Exam 2: Polynomial, Power, and Rational Functions

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. - $f ( x ) = \frac { x + 1 } { x ^ { 2 } + 4 x - 5 }$ $Write the word or phrase that best completes each statement or answers the question. Graph the rational function and analyze it in the following way: find the intercepts, asymptotes, use limits to describe the behavior at the vertical asymptotes and the end behavior. Find the domain and range. Determine where the function is continuous and where it is increasing and decreasing. Find any local extrema. - f ( x ) = \frac { x + 1 } { x ^ { 2 } + 4 x - 5 }$ $Write the word or phrase that best completes each statement or answers the question. Graph the rational function and analyze it in the following way: find the intercepts, asymptotes, use limits to describe the behavior at the vertical asymptotes and the end behavior. Find the domain and range. Determine where the function is continuous and where it is increasing and decreasing. Find any local extrema. - f ( x ) = \frac { x + 1 } { x ^ { 2 } + 4 x - 5 }$

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Question 1

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$\mathrm { x }$ -intercept: $( - 1,0 )$ $\mathrm { y }$ -intercept: $\left( 0 , - \frac { 1 } { 5 } \right)$
Vertical asymptotes: $x = - 5 , x = 1$
Horizontal asymptote: $\mathrm { y } = 0$
$\mathrm { x } -intercept: ( - 1,0 ) \mathrm { y } -intercept: \left( 0 , - \frac { 1 } { 5 } \right) Vertical asymptotes: x = - 5 , x = 1 Horizontal asymptote: \mathrm { y } = 0 \lim _ { x \rightarrow \infty } f ( x ) = 0 , \lim _ { x \rightarrow \infty } f ( x ) = 0 Domain: ( - \infty , - 5 ) \cup ( - 5,1 ) \cup ( 1 , \infty ) Range: ( - \infty , \infty ) Continuity: all x \neq - 5,1 No local extrema Decreasing on: ( - \infty , - 5 ) , ( - 5,1 ) , ( 1 , \infty )$

$\lim _ { x \rightarrow \infty } f ( x ) = 0 , \lim _ { x \rightarrow \infty } f ( x ) = 0$
Domain: $( - \infty , - 5 ) \cup ( - 5,1 ) \cup ( 1 , \infty )$
Range: $( - \infty , \infty )$
Continuity: all $x \neq - 5,1$
No local extrema
Decreasing on: $( - \infty , - 5 ) , ( - 5,1 ) , ( 1 , \infty )$

Find the axis of the graph of the function. - $f ( x ) = 3 x ^ { 2 } - 18 x + 28$

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Question 2

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Solve the problem. -An open-top rectangular box has a square base and it will hold 106 cubic centimeters (cc). Each side has length $x$ $\mathrm { cm }$ , and it has a height of $\mathrm { y } \mathrm { cm }$ . Its surface area is given by $S ( x ) = \frac { 424 } { x } + x ^ { 2 }$ Graph the function on the interval $( 0 , \infty )$ .

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Question 3

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Use limits to describe the behavior of the rational function near the indicated asymptote. - $f ( x ) = \frac { 3 } { x - 4 }$ Describe the behavior of the function near its vertical asymptote.

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Question 4

Find the vertex of the graph of the function. - $f ( x ) = 3 x ^ { 2 } + 18 x + 23$

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Question 5

Solve the problem. -A rectangular piece of cardboard measuring 17 inches by 45 inches is to be made into a box with an open top by cutting equal size squares from each corner and folding up the sides. Let $x$ represent the length of a side of each such square. For what value of $x$ will the volume be a maximum? If necessary, round to 2 decimal places.

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Question 6

Match the equation to the correct graph. - $f ( x ) = 2 ( x + 5 ) ^ { 2 } - 3$

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Question 7

Find the vertex of the graph of the function. - $f ( x ) = ( x - 1 ) ^ { 2 } - 10$

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Question 8

Choose the one alternative that best completes the statement or answers the question. Describe how to obtain the graph of the given monomial function from the graph of g(x) = xn with the same power n. - $f ( x ) = \frac { 1 } { 4 } x ^ { 8 }$

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Question 9

Graph the function in a viewing window that shows all of its extrema and x-intercepts. - $f ( x ) = 2 x ^ { 4 } - x ^ { 3 } + 5 x ^ { 2 } + 3 x + 2$ $Graph the function in a viewing window that shows all of its extrema and x-intercepts. - f ( x ) = 2 x ^ { 4 } - x ^ { 3 } + 5 x ^ { 2 } + 3 x + 2$

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Question 10

Graph the function. - $f ( x ) = ( x + 2 ) ^ { 2 } ( x - 3 ) ^ { 3 }$ $Graph the function. - f ( x ) = ( x + 2 ) ^ { 2 } ( x - 3 ) ^ { 3 }$

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Question 11

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions. -Zeros: $- 4 , \mathrm { i } , - \mathrm { i } ; \mathrm { f } ( - 3 ) = 60$

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Question 12

Describe how the graph of the given function can be obtained by transforming the graph of the reciprocal function f(x) = 1/x. - $f ( x ) = \frac { 2 } { x - 2 }$

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Question 13

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

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Question 14

Write the quadratic function in vertex form. - $y = x ^ { 2 } - 20 x + 91$

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Question 15

Match the given graph with its polynomial function.Choose the one alternative that best completes the statement or answers the question. -

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Question 16

Find the requested function. -Find the cubic function with the given table of values.

