graph of the given function can be obtained by transforming the graph of the reciprocal function f(x) = 1/x. - f(x)=1x−5f ( x ) = \frac { 1 } { x - 5 }f(x)=x−51​

Evaluate the limit based on the graph of f shown. - lim⁡x→∞f(x)\lim _ { x \rightarrow \infty } f ( x )limx→∞​f(x)

State the power and constant of variation for the function, and then analyze it. - f(x)=4x3f ( x ) = 4 x ^ { 3 }f(x)=4x3

Use limits to describe the behavior of the rational function near the indicated asymptote. - f(x)=−8x+2f ( x ) = - \frac { 8 } { x + 2 }f(x)=−x+28​ Describe the behavior of the function near its vertical asymptote.

Match the equation to one of the curves (for x ≥ 0). - f(x)=−37x4f ( x ) = - \frac { 3 } { 7 } x ^ { 4 }f(x)=−73​x4