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Precalculus

Ron Larson

Chapter 2

Polynomial and Rational Functions - all with Video Answers

Educators

Section 1

Quadratic Functions and Models

01:44

Problem 1

Fill in the blanks.
Linear, constant, and squaring functions are examples of ____ functions.

JH
J Hardin
Numerade Educator
01:05

Problem 2

Fill in the blanks.
A polynomial function of $x$ with degree $n$ has the form $f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots \cdot+a_{1} x+a_{0}$
$\left(a_{n} \neq 0\right),$ where $n$ is a _____ ______ and $a_{n}, a_{n-1}, \ldots, a_{1}, a_{0}$ are
____ numbers.

Vanessa Lamar
Vanessa Lamar
Numerade Educator
00:53

Problem 3

Fill in the blanks.
A ____ function is a second-degree polynomial function, and its graph is called a _____.

JH
J Hardin
Numerade Educator
00:59

Problem 4

Fill in the blanks.
When the graph of a quadratic function opens downward, its leading coefficient is ____ and the vertex of the graph is a_____.

JH
J Hardin
Numerade Educator
00:27

Problem 5

Match the quadratic function with its graph. [The graphs are labeled (a), (b),
(c), and (d). $]$
(GRAPH CANNOT COPY)
$$f(x)=x^{2}-2$$

AG
Ankit Gupta
Numerade Educator
01:27

Problem 6

Match the quadratic function with its graph. [The graphs are labeled (a), (b),
(c), and (d). $]$
(GRAPH CANNOT COPY)
$$f(x)=(x+1)^{2}-2$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
00:30

Problem 7

Match the quadratic function with its graph. [The graphs are labeled (a), (b),
(c), and (d). $]$
(GRAPH CANNOT COPY)
$$f(x)=-(x-4)^{2}$$

AG
Ankit Gupta
Numerade Educator
01:16

Problem 8

Match the quadratic function with its graph. [The graphs are labeled (a), (b),
(c), and (d). $]$
(GRAPH CANNOT COPY)
$$f(x)=4-(x-2)^{2}$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
05:14

Problem 9

Sketch the graph of each quadratic function and compare it with the graph of $y=x^{2}$.
(a) $f(x)=\frac{1}{2} x^{2}$
(b) $g(x)=-\frac{1}{8} x^{2}$
(c) $h(x)=\frac{3}{2} x^{2}$
(d) $k(x)=-3 x^{2}$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
04:29

Problem 10

Sketch the graph of each quadratic function and compare it with the graph of $y=x^{2}$.
(a) $f(x)=x^{2}+1$
(b) $g(x)=x^{2}-1$
(c) $h(x)=x^{2}+3$
(d) $k(x)=x^{2}-3$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
04:25

Problem 11

Sketch the graph of each quadratic function and compare it with the graph of $y=x^{2}$.
(a) $f(x)=(x-1)^{2}$
(b) $g(x)=(3 x)^{2}+1$
(c) $h(x)=\left(\frac{1}{3} x\right)^{2}-3$
(d) $k(x)=(x+3)^{2}$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
04:21

Problem 12

Sketch the graph of each quadratic function and compare it with the graph of $y=x^{2}$.
(a) $f(x)=-\frac{1}{2}(x-2)^{2}+1$
(b) $g(x)=\left[\frac{1}{2}(x-1)\right]^{2}-3$
(c) $h(x)=-\frac{1}{2}(x+2)^{2}-1$
(d) $k(x)=[2(x+1)]^{2}+4$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
03:58

Problem 13

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=x^{2}-6 x$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
02:48

Problem 14

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$g(x)=x^{2}-8 x$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:22

Problem 15

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$h(x)=x^{2}-8 x+16$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:38

Problem 16

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$g(x)=x^{2}+2 x+1$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
02:10

Problem 17

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=x^{2}-6 x+2$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
03:26

Problem 18

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=x^{2}+16 x+61$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
02:58

Problem 19

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=x^{2}-8 x+21$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
02:37

Problem 20

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=x^{2}+12 x+40$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
02:19

Problem 21

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=x^{2}-x+\frac{5}{4}$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
04:23

Problem 22

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=x^{2}+3 x+\frac{1}{4}$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
04:23

Problem 23

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=-x^{2}+2 x+5$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
03:05

Problem 24

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=-x^{2}-4 x+1$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
04:05

Problem 25

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$h(x)=4 x^{2}-4 x+21$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
04:09

