Unit 1: Polynomial and Rational Functions: practice

Question 1

The graph of the function

f

is shown for

-5

\le x

\le 5

. What are all intervals on which

f

is increasing and the graph of

f

is concave down?

Question 2

Values of h(t): t = 0, 2, 5, 9, 12 hours give h(t) = 40, 46, 55, 61, 58 degrees Fahrenheit.
$t$ (hours)	$h(t)$ (degrees Fahrenheit)
0	40
2	46
5	55
9	61
12	58

The temperature of a greenhouse, in degrees Fahrenheit, is given by

h(t)

, where

t

is the number of hours after sunrise. The table gives values of

h(t)

at selected values of

t

. What is the average rate of change of

h

over the interval

2

\le t

\le 9

, in degrees Fahrenheit per hour?

Question 3

The function

V

gives the volume of water, in liters, in a tank

t

minutes after a valve is opened. The average rate of change of

V

over the interval

4

\le t

\le 10

is

-12

liters per minute. Which of the following statements must be true?

Question 4

The graph of the function

g

is shown, with the points

A

and

B

on the graph labeled. What is the average rate of change of

g

over the interval

0

\le x

\le 4

?

Question 5

Values of f(x): x = 0, 1, 2, 3 give f(x) = 1, 0, 3, 10.
$x$	$f(x)$
0	1
1	0
2	3
3	10

The function

f

is a quadratic function. The table gives values of

f(x)

for

x

= 0, 1, 2, 3

. What is the value of

f(4)

?

Question 6

Values of g(x): x = 0, 2, 4, 6, 8 give g(x) = 3, 11, 17, 21, 23.
$x$	$g(x)$
0	3
2	11
4	17
6	21
8	23

The table gives values of the function

g

at selected values of

x

. Based on the table, which of the following best describes

g

on the interval

0

< x

< 8

?

Question 7

Values of f(x): x = 1.9, 2, 2.1, 4.9, 5, 5.1 give f(x) = 6.2, 6.4, 6.6, 10.0, 9.85, 9.7.
$x$	$f(x)$
1.9	6.2
2	6.4
2.1	6.6
4.9	10.0
5	9.85
5.1	9.7

The table gives values of the function

f

at selected values of

x

. Using average rates of change over the intervals

1.9

\le x

\le 2.1

and

4.9

\le x

\le 5.1

, which of the following gives the best estimates of the rate of change of

f

at

x

= 2

and at

x

= 5

?

Question 8

Water drains from a tank. The volume of water in the tank decreases over time, and it decreases more and more slowly as time passes. The function

V

gives the volume of water

t

minutes after the water starts to drain. Which of the following describes the graph of

V

?

Question 9

The graph of the function

f

, with domain

-3

\le x

\le 4

, is shown. What is the range of

f

?

Question 10

Values of f(x): x = −2, −1, 0, 1, 2, 3, 4 give f(x) = 4, 1, 0, 1, 4, 9, 16.
$x$	$f(x)$
−2	4
−1	1
0	0
1	1
2	4
3	9
4	16

The table gives values of the function

f

for every input in its domain,

\{-2, -1, 0, 1, 2, 3, 4\}

. What is the preimage of the output value 4?

Question 11Graphing calculator

The temperature of a liquid, in degrees Celsius, is modeled by

T(t)

= 0.037t^3

{}- 0.81t^2

{}+ 3.4t

{}+ 18.6

, where

t

is the time in minutes and

0

\le t

\le 12

. What is the average rate of change of the temperature over the interval

1.5

\le t

\le 6.2

, in degrees Celsius per minute?

Question 12Graphing calculator

The number of members of a club, in hundreds, is modeled by

M(t)

= -0.0173t^3

{}+ 0.52t^2

{}+ 1.9t

{}+ 48.6

, where

t

is the number of months since the club opened. Over which of the following intervals is the average rate of change of

M

greatest?

Question 13

For a function

g

, the average rate of change over every interval of length 2 is equal to 6. What is the value of

g(5) - g(1)

?

Question 14

The function

f

is given by

f(x)

= -3x^2 + 5x

. Which of the following describes the average rates of change of

f

over the consecutive intervals

[0, 1]

,

[1, 2]

and

[2, 3]

?

Question 15Graphing calculator

The speed of a cyclist, in meters per second, is modeled by

s(t)

= 40

{}- \dfrac{25}{t + 1}

, where

t

is the time in seconds after a race starts. Using the average rate of change of

s

over the interval

2.99

\le t

\le 3.01

, what is an estimate of the rate of change of the cyclist's speed at

t

= 3

, in meters per second per second?

