Question 1

Use the graph to find the oblique asymptote, if any, of the function.
-

Accepted Answer

A)  $y = - x$ 
B)  $y = 2 x$ 
C)  $y = x$ 
D)  no oblique asymptote 
A)  $y = - x$ 
B)  $y = 2 x$ 
C)  $y = x$ 
D)  no oblique asymptote C

Question 2

Solve the inequality.
-$$( b - 2 ) ( b - 3 ) ( b - 4 ) < 0$$

Accepted Answer

A)  $( - \infty , 3 )$ 
B)  $( - \infty , 2 )$ or $( 3,4 )$ 
C)  $( 4 , \infty )$ 
D)  $( 2,3 )$ or $( 4 , \infty )$ 
A)  $( - \infty , 3 )$ 
B)  $( - \infty , 2 )$ or $( 3,4 )$ 
C)  $( 4 , \infty )$ 
D)  $( 2,3 )$ or $( 4 , \infty )$ B

Question 3

Information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f.
-Degree 5; zeros: $8,2 + 5 \mathrm { i } , - 8 \mathrm { i }$

Accepted Answer

A)  $- 8,2 - 5 i , 8 i$ 
B)  $- 2 - 5 \mathrm { i } , 8 \mathrm { i }$ 
C)  $- 2 + 5 i , 8 i$ 
D)  $2 - 5 i , 8 i$ 
A)  $- 8,2 - 5 i , 8 i$ 
B)  $- 2 - 5 \mathrm { i } , 8 \mathrm { i }$ 
C)  $- 2 + 5 i , 8 i$ 
D)  $2 - 5 i , 8 i$ D

Question 4

Find the x- and y-intercepts of f.
-$$f ( x ) = ( x - 2 ) ^ { 2 } \left( x ^ { 2 } - 25 \right)$$

Accepted Answer

A)  $x$-intercepts: $- 5,2,5$; y-intercept: 100 
B)  $x$-intercepts: 2, 25; y-intercept: 50 
C)  x-intercepts: $- 2 , - 25$; y-intercept: 50 
D)  $x$-intercepts: $- 5,2,5$; y-intercept: $- 100$ 
A)  $x$-intercepts: $- 5,2,5$; y-intercept: 100 
B)  $x$-intercepts: 2, 25; y-intercept: 50 
C)  x-intercepts: $- 2 , - 25$; y-intercept: 50 
D)  $x$-intercepts: $- 5,2,5$; y-intercept: $- 100$

Question 5

Solve the inequality.
-$$x ^ { 2 } + 7 x \geq 0$$

Accepted Answer

A)  $[ - 7,0 ]$ 
B)  $( - \infty , 0 ]$ or $[ 7 , \infty )$ 
C)  $[ 0,7 ]$ 
D)  $( - \infty , - 7 ]$ or $[ 0 , \infty )$ 
A)  $[ - 7,0 ]$ 
B)  $( - \infty , 0 ]$ or $[ 7 , \infty )$ 
C)  $[ 0,7 ]$ 
D)  $( - \infty , - 7 ]$ or $[ 0 , \infty )$

Question 6

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the
zeros to factor f over the real numbers.
-$$f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 9 x - 18$$

Accepted Answer

A)  $- 3 , - 2,3 ; f ( x ) = ( x + 3 ) ( x + 2 ) ( x - 3 )$ 
B)  $- 3,2,3 ; f ( x ) = ( x + 3 ) ( x - 2 ) ( x - 3 )$ 
C)  $- 2 ; f ( x ) = ( x + 2 ) \left( x ^ { 2 } + x - 9 \right)$ 
D)  $- 3 ; f ( x ) = ( x + 3 ) \left( x ^ { 2 } - x - 6 \right)$ 
A)  $- 3 , - 2,3 ; f ( x ) = ( x + 3 ) ( x + 2 ) ( x - 3 )$ 
B)  $- 3,2,3 ; f ( x ) = ( x + 3 ) ( x - 2 ) ( x - 3 )$ 
C)  $- 2 ; f ( x ) = ( x + 2 ) \left( x ^ { 2 } + x - 9 \right)$ 
D)  $- 3 ; f ( x ) = ( x + 3 ) \left( x ^ { 2 } - x - 6 \right)$

Question 7

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at
each x -intercept.
-$$f ( x ) = 3 ( x - 6 ) ( x + 7 ) ^ { 3 }$$