Problem 26

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=2 x^{2}-x+1$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
02:04

Problem 27

Use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and $x$ -intercept(s). Then check your results algebraically by writing the quadratic function in standard form.
$$f(x)=-\left(x^{2}+2 x-3\right)$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
03:19

Problem 28

Use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and $x$ -intercept(s). Then check your results algebraically by writing the quadratic function in standard form.
$$f(x)=-\left(x^{2}+x-30\right)$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
02:19

Problem 29

Use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and $x$ -intercept(s). Then check your results algebraically by writing the quadratic function in standard form.
$$g(x)=x^{2}+8 x+11$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
02:01

Problem 30

Use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and $x$ -intercept(s). Then check your results algebraically by writing the quadratic function in standard form.
$$f(x)=x^{2}+10 x+14$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
02:08

Problem 31

Use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and $x$ -intercept(s). Then check your results algebraically by writing the quadratic function in standard form.
$$f(x)=-2 x^{2}+12 x-18$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
02:45

Problem 32

Use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and $x$ -intercept(s). Then check your results algebraically by writing the quadratic function in standard form.
$$f(x)=-4 x^{2}+24 x-41$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
02:45

Problem 33

Use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and $x$ -intercept(s). Then check your results algebraically by writing the quadratic function in standard form.
$$g(x)=\frac{1}{2}\left(x^{2}+4 x-2\right)$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
02:59

Problem 34

Use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and $x$ -intercept(s). Then check your results algebraically by writing the quadratic function in standard form.
$$f(x)=\frac{3}{5}\left(x^{2}+6 x-5\right)$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:16

Problem 35

Write the standard form of the quadratic function whose graph is the parabola shown.
(GRAPH CANNOT COPY)

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:14

Problem 36

Write the standard form of the quadratic function whose graph is the parabola shown.
(GRAPH CANNOT COPY)

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:23

Problem 37

Write the standard form of the quadratic function whose graph is a parabola with the given vertex and that passes through the given point.
Vertex: $(-2,5) ;$ point: $(0,9)$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:18

Problem 38

Write the standard form of the quadratic function whose graph is a parabola with the given vertex and that passes through the given point.
Vertex: $(-3,-10) ;$ point: $(0,8)$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:21

Problem 39

Write the standard form of the quadratic function whose graph is a parabola with the given vertex and that passes through the given point.
Vertex: $(1,-2) ;$ point: $(-1,14)$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:23

Problem 40

Write the standard form of the quadratic function whose graph is a parabola with the given vertex and that passes through the given point.
Vertex: $(2,3) ;$ point: $(0,2)$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:23

Problem 41

Write the standard form of the quadratic function whose graph is a parabola with the given vertex and that passes through the given point.
Vertex: $(5,12) ;$ point: $(7,15)$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:28

Problem 42

Write the standard form of the quadratic function whose graph is a parabola with the given vertex and that passes through the given point.
Vertex: $(-2,-2) ;$ point: $(-1,0)$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
02:39

Problem 43

Write the standard form of the quadratic function whose graph is a parabola with the given vertex and that passes through the given point.
Vertex: $\left(-\frac{1}{4}, \frac{3}{2}\right) ;$ point: $(-2,0)$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
02:04

Problem 44

Write the standard form of the quadratic function whose graph is a parabola with the given vertex and that passes through the given point.
Vertex: $\left(\frac{5}{2},-\frac{3}{4}\right) ;$ point: $(-2,4)$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:21

Problem 45

Write the standard form of the quadratic function whose graph is a parabola with the given vertex and that passes through the given point.
Vertex: $\left(-\frac{5}{2}, 0\right) ;$ point: $\left(-\frac{7}{2},-\frac{16}{3}\right)$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
02:00

Problem 46

Write the standard form of the quadratic function whose graph is a parabola with the given vertex and that passes through the given point.
Vertex: $(6,6) ;$ point: $\left(\frac{61}{10}, \frac{3}{2}\right)$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:22

Problem 47

Determine the $x$ -intercept(s) of the graph visually. Then find the $x$ -intercept(s) algebraically to confirm your results.
(GRAPH CANNOT COPY)
$$y=x^{2}-2 x-3$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:22

Problem 48

Determine the $x$ -intercept(s) of the graph visually. Then find the $x$ -intercept(s) algebraically to confirm your results.
(GRAPH CANNOT COPY)
$$y=x^{2}-2 x-3$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:13