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Question 17

Determine if the function is a power function. If it is, then state the power and constant of variation. - $f ( x ) = 6 x ^ { - 3 / 4 }$

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Question 18

Match the equation to the correct graph. - $f ( x ) = 3 - 4 ( x - 5 ) ^ { 2 }$

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Question 19

Divide using synthetic division, and write a summary statement in fraction form. - $\frac { 2 x ^ { 3 } + 3 x ^ { 2 } + 4 x - 10 } { x + 1 }$

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Question 20

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Find the axis of the graph of the function. - $f ( x ) = 3 x ^ { 2 } - 18 x + 28$

Use limits to describe the behavior of the rational function near the indicated asymptote. - $f ( x ) = \frac { 3 } { x - 4 }$ Describe the behavior of the function near its vertical asymptote.

Find the vertex of the graph of the function. - $f ( x ) = 3 x ^ { 2 } + 18 x + 23$

Match the equation to the correct graph. - $f ( x ) = 2 ( x + 5 ) ^ { 2 } - 3$

Find the vertex of the graph of the function. - $f ( x ) = ( x - 1 ) ^ { 2 } - 10$

Choose the one alternative that best completes the statement or answers the question. Describe how to obtain the graph of the given monomial function from the graph of g(x) = xn with the same power n. - $f ( x ) = \frac { 1 } { 4 } x ^ { 8 }$

Graph the function in a viewing window that shows all of its extrema and x-intercepts. - $f ( x ) = 2 x ^ { 4 } - x ^ { 3 } + 5 x ^ { 2 } + 3 x + 2$ $Graph the function in a viewing window that shows all of its extrema and x-intercepts. - f ( x ) = 2 x ^ { 4 } - x ^ { 3 } + 5 x ^ { 2 } + 3 x + 2$

Graph the function. - $f ( x ) = ( x + 2 ) ^ { 2 } ( x - 3 ) ^ { 3 }$ $Graph the function. - f ( x ) = ( x + 2 ) ^ { 2 } ( x - 3 ) ^ { 3 }$

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions. -Zeros: $- 4 , \mathrm { i } , - \mathrm { i } ; \mathrm { f } ( - 3 ) = 60$

Describe how the graph of the given function can be obtained by transforming the graph of the reciprocal function f(x) = 1/x. - $f ( x ) = \frac { 2 } { x - 2 }$

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

Write the quadratic function in vertex form. - $y = x ^ { 2 } - 20 x + 91$

Match the given graph with its polynomial function.Choose the one alternative that best completes the statement or answers the question. -

Find the requested function. -Find the cubic function with the given table of values.

Determine if the function is a power function. If it is, then state the power and constant of variation. - $f ( x ) = 6 x ^ { - 3 / 4 }$

Match the equation to the correct graph. - $f ( x ) = 3 - 4 ( x - 5 ) ^ { 2 }$

Divide using synthetic division, and write a summary statement in fraction form. - $\frac { 2 x ^ { 3 } + 3 x ^ { 2 } + 4 x - 10 } { x + 1 }$

Functions and Graphs

Exponential, Logistic, and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems and Matrices

Analytic Geometry in Two and Three Dimensions

Discrete Mathematics

Statistics and Probability

An Introduction to Calculus: Limits, Derivatives, and Integrals

Prerequisites

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Exam 2: Polynomial, Power, and Rational Functions

Find the axis of the graph of the function. - f(x)=3x2−18x+28f ( x ) = 3 x ^ { 2 } - 18 x + 28f(x)=3x2−18x+28

Use limits to describe the behavior of the rational function near the indicated asymptote. - f(x)=3x−4f ( x ) = \frac { 3 } { x - 4 }f(x)=x−43​ Describe the behavior of the function near its vertical asymptote.