Question 16

The polynomial function

p

is given by

p(x)

= 3x^2 - 2x^5 + 7

. Which of the following describes the end behavior of

p

?

Question 17

The polynomial function

p

is given by

p(x)

= (1 - x)^3(x + 2)^2

. Which of the following describes the end behavior of

p

?

Question 18

The graph of the polynomial function

p

is shown. Which of the following could be an expression for

p(x)

?

Question 19

The polynomial function

p

has degree 5 and real coefficients. Its zeros include

x

= 2

,

x

= -1

with multiplicity 2, and

x

= 3i

. Which of the following could be an expression for

p(x)

?

Question 20

The polynomial function

p

has degree 4 and real coefficients. Two of its zeros are

x

= 1 + 2i

and

x

= 5

. Which of the following must be true?

Question 21

The polynomial function

p

is an even function, and

p(2)

= -5

is a relative minimum of

p

. Which of the following statements about

p(-2)

must be true?

Question 22

Which of the following functions is an odd function?

Question 23

Values of p(x): x = 0, 1, 2, 3, 4, 5 give p(x) = 4, 2, 2, 16, 56, 134.
$x$	$p(x)$
0	4
1	2
2	2
3	16
4	56
5	134

The table gives values of a polynomial function

p

for

x

= 0, 1, 2, 3, 4, 5

. What is the least possible degree of

p

?

Question 24

The function

p

is given by

p(x)

= (x + 2)(x - 1)^2(x - 4)

. What are all values of

x

for which

p(x)

< 0

?

Question 25

The polynomial function

f

has real zeros at

x

= -1

,

x

= 2

and

x

= 6

and no other real zeros. Which of the following must be true?

Question 26

The graph of the polynomial function

p

of degree 4 is shown. Which of the following statements about

p

is true?

Question 27

The graph of the polynomial function

p

, with domain

-1

\le x

\le 5

, is shown. What is the global maximum value of

p

on its domain?

Question 28Graphing calculator

The function

f

is given by

f(x)

= x^3 - 4.2x^2 + 3x + 1

. On which of the following intervals is

f

decreasing?

Question 29Graphing calculator

The polynomial function

p

is given by

p(x)

= x^4 - 3x^3 + x - 5

. What is the least real zero of

p

?

Question 30Graphing calculator

The function

g

is given by

g(x)

= -0.5x^4

{}+ 2x^3

{}- x

{}+ 3

. What is the maximum value of

g

on the interval

0

\le x

\le 5

?

Question 31

The graph of the cubic polynomial function

p

is shown. At which of the following values of

x

does the rate of change of

p

change from decreasing to increasing?

Question 32

The polynomial function

p

is given by

p(x)

= (x - 3)^2(x + 1)^3

. Which of the following describes the graph of

p

at its

x

-intercepts?

Question 33

The polynomial function

p

is given by

p(x)

= x^4 + 5x^2 - 36

. What are all the complex zeros of

p

?

Question 34Graphing calculator

The function

p

is given by

p(x)

= x^3 - 2x^2 - 5x + 3

. What are all values of

x

for which

p(x)

> 0

?

Question 35

The graph of the polynomial function

p

is shown. What is the least possible degree of

p

?

Question 36

The function

r

is given by

r(x)

= \dfrac{6x^2 - x + 2}{3 - 2x^2}

. Which of the following is an equation of the horizontal asymptote of the graph of

r

?

Question 37

The function

r

is given by

r(x)

= \dfrac{x^2 + 3x - 1}{x - 2}

. Which of the following is an equation of the slant asymptote of the graph of

r

?

Question 38

The function

r

is given by

r(x)

= \dfrac{x - 5}{(x - 2)(x + 3)}

. What is

\lim_{x \to 2^+} r(x)

?

Question 39

The function

r

is given by

r(x)

= \dfrac{(x - 1)^2(x + 2)}{(x - 1)^3(x - 4)}

. What are all vertical asymptotes of the graph of

r

?

Question 40

The function

r

is given by

r(x)

= \dfrac{x^2 - 4x + 3}{x^2 - 1}

. The graph of

r

has a hole. What are the coordinates of the hole?

Question 41

The function

r

is given by

r(x)

= \dfrac{x^2 - 2x - 8}{x^2 - 16}

. What are all zeros of

r

?

Question 42

The function

r

is given by

r(x)

= \dfrac{x - 3}{(x + 1)(x - 5)}

. What are all values of

x

for which

r(x)

\ge 0

?