Problem 48

Determine the $x$ -intercept(s) of the graph visually. Then find the $x$ -intercept(s) algebraically to confirm your results.
(GRAPH CANNOT COPY)
$$y=x^{2}-4 x-5$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:37

Problem 49

Determine the $x$ -intercept(s) of the graph visually. Then find the $x$ -intercept(s) algebraically to confirm your results.
(GRAPH CANNOT COPY)
$$y=2 x^{2}+5 x-3$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:54

Problem 50

Determine the $x$ -intercept(s) of the graph visually. Then find the $x$ -intercept(s) algebraically to confirm your results.
(GRAPH CANNOT COPY)
$$y=-2 x^{2}+5 x+3$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:16

Problem 51

Use a graphing utility to graph the quadratic function. Find the $x$ -intercept(s) of the graph and compare them with the solutions of the corresponding quadratic equation when $f(x)=0$.
$$f(x)=x^{2}-4 x$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:26

Problem 52

Use a graphing utility to graph the quadratic function. Find the $x$ -intercept(s) of the graph and compare them with the solutions of the corresponding quadratic equation when $f(x)=0$.
$$f(x)=-2 x^{2}+10 x$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:40

Problem 53

Use a graphing utility to graph the quadratic function. Find the $x$ -intercept(s) of the graph and compare them with the solutions of the corresponding quadratic equation when $f(x)=0$.
$$f(x)=x^{2}-9 x+18$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:28

Problem 54

Use a graphing utility to graph the quadratic function. Find the $x$ -intercept(s) of the graph and compare them with the solutions of the corresponding quadratic equation when $f(x)=0$.
$$f(x)=x^{2}-8 x-20$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:46

Problem 55

Use a graphing utility to graph the quadratic function. Find the $x$ -intercept(s) of the graph and compare them with the solutions of the corresponding quadratic equation when $f(x)=0$.
$$f(x)=2 x^{2}-7 x-30$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:38

Problem 56

Use a graphing utility to graph the quadratic function. Find the $x$ -intercept(s) of the graph and compare them with the solutions of the corresponding quadratic equation when $f(x)=0$.
$$f(x)=\frac{7}{10}\left(x^{2}+12 x-45\right)$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
02:08

Problem 57

Find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given $x$ -intercepts. (There are many correct answers.)
$$(-3,0),(3,0)$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:35

Problem 58

Find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given $x$ -intercepts. (There are many correct answers.)
$$(-5,0),(5,0)$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
02:07

Problem 59

Find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given $x$ -intercepts. (There are many correct answers.)
$$(-1,0),(4,0)$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:31

Problem 60

Find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given $x$ -intercepts. (There are many correct answers.)
$$(-2,0),(3,0)$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
02:58

Problem 61

Find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given $x$ -intercepts. (There are many correct answers.)
$$(-3,0),\left(-\frac{1}{2}, 0\right)$$

AG
Ankit Gupta
Numerade Educator
02:09

Problem 62

Find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given $x$ -intercepts. (There are many correct answers.)
$$\left(-\frac{3}{2}, 0\right),(-5,0)$$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:54

Problem 63

Find two positive real numbers whose product is a maximum.
The sum is 110 .

AG
Ankit Gupta
Numerade Educator
02:39

Problem 64

Find two positive real numbers whose product is a maximum.
The sum is $S$.

Vanessa Lamar
Vanessa Lamar
Numerade Educator
02:45

Problem 65

Find two positive real numbers whose product is a maximum.
The sum of the first and twice the second is 24

AG
Ankit Gupta
Numerade Educator
02:47

Problem 66

Find two positive real numbers whose product is a maximum.
The sum of the first and three times the second is 42.

AG
Ankit Gupta
Numerade Educator
02:20

Problem 67

The path of a diver is modeled by
$f(x)=-\frac{4}{9} x^{2}+\frac{24}{9} x+12$
where $f(x)$ is the height (in feet) and $x$ is the horizontal
distance (in feet) from the end of the diving board. What is the maximum
height of the diver?

Vanessa Lamar
Vanessa Lamar
Numerade Educator
View

Problem 68

The path of a punted football is modeled by
$f(x)=-\frac{16}{2025} x^{2}+\frac{9}{5} x+1.5$
where $f(x)$ is the height (in feet) and $x$ is the horizontal distance (in feet) from the point at which the ball is punted.
(a) How high is the ball when it is punted?
(b) What is the maximum height of the punt?
(c) How long is the punt?