Accepted Answer

A)  6 , multiplicity 1 , crosses $x$-axis; $- 7$, multiplicity 3 , crosses $x$-axis 
B)  $- 6$, multiplicity 1 , touches $x$-axis; 7 , multiplicity 3 
C)  $- 6$, multiplicity 1 , crosses $x$-axis; 7 , multiplicity 3 , crosses $x$-axis 
D)  6 , multiplicity 1 , touches $x$-axis; $- 7$, multiplicity 3 
A)  6 , multiplicity 1 , crosses $x$-axis; $- 7$, multiplicity 3 , crosses $x$-axis 
B)  $- 6$, multiplicity 1 , touches $x$-axis; 7 , multiplicity 3 
C)  $- 6$, multiplicity 1 , crosses $x$-axis; 7 , multiplicity 3 , crosses $x$-axis 
D)  6 , multiplicity 1 , touches $x$-axis; $- 7$, multiplicity 3

Question 8

Solve the inequality.
-$$x ^ { 2 } + 6 x \leq 0$$

Accepted Answer

A)  $( - \infty , 0 ]$ or $[ 6 , \infty )$ 
B)  $[ - 6,0 ]$ 
C)  $( - \infty , - 6 ]$ or $[ 0 , \infty )$ 
D)  $[ 0,6 ]$ 
A)  $( - \infty , 0 ]$ or $[ 6 , \infty )$ 
B)  $[ - 6,0 ]$ 
C)  $( - \infty , - 6 ]$ or $[ 0 , \infty )$ 
D)  $[ 0,6 ]$

Question 9

Find the indicated intercept(s) of the graph of the function.
-$x$-intercepts of $f ( x ) = \frac { ( x - 8 ) ( 2 x + 7 ) } { x ^ { 2 } + 3 x - 4 }$

Accepted Answer

A)  $( - 8,0 ) , \left( \frac { 7 } { 2 } , 0 \right)$ 
B)  $( 8,0 ) , ( - 7,0 )$ 
C)  $( 8,0 ) , \left( - \frac { 7 } { 2 } , 0 \right)$ 
D)  none 
A)  $( - 8,0 ) , \left( \frac { 7 } { 2 } , 0 \right)$ 
B)  $( 8,0 ) , ( - 7,0 )$ 
C)  $( 8,0 ) , \left( - \frac { 7 } { 2 } , 0 \right)$ 
D)  none

Question 10

Find the vertical asymptotes of the rational function.
-$$f ( x ) = \frac { - 2 x ( x + 2 ) } { 3 x ^ { 2 } - 5 x - 8 }$$

Accepted Answer

A)  $x = \frac { 8 } { 3 } , x = - 1$ 
B)  $x = - \frac { 8 } { 3 } , x = 1$ 
C)  $x = - \frac { 3 } { 8 } , x = 1$ 
D)  $x = \frac { 3 } { 8 } , x = - 1$ 
A)  $x = \frac { 8 } { 3 } , x = - 1$ 
B)  $x = - \frac { 8 } { 3 } , x = 1$ 
C)  $x = - \frac { 3 } { 8 } , x = 1$ 
D)  $x = \frac { 3 } { 8 } , x = - 1$

Question 11

Give the equation of the oblique asymptote, if any, of the function.
-$$h ( x ) = \frac { 3 x ^ { 2 } - 8 x - 4 } { 9 x ^ { 2 } - 4 x + 6 }$$

Accepted Answer

A)  $y = x + \frac { 1 } { 3 }$ 
B)  $y = \frac { 1 } { 3 }$ 
C)  $y = \frac { 1 } { 3 } x$ 
D)  no oblique asymptote 
A)  $y = x + \frac { 1 } { 3 }$ 
B)  $y = \frac { 1 } { 3 }$ 
C)  $y = \frac { 1 } { 3 } x$ 
D)  no oblique asymptote

Question 12

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not.
-$$f ( x ) = \frac { x ^ { 4 } - 7 } { x ^ { 3 } }$$

Accepted Answer

A)  Yes; degree -3 
B)  Yes; degree 3 
C)  No; it is a ratio of polynomials 
D)  Yes; degree 4 
A)  Yes; degree -3 
B)  Yes; degree 3 
C)  No; it is a ratio of polynomials 
D)  Yes; degree 4