Find the vertex of the graph of the function. - f(x)=3x2+18x+23f ( x ) = 3 x ^ { 2 } + 18 x + 23f(x)=3x2+18x+23

Match the equation to the correct graph. - f(x)=2(x+5)2−3f ( x ) = 2 ( x + 5 ) ^ { 2 } - 3f(x)=2(x+5)2−3

Find the vertex of the graph of the function. - f(x)=(x−1)2−10f ( x ) = ( x - 1 ) ^ { 2 } - 10f(x)=(x−1)2−10

Choose the one alternative that best completes the statement or answers the question. Describe how to obtain the graph of the given monomial function from the graph of g(x) = xn with the same power n. - f(x)=14x8f ( x ) = \frac { 1 } { 4 } x ^ { 8 }f(x)=41​x8

Graph the function in a viewing window that shows all of its extrema and x-intercepts. - f(x)=2x4−x3+5x2+3x+2f ( x ) = 2 x ^ { 4 } - x ^ { 3 } + 5 x ^ { 2 } + 3 x + 2f(x)=2x4−x3+5x2+3x+2

Graph the function. - f(x)=(x+2)2(x−3)3f ( x ) = ( x + 2 ) ^ { 2 } ( x - 3 ) ^ { 3 }f(x)=(x+2)2(x−3)3

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions. -Zeros: −4,i,−i;f(−3)=60- 4 , \mathrm { i } , - \mathrm { i } ; \mathrm { f } ( - 3 ) = 60−4,i,−i;f(−3)=60

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)=2x−2f ( x ) = \frac { 2 } { x - 2 }f(x)=x−22​

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​

Write the quadratic function in vertex form. - y=x2−20x+91y = x ^ { 2 } - 20 x + 91y=x2−20x+91

Match the given graph with its polynomial function.Choose the one alternative that best completes the statement or answers the question. -

Find the requested function. -Find the cubic function with the given table of values.

Determine if the function is a power function. If it is, then state the power and constant of variation. - f(x)=6x−3/4f ( x ) = 6 x ^ { - 3 / 4 }f(x)=6x−3/4

Match the equation to the correct graph. - f(x)=3−4(x−5)2f ( x ) = 3 - 4 ( x - 5 ) ^ { 2 }f(x)=3−4(x−5)2

Divide using synthetic division, and write a summary statement in fraction form. - 2x3+3x2+4x−10x+1\frac { 2 x ^ { 3 } + 3 x ^ { 2 } + 4 x - 10 } { x + 1 }x+12x3+3x2+4x−10​

Functions and Graphs

Exponential, Logistic, and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems and Matrices

Analytic Geometry in Two and Three Dimensions

Discrete Mathematics

Statistics and Probability

An Introduction to Calculus: Limits, Derivatives, and Integrals

Prerequisites

Filters

Find the axis of the graph of the function. - $f ( x ) = 3 x ^ { 2 } - 18 x + 28$

Use limits to describe the behavior of the rational function near the indicated asymptote. - $f ( x ) = \frac { 3 } { x - 4 }$ Describe the behavior of the function near its vertical asymptote.

Find the vertex of the graph of the function. - $f ( x ) = 3 x ^ { 2 } + 18 x + 23$

Match the equation to the correct graph. - $f ( x ) = 2 ( x + 5 ) ^ { 2 } - 3$

Find the vertex of the graph of the function. - $f ( x ) = ( x - 1 ) ^ { 2 } - 10$

Choose the one alternative that best completes the statement or answers the question. Describe how to obtain the graph of the given monomial function from the graph of g(x) = xn with the same power n. - $f ( x ) = \frac { 1 } { 4 } x ^ { 8 }$

Graph the function in a viewing window that shows all of its extrema and x-intercepts. - $f ( x ) = 2 x ^ { 4 } - x ^ { 3 } + 5 x ^ { 2 } + 3 x + 2$ $Graph the function in a viewing window that shows all of its extrema and x-intercepts. - f ( x ) = 2 x ^ { 4 } - x ^ { 3 } + 5 x ^ { 2 } + 3 x + 2$

Graph the function. - $f ( x ) = ( x + 2 ) ^ { 2 } ( x - 3 ) ^ { 3 }$ $Graph the function. - f ( x ) = ( x + 2 ) ^ { 2 } ( x - 3 ) ^ { 3 }$

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions. -Zeros: $- 4 , \mathrm { i } , - \mathrm { i } ; \mathrm { f } ( - 3 ) = 60$

Describe how the graph of the given function can be obtained by transforming the graph of the reciprocal function f(x) = 1/x. - $f ( x ) = \frac { 2 } { x - 2 }$

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

Write the quadratic function in vertex form. - $y = x ^ { 2 } - 20 x + 91$

Determine if the function is a power function. If it is, then state the power and constant of variation. - $f ( x ) = 6 x ^ { - 3 / 4 }$

Match the equation to the correct graph. - $f ( x ) = 3 - 4 ( x - 5 ) ^ { 2 }$

Divide using synthetic division, and write a summary statement in fraction form. - $\frac { 2 x ^ { 3 } + 3 x ^ { 2 } + 4 x - 10 } { x + 1 }$