Question 43

Values of r(x): x = 10, 100, 1000, 10000 give r(x) = 2.6474, 2.5127, 2.5013, 2.5001.
$x$	$r(x)$
10	2.6474
100	2.5127
1000	2.5013
10000	2.5001

The table gives values of a rational function

r

for selected large values of

x

. The values continue the same pattern as

x

increases. Which of the following is the best estimate of

\lim_{x \to \infty} r(x)

?

Question 44

The graph of the rational function

r

is shown. Which of the following could be an expression for

r(x)

?

Question 45

The function

r

is given by

r(x)

= \dfrac{(x + 1)^3(x - 2)}{(x + 1)^2(x - 5)}

. Which of the following is true about the graph of

r

at

x

= -1

?

Question 46

The function

r

is given by

r(x)

= \dfrac{3x^2 + 1}{x^3 - 8}

. What is

\lim_{x \to -\infty} r(x)

?

Question 47Graphing calculator

The function

r

is given by

r(x)

= \dfrac{x^3 + 2}{x^2 + 1}

. What is the value of

x

for which

r(x)

= 4

?

Question 48

The graph of the rational function

r

is shown. Which of the following could be an expression for

r(x)

?

Question 49Graphing calculator

The function

r

is given by

r(x)

= \dfrac{x^2 + 3}{x + 1}

. What are all values of

x

for which

r(x)

< 2.5

?

Question 50Graphing calculator

The concentration of a medicine in a patient's blood, in milligrams per liter, is modeled by

C(t)

= \dfrac{40t}{t^2 + 9}

, where

t

is the number of hours after the medicine is taken. The concentration rises and then falls. At what time

t

, in hours, has the concentration fallen back to 5 milligrams per liter?

Question 51

Which of the following is equivalent to

\dfrac{2x^3 - 3x^2 + 4x - 5}{x - 2}

for

x

\ne 2

?

Question 52

When

(x - 3)^5

is expanded and written in standard form, what is the coefficient of

x^3

?

Question 53

The profit of a company, in thousands of dollars, is modeled by

P(x)

= -2x^2 + 28x - 80

, where

x

is the number of items made, in hundreds. Each of the following is an equivalent form of

P(x)

. Which form shows, as constants in the expression, the numbers of items at which the profit is zero?

Question 54

The point

(4, -6)

is on the graph of the function

f

. The function

g

is given by

g(x)

= -2f(x + 3) + 1

. Which of the following points must be on the graph of

g

?

Question 55

The graphs of the functions

f

and

g

are shown. The graph of

g

is a translation of the graph of

f

. Which of the following is an expression for

g(x)

?

Question 56

The function

g

is given by

g(x) = f\left(\dfrac{x}{4}\right)

. Which of the following describes how the graph of

g

is obtained from the graph of

f

?

Question 57

The graph of the function

f

, made of three line segments, is shown for its whole domain,

-2

\le x

\le 6

. The function

g

is given by

g(x)

= 2f(x - 3) + 4

. What is the range of

g

?

Question 58

The graph of the function

g

is obtained by reflecting the graph of

f

over the

y

-axis and then translating it 2 units up. Which of the following is an expression for

g(x)

?

Question 59

Values of f(x): x = −4, −2, 0, 2, 4, 6 give f(x) = 9, 5, 1, −3, 0, 8.
$x$	$f(x)$
−4	9
−2	5
0	1
2	−3
4	0
6	8

The table gives values of the function

f

at selected values of

x

. The function

g

is given by

g(x)

= 2f(x - 3) + 1

. Using the table, for what value of

x

is

g(x)

= 11

?

Question 60

The average rate of change of the function

f

over the interval

1

\le x

\le 5

is 3. The function

g

is given by

g(x)

= -2f(x) + 7

. What is the average rate of change of

g

over the interval

1

\le x

\le 5

?

Question 61

Which of the following is equivalent to

\dfrac{x^2 - 1}{x^2 + 2x - 3}

for all

x

in its domain?

Question 62

The graph of the polynomial function

p

is shown. The function

q

is given by

q(x)

= p(x + 2)

. What are the zeros of

q

?

Question 63

The function

r

is given by

r(x)

= \dfrac{x^2 - 9}{x^3 - 4x}

. What is the domain of

r

?

Question 64Graphing calculator

The function

f

is given by

f(x)

= x^3 - 5x + 1

, and the function

g

is given by

g(x)

= f(2x)

. What is the greatest zero of

g

?

Question 65Graphing calculator

The function

f

is given by

f(x)

= \dfrac{3x + 1}{x^2 + 2}

, and the function

g

is given by

g(x)

= f(x + 2) - 1

. What is the maximum value of

g

?