Question 13

Find the indicated intercept(s) of the graph of the function.
-$y$-intercept of $f ( x ) = \frac { x ^ { 2 } - 10 x - 4 } { x ^ { 2 } + 13 x + 4 }$

Accepted Answer

A)  $( 0,13 )$ 
B)  $( 0 , - 1 )$ 
C)  $( 0 , - 4 )$ 
D)  none 
A)  $( 0,13 )$ 
B)  $( 0 , - 1 )$ 
C)  $( 0 , - 4 )$ 
D)  none

Question 14

Graph the function.
-$$f ( x ) = \frac { x ^ { 2 } - 4 x } { ( x - 2 ) ^ { 2 } }$$

Accepted Answer

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

Question 15

Choose the one alternative that best completes the statement or answers the question.
Find a bound on the real zeros of the polynomial function.
-$$f ( x ) = x ^ { 4 } + 2 x ^ { 2 } - 3$$

Accepted Answer

A)  $- 3$ and 3 
B)  $- 6$ and 6 
C)  $- 4$ and 4 
D)  $- 5$ and 5 
A)  $- 3$ and 3 
B)  $- 6$ and 6 
C)  $- 4$ and 4 
D)  $- 5$ and 5

Question 16

Give the equation of the horizontal asymptote, if any, of the function.
-$$h ( x ) = \frac { 2 x ^ { 3 } - 8 x - 9 } { 7 x + 8 }$$

Accepted Answer

A)  $y = \frac { 2 } { 7 }$ 
B)  $y = 2$ 
C)  $y = 0$ 
D)  no horizontal asymptotes 
A)  $y = \frac { 2 } { 7 }$ 
B)  $y = 2$ 
C)  $y = 0$ 
D)  no horizontal asymptotes

Question 17

Solve the problem.
-Determine which rational function $\mathrm { R } ( \mathrm { x } )$ has a graph that crosses the $x$-axis at $- 1$, touches the $x$-axis at $- 4$, has vertical asymptotes at $x = - 2$ and $x = 3$, and has one horizontal asymptote at $y = - 2$.

Accepted Answer

A)  $R ( x ) = \frac { - 2 ( x + 1 ) ( x + 4 ) ^ { 2 } } { ( x + 2 ) ^ { 2 } ( x - 3 ) } , x \neq z - 2,3$ 
B)  $R ( x ) = \frac { - 2 ( x + 1 ) ( x + 4 ) } { ( x + 2 ) ( x - 3 ) } , x \neq - 2,3$ 
C)  $R ( x ) = \frac { - ( x + 1 ) ( x + 4 ) ^ { 2 } } { 2 ( x - 2 ) ^ { 2 } ( x + 3 ) } , x \neq 2 , - 3$ 
D)  $R ( x ) = \frac { - 2 ( x - 3 ) ( x + 2 ) ^ { 2 } } { ( x + 4 ) ^ { 2 } ( x + 1 ) } , x \neq - 4 , - 1$ 
A)  $R ( x ) = \frac { - 2 ( x + 1 ) ( x + 4 ) ^ { 2 } } { ( x + 2 ) ^ { 2 } ( x - 3 ) } , x \neq z - 2,3$ 
B)  $R ( x ) = \frac { - 2 ( x + 1 ) ( x + 4 ) } { ( x + 2 ) ( x - 3 ) } , x \neq - 2,3$ 
C)  $R ( x ) = \frac { - ( x + 1 ) ( x + 4 ) ^ { 2 } } { 2 ( x - 2 ) ^ { 2 } ( x + 3 ) } , x \neq 2 , - 3$ 
D)  $R ( x ) = \frac { - 2 ( x - 3 ) ( x + 2 ) ^ { 2 } } { ( x + 4 ) ^ { 2 } ( x + 1 ) } , x \neq - 4 , - 1$

Question 18

Use the Factor Theorem to determine whether x - c is a factor of f(x).
-$f ( x ) = x ^ { 3 } + 7 x ^ { 2 } - 16 x + 18 ; x + 9$

Accepted Answer

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

Question 19

Graph the function.
-$$f ( x ) = \frac { ( x + 2 ) ( x - 4 ) } { x ^ { 2 } - 36 }$$

