Precalculus

Margaret L. Lial, John Hornsby, David I. Schneide

Chapter 3

Polynomial and Rational Functions - all with Video Answers

Educators

+ 1 more educators

Section 1

Quadratic Functions and Models

Problem 1

Fill in the blank(s) to correctly complete each sentence.
A polynomial function with leading term $3 x^{5}$ has degree ______________ .

Kian Manafi

Kian Manafi

Numerade Educator

Problem 2

Fill in the blank(s) to correctly complete each sentence.
The lowest point on the graph of a parabola that opens up is the ___________ of the parabola.

Kian Manafi

Numerade Educator

Problem 3

Fill in the blank(s) to correctly complete each sentence.
The highest point on the graph of a parabola that opens down is the _______________ of the parabola.

Kian Manafi

Numerade Educator

Problem 4

Fill in the blank(s) to correctly complete each sentence.
The axis of symmetry of the graph of $f(x)=2(x+4)^{2}-6$ has equation $x=$ ___________ .

Kian Manafi

Numerade Educator

Problem 5

The axis of symmetry of the graph of $f(x)=2(x+4)^{2}-6$ has equation $x=$
The vertex of the graph of $f(x)=x^{2}+2 x+4$ has $x$ -coordinate ______________ .

Kian Manafi

Numerade Educator

Problem 6

Fill in the blank(s) to correctly complete each sentence.
The graph of $f(x)=-2 x^{2}-6 x+5$ opens down with $y$ -intercept $(0,$ _____________ ), so it has $\underline{\text{(no/one/two)}}$ $x$ -intercept $(\mathrm{s})$.

Kian Manafi

Numerade Educator

Problem 7

Match each equation in Column I with the description of the parabola that is its graph in Column II.
$$\mathbf{I}$$
$$y=(x+4)^{2}+2$$
$$\mathbf{II}$$
A. vertex $(-2,4),$ opens up
B. vertex $(-2,4),$ opens down
C. vertex $(-4,2),$ opens up
D. vertex $(-4,2),$ opens down

Kian Manafi

Numerade Educator

Problem 8

Match each equation in Column I with the description of the parabola that is its graph in Column II.
$$\mathbf{I}$$
$$y=(x+2)^{2}+4$$
$$\mathbf{II}$$
A. vertex $(-2,4),$ opens up
B. vertex $(-2,4),$ opens down
C. vertex $(-4,2),$ opens up
D. vertex $(-4,2),$ opens down

Kian Manafi

Numerade Educator

Problem 9

Match each equation in Column I with the description of the parabola that is its graph in Column II.
$$\mathbf{I}$$
$$y=-(x+4)^{2}+2$$
$$\mathbf{II}$$
A. vertex $(-2,4),$ opens up
B. vertex $(-2,4),$ opens down
C. vertex $(-4,2),$ opens up
D. vertex $(-4,2),$ opens down

Kian Manafi

Numerade Educator

Problem 10

Match each equation in Column I with the description of the parabola that is its graph in Column II.
$$\mathbf{I}$$
$$y=-(x+2)^{2}+4$$
$$\mathbf{II}$$
A. vertex $(-2,4),$ opens up
B. vertex $(-2,4),$ opens down
C. vertex $(-4,2),$ opens up
D. vertex $(-4,2),$ opens down

Kian Manafi

Numerade Educator

Problem 11

Consider the graph of each quadratic function. Do the following.
(a) Give the domain and range.
(b) Give the coordinates of the vertex.
(c) Give the equation of the axis.
(d) Find the $y$ -intercept.
(e) Find the $x$ -intercepts.
(GRAPH CAN'T COPY)

Kian Manafi

Numerade Educator

Problem 12

Consider the graph of each quadratic function. Do the following.
(a) Give the domain and range.
(b) Give the coordinates of the vertex.
(c) Give the equation of the axis.
(d) Find the $y$ -intercept.
(e) Find the $x$ -intercepts.
(GRAPH CAN'T COPY)

Swati Agarwal

Numerade Educator

Problem 13

Consider the graph of each quadratic function. Do the following.
(a) Give the domain and range.
(b) Give the coordinates of the vertex.
(c) Give the equation of the axis.
(d) Find the $y$ -intercept.
(e) Find the $x$ -intercepts.
(GRAPH CAN'T COPY)

Swati Agarwal

Numerade Educator

Problem 14

Consider the graph of each quadratic function. Do the following.
(a) Give the domain and range.
(b) Give the coordinates of the vertex.
(c) Give the equation of the axis.
(d) Find the $y$ -intercept.
(e) Find the $x$ -intercepts.
(GRAPH CAN'T COPY)

Swati Agarwal

Numerade Educator

Problem 15

Match each function with its graph without actually entering it into a calculator. Then, after completing the exercises, check the answers with a calculator. Use the standard viewing window. (GRAPH CAN'T COPY)
$$f(x)=(x-4)^{2}-3$$