Danielle Fairburn
Danielle Fairburn
Numerade Educator
01:41

Problem 69

A manufacturer of lighting fixtures has daily production costs of $C=800-10 x+0.25 x^{2}$ where $C$ is the total cost (in dollars) and $x$ is the number of units produced. What daily production number yields a minimum cost?

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:37

Problem 70

The profit $P$ (in hundreds of dollars) that a company makes depends on the amount $x$ (in hundreds of dollars) the company spends on advertising according to the model $P=230+20 x-0.5 x^{2} .$ What expenditure for advertising yields a maximum profit?

Vanessa Lamar
Vanessa Lamar
Numerade Educator
04:42

Problem 71

The total revenue $R$ earned (in thousands of dollars) from manufacturing handheld video games is given by $R(p)=-25 p^{2}+1200 p$ where $p$ is the price per unit (in dollars).
(a) Find the revenues when the prices per unit are $\$ 20$, $\$ 25,$ and $\$ 30$
(b) Find the unit price that yields a maximum revenue. What is the maximum revenue? Explain.

Vanessa Lamar
Vanessa Lamar
Numerade Educator
05:45

Problem 72

The total revenue $R$ earned per day (in dollars) from a pet-sitting service is given by $R(p)=-12 p^{2}+150 p,$ where $p$ is the price charged per pet (in dollars).
(a) Find the revenues when the prices per pet are $\$ 4$ $\$ 6,$ and $\$ 8$
(b) Find the unit price that yields a maximum revenue. What is the maximum revenue? Explain.

Vanessa Lamar
Vanessa Lamar
Numerade Educator
03:39

Problem 73

A rancher has 200 feet of fencing to enclose two adjacent rectangular corrals (see figure).
(FIGURE CANNOT COPY)
(a) Write the area $A$ of the corrals as a function of $x .$
(b) What dimensions produce a maximum enclosed area?

Nicole Smina
Nicole Smina
Numerade Educator
06:49

Problem 74

A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). The perimeter of the window is 16 feet.
(FIGURE CANNOT COPY)
(a) Write the area $A$ of the window as a function of $x .$
(b) What dimensions produce a window of maximum area?

JH
J Hardin
Numerade Educator
01:02

Problem 75

Determine whether the statement is true or false. Justify your answer.
The graph of $f(x)=-12 x^{2}-1$ has no $x$ -intercepts.

Vanessa Lamar
Vanessa Lamar
Numerade Educator
01:42

Problem 76

Determine whether the statement is true or false. Justify your answer.
The graphs of $f(x)=-4 x^{2}-10 x+7 \quad$ and $g(x)=12 x^{2}+30 x+1$ have the same axis of symmetry.

AG
Ankit Gupta
Numerade Educator
01:52

Problem 77

Find the values of $b$ such that the function has the given maximum or minimum value.
$f(x)=-x^{2}+b x-75 ;$ Maximum value: 25

Vanessa Lamar
Vanessa Lamar
Numerade Educator
02:04

Problem 78

Find the values of $b$ such that the function has the given maximum or minimum value.
$f(x)=x^{2}+b x-25 ;$ Minimum value: $-50$

Vanessa Lamar
Vanessa Lamar
Numerade Educator
03:45

Problem 79

Write the quadratic function
$f(x)=a x^{2}+b x+c$
in standard form to verify that the vertex occurs at
$\left(-\frac{b}{2 a}, f\left(-\frac{b}{2 a}\right)\right)$.

Vanessa Lamar
Vanessa Lamar
Numerade Educator
02:49

Problem 80

The graph shows a quadratic function of the form
$P(t)=a t^{2}+b t+c$
which represents the yearly profit for a company, where $P(t)$ is the profit in year $t$.\
(GRAPH CANNOT COPY)(a) Is the value of $a$ positive, negative, or zero? Explain.
(b) Write an expression in terms of $a$ and $b$ that represents the year $t$ when the company made the least profit.
(c) The company made the same yearly profits in 2008 and $2016 .$ Estimate the year in which the company made the least profit.

Vanessa Lamar
Vanessa Lamar
Numerade Educator
04:30

Problem 81

Assume that the function
$f(x)=a x^{2}+b x+c, a \neq 0$
has two real zeros. Prove that the $x$ -coordinate of the vertex of the graph is the average of the zeros of $f$ (Hint: Use the Quadratic Formula.)

Vanessa Lamar
Vanessa Lamar
Numerade Educator