Accepted Answer

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

Question 20

Write the word or phrase that best completes each statement or answers the question.
Analyze the graph of the given function f as follows:
(a) Determine the end behavior: find the power function that the graph of f resembles for large values of |x|.
(b) Find the x- and y-intercepts of the graph.
(c) Determine whether the graph crosses or touches the x-axis at each x-intercept.
(d) Graph f using a graphing utility.
(e) Use the graph to determine the local maxima and local minima, if any exist. Round turning points to two decimal places.
(f) Use the information obtained in (a) - (e) to draw a complete graph of f by hand. Label all intercepts and turning points.
(g) Find the domain of f. Use the graph to find the range of f.
(h) Use the graph to determine where f is increasing and where f is decreasing.
-For the polynomial function $f ( x ) = 2 x ^ { 4 } - 7 x ^ { 3 } + 11 x - 4$
a) Find the $x$ - and $y$-intercepts of the graph of $f$. Round to two decimal places, if necessary.
b) Determine whether the graph crosses or touches the $x$-axis at each $x$-intercept.
c) End behavior: find the power function that the graph of $f$ resembles for large values of $| x |$.
d) Use a graphing utility to graph the function.Approximate the local maxima rounded to two decimal places, i necessary. Approximate the local minima rounded to two decimal places, if necessary.
e) Determine the number of turning points on the graph.
f) Put all the information together, and connect the points with a smooth, continuous curve to obtain the graph

Accepted Answer

a) The @#LAT-DLM&-intercepts are @#LAT-DLM&, and 2.94. The @#LAT-DLM&-

Use the graph to find the oblique asymptote, if any, of the function. -

Solve the inequality. - $( b - 2 ) ( b - 3 ) ( b - 4 ) < 0$

Information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. -Degree 5; zeros: $8,2 + 5 \mathrm { i } , - 8 \mathrm { i }$

Find the x- and y-intercepts of f. - $f ( x ) = ( x - 2 ) ^ { 2 } \left( x ^ { 2 } - 25 \right)$

Solve the inequality. - $x ^ { 2 } + 7 x \geq 0$

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. - $f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 9 x - 18$

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. - $f ( x ) = 3 ( x - 6 ) ( x + 7 ) ^ { 3 }$

Solve the inequality. - $x ^ { 2 } + 6 x \leq 0$

Find the indicated intercept(s) of the graph of the function. - $x$ -intercepts of $f ( x ) = \frac { ( x - 8 ) ( 2 x + 7 ) } { x ^ { 2 } + 3 x - 4 }$

Find the vertical asymptotes of the rational function. - $f ( x ) = \frac { - 2 x ( x + 2 ) } { 3 x ^ { 2 } - 5 x - 8 }$

Give the equation of the oblique asymptote, if any, of the function. - $h ( x ) = \frac { 3 x ^ { 2 } - 8 x - 4 } { 9 x ^ { 2 } - 4 x + 6 }$

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. - $f ( x ) = \frac { x ^ { 4 } - 7 } { x ^ { 3 } }$

Find the indicated intercept(s) of the graph of the function. - $y$ -intercept of $f ( x ) = \frac { x ^ { 2 } - 10 x - 4 } { x ^ { 2 } + 13 x + 4 }$

Graph the function. - $f ( x ) = \frac { x ^ { 2 } - 4 x } { ( x - 2 ) ^ { 2 } }$ $Graph the function. - f ( x ) = \frac { x ^ { 2 } - 4 x } { ( x - 2 ) ^ { 2 } }$

Choose the one alternative that best completes the statement or answers the question. Find a bound on the real zeros of the polynomial function. - $f ( x ) = x ^ { 4 } + 2 x ^ { 2 } - 3$

Give the equation of the horizontal asymptote, if any, of the function. - $h ( x ) = \frac { 2 x ^ { 3 } - 8 x - 9 } { 7 x + 8 }$

Solve the problem. -Determine which rational function $\mathrm { R } ( \mathrm { x } )$ has a graph that crosses the $x$ -axis at $- 1$ , touches the $x$ -axis at $- 4$ , has vertical asymptotes at $x = - 2$ and $x = 3$ , and has one horizontal asymptote at $y = - 2$ .