Question 66

Data: x = 0, 1, 2, 3, 4, 5 give y = 3.0, 4.9, 8.6, 14.1, 21.5, 30.8.
$x$	$y$
0	3.0
1	4.9
2	8.6
3	14.1
4	21.5
5	30.8

The table shows data collected at equally spaced values of

x

. Which type of function is most appropriate to model the data, and why?

Question 67

An open box is made from a 20-inch by 30-inch sheet of cardboard by cutting a square with side length

x

inches from each corner and folding up the sides. The volume of the box, in cubic inches, is modeled by

V(x)

= x(20 - 2x)(30 - 2x)

. Which of the following is the most appropriate domain for

V

in this context?

Question 68Graphing calculator

Visitors in thousands: t = 0, 1, 2, 3, 4, 5, 6 give 20.3, 24.6, 24.1, 25.2, 26.0, 33.4, 46.8.
$t$ (years since 2018)	Visitors (thousands)
0	20.3
1	24.6
2	24.1
3	25.2
4	26.0
5	33.4
6	46.8

The table gives the number of visitors to a museum, in thousands, in selected years, where

t

is the number of years since 2018. A cubic regression is used to model the number of visitors as a function of

t

. Based on the model, what is the predicted number of visitors, in thousands, in 2025?

Question 69Graphing calculator

An open box is made from a 20-inch by 30-inch sheet of cardboard by cutting a square with side length

x

inches from each corner and folding up the sides, so its volume, in cubic inches, is

V(x)

= x(20 - 2x)(30 - 2x)

for

0

< x

< 10

. What side length

x

, in inches, gives the box with the greatest volume?

Question 70

The intensity of light from a lamp is inversely proportional to the square of the distance from the lamp. At a distance of 2 meters the intensity is 45 lux. Based on this model, what is the intensity at a distance of 6 meters, in lux?

Question 71

A parking garage charges 3 dollars for a stay of 1 hour or less. For a longer stay, it charges the 3 dollars plus 2 dollars per hour for the time beyond the first hour, charged in proportion for parts of an hour. Which of the following models the charge

f(t)

, in dollars, for a stay of

t

hours?

Question 72

A cafe owner models the number of iced drinks sold in a day as a linear function of the day's high temperature, in degrees Fahrenheit, for temperatures from 60 to 95. Which of the following is an assumption of this model?

Question 73Graphing calculator

A ball is thrown upward from a platform. Its height above the ground, in meters, is modeled by

h(t)

= -4.9t^2 + 19.6t + 2

, where

t

is the time in seconds after it is thrown. The model applies from the moment the ball is thrown until it hits the ground. Which of the following is the domain of the model?

Question 74

The water level in a pond, in centimeters above a marker, is modeled by a quadratic function

f

of time

t

, in hours. The greatest value of

f

is 30, at

t

= 4

, and

f(0)

= 14

. Which of the following is an expression for

f(t)

?

Question 75

The scatterplot shows a data set of 5 points, all with different

x

-values. Which of the following statements must be true?

Question 76Graphing calculator

Depth in meters: months m = 1 to 8 give 12.1, 15.8, 14.9, 12.2, 10.6, 11.9, 16.3, 23.0.
$m$ (month)	Depth (meters)
1	12.1
2	15.8
3	14.9
4	12.2
5	10.6
6	11.9
7	16.3
8	23.0

The table gives the depth of water in a reservoir, in meters, at the end of month

m

for

m

= 1

to

m

= 8

. A quartic regression is used to model the depth as a function of

m

. Based on the model, what is the predicted depth, in meters, at the end of month 9?

Question 77

The average cost per item, in dollars, of making

x

items is modeled by

A(x)

= \dfrac{1200 + 8x}{x}

for

x

> 0

. Which of the following describes

A(x)

as

x

increases without bound?

Question 78Graphing calculator

The number of people on a hiking trail, in hundreds, is modeled by

R(t)

= -0.08t^3

{}+ 1.2t^2

{}- 3t

{}+ 4

, where

t

is the number of hours after 6 a.m. and

0

\le t

\le 12

. At what value of

t

is the number of people on the trail greatest?

Question 79Graphing calculator

Weekly sales at two stores, in thousands of dollars, are modeled by

A(t)

= 0.3t^3

{}- 2t^2

{}+ 5t

{}+ 10

and

B(t)

= 2.5t + 12

, where

t

is the number of weeks after a sale begins,

t

\ge 0

. For what value of

t

are the modeled sales at the two stores equal?

Question 80

The scatterplot shows a data set. Which type of function is most appropriate to model the data, and why?