Use the Factor Theorem to determine whether x - c is a factor of f(x). - $f ( x ) = x ^ { 3 } + 7 x ^ { 2 } - 16 x + 18 ; x + 9$

Graph the function. - $f ( x ) = \frac { ( x + 2 ) ( x - 4 ) } { x ^ { 2 } - 36 }$ $Graph the function. - f ( x ) = \frac { ( x + 2 ) ( x - 4 ) } { x ^ { 2 } - 36 }$

Functions and Their Graphs

Linear and Quadratic Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometric Functions

Polar Coordinates; Vectors

Analytic Geometry

Systems of Equations and Inequalities

Sequences; Induction; the Binomial Theorem

Counting and Probability

A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Foundations: a Prelude to Functions

Graphing Utilities

Filters

Exam 3: Polynomial and Rational Functions

Use the graph to find the oblique asymptote, if any, of the function. -

Solve the inequality. - (b−2)(b−3)(b−4)<0( b - 2 ) ( b - 3 ) ( b - 4 ) < 0(b−2)(b−3)(b−4)<0

Information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. -Degree 5; zeros: 8,2+5i,−8i8,2 + 5 \mathrm { i } , - 8 \mathrm { i }8,2+5i,−8i

Find the x- and y-intercepts of f. - f(x)=(x−2)2(x2−25)f ( x ) = ( x - 2 ) ^ { 2 } \left( x ^ { 2 } - 25 \right)f(x)=(x−2)2(x2−25)

Solve the inequality. - x2+7x≥0x ^ { 2 } + 7 x \geq 0x2+7x≥0

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. - f(x)=x3+2x2−9x−18f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 9 x - 18f(x)=x3+2x2−9x−18

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. - f(x)=3(x−6)(x+7)3f ( x ) = 3 ( x - 6 ) ( x + 7 ) ^ { 3 }f(x)=3(x−6)(x+7)3

Solve the inequality. - x2+6x≤0x ^ { 2 } + 6 x \leq 0x2+6x≤0

Find the indicated intercept(s) of the graph of the function. - xxx -intercepts of f(x)=(x−8)(2x+7)x2+3x−4f ( x ) = \frac { ( x - 8 ) ( 2 x + 7 ) } { x ^ { 2 } + 3 x - 4 }f(x)=x2+3x−4(x−8)(2x+7)​

Find the vertical asymptotes of the rational function. - f(x)=−2x(x+2)3x2−5x−8f ( x ) = \frac { - 2 x ( x + 2 ) } { 3 x ^ { 2 } - 5 x - 8 }f(x)=3x2−5x−8−2x(x+2)​

Give the equation of the oblique asymptote, if any, of the function. - h(x)=3x2−8x−49x2−4x+6h ( x ) = \frac { 3 x ^ { 2 } - 8 x - 4 } { 9 x ^ { 2 } - 4 x + 6 }h(x)=9x2−4x+63x2−8x−4​

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. - f(x)=x4−7x3f ( x ) = \frac { x ^ { 4 } - 7 } { x ^ { 3 } }f(x)=x3x4−7​

Find the indicated intercept(s) of the graph of the function. - yyy -intercept of f(x)=x2−10x−4x2+13x+4f ( x ) = \frac { x ^ { 2 } - 10 x - 4 } { x ^ { 2 } + 13 x + 4 }f(x)=x2+13x+4x2−10x−4​

Graph the function. - f(x)=x2−4x(x−2)2f ( x ) = \frac { x ^ { 2 } - 4 x } { ( x - 2 ) ^ { 2 } }f(x)=(x−2)2x2−4x​

Choose the one alternative that best completes the statement or answers the question. Find a bound on the real zeros of the polynomial function. - f(x)=x4+2x2−3f ( x ) = x ^ { 4 } + 2 x ^ { 2 } - 3f(x)=x4+2x2−3

Give the equation of the horizontal asymptote, if any, of the function. - h(x)=2x3−8x−97x+8h ( x ) = \frac { 2 x ^ { 3 } - 8 x - 9 } { 7 x + 8 }h(x)=7x+82x3−8x−9​

Use the Factor Theorem to determine whether x - c is a factor of f(x). - f(x)=x3+7x2−16x+18;x+9f ( x ) = x ^ { 3 } + 7 x ^ { 2 } - 16 x + 18 ; x + 9f(x)=x3+7x2−16x+18;x+9

Graph the function. - f(x)=(x+2)(x−4)x2−36f ( x ) = \frac { ( x + 2 ) ( x - 4 ) } { x ^ { 2 } - 36 }f(x)=x2−36(x+2)(x−4)​