Kian Manafi

Numerade Educator

Problem 16

Match each function with its graph without actually entering it into a calculator. Then, after completing the exercises, check the answers with a calculator. Use the standard viewing window. (GRAPH CAN'T COPY)
$$f(x)=-(x-4)^{2}+3$$

Kian Manafi

Numerade Educator

Problem 17

Match each function with its graph without actually entering it into a calculator. Then, after completing the exercises, check the answers with a calculator. Use the standard viewing window. (GRAPH CAN'T COPY)
$$f(x)=(x+4)^{2}-3$$

Kian Manafi

Numerade Educator

Problem 18

Match each function with its graph without actually entering it into a calculator. Then, after completing the exercises, check the answers with a calculator. Use the standard viewing window. (GRAPH CAN'T COPY)
$$f(x)=-(x+4)^{2}+3$$

Kian Manafi

Numerade Educator

Problem 19

Graph the following on the same coordinate system.
(a) $y=x^{2}$
(b) $y=3 x^{2}$
(c) $y=\frac{1}{3} x^{2}$
(d) How does the coefficient of $x^{2}$ affect the shape of the graph?

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator

Problem 20

Graph the following on the same coordinate system.
(a) $y=x^{2}$
(b) $y=x^{2}-2$
(c) $y=x^{2}+2$
(d) How do the graphs in parts (b) and (c) differ from the graph of $y=x^{2} ?$

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator

Problem 21

Graph the following on the same coordinate system.
(a) $y=(x-2)^{2}$
(b) $y=(x+1)^{2}$
(c) $y=(x+3)^{2}$
(d) How do these graphs differ from the graph of $y=x^{2} ?$

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator

Problem 22

A quadratic function $f(x)$ has vertex $(0,0),$ and all of its intercepts are the same point. What is the general form of its equation?

Kian Manafi

Numerade Educator

Problem 23

Graph each quadratic function. Give the ( $a$ ) vertex, ( $b$ ) axis, ( $c$ ) domain, and ( $d$ ) range.
Then determine ( $e$ ) the largest open interval of the domain over which the function is increasing and ( $f$ ) the largest open interval over which the function is decreasing.
$$f(x)=(x-2)^{2}$$

Adam Dehollander

Adam Dehollander

Numerade Educator

Problem 24

Graph each quadratic function. Give the ( $a$ ) vertex, ( $b$ ) axis, ( $c$ ) domain, and ( $d$ ) range.
Then determine ( $e$ ) the largest open interval of the domain over which the function is increasing and ( $f$ ) the largest open interval over which the function is decreasing.
$$f(x)=(x+4)^{2}$$

Adam Dehollander

Adam Dehollander

Numerade Educator

Problem 25

Graph each quadratic function. Give the ( $a$ ) vertex, ( $b$ ) axis, ( $c$ ) domain, and ( $d$ ) range.
Then determine ( $e$ ) the largest open interval of the domain over which the function is increasing and ( $f$ ) the largest open interval over which the function is decreasing.
$$f(x)=(x+3)^{2}-4$$

Adam Dehollander

Adam Dehollander

Numerade Educator

Problem 26

Graph each quadratic function. Give the ( $a$ ) vertex, ( $b$ ) axis, ( $c$ ) domain, and ( $d$ ) range.
Then determine ( $e$ ) the largest open interval of the domain over which the function is increasing and ( $f$ ) the largest open interval over which the function is decreasing.
$$f(x)=(x-5)^{2}-4$$

Adam Dehollander

Adam Dehollander

Numerade Educator

Problem 27

Graph each quadratic function. Give the ( $a$ ) vertex, ( $b$ ) axis, ( $c$ ) domain, and ( $d$ ) range.
Then determine ( $e$ ) the largest open interval of the domain over which the function is increasing and ( $f$ ) the largest open interval over which the function is decreasing.
$$f(x)=-\frac{1}{2}(x+1)^{2}-3$$

Adam Dehollander

Adam Dehollander

Numerade Educator

Problem 28

Graph each quadratic function. Give the ( $a$ ) vertex, ( $b$ ) axis, ( $c$ ) domain, and ( $d$ ) range.
Then determine ( $e$ ) the largest open interval of the domain over which the function is increasing and ( $f$ ) the largest open interval over which the function is decreasing.
$$f(x)=-3(x-2)^{2}+1$$

Adam Dehollander

Adam Dehollander

Numerade Educator

Problem 29

Graph each quadratic function. Give the ( $a$ ) vertex, ( $b$ ) axis, ( $c$ ) domain, and ( $d$ ) range.
Then determine ( $e$ ) the largest open interval of the domain over which the function is increasing and ( $f$ ) the largest open interval over which the function is decreasing.
$$f(x)=x^{2}-2 x+3$$

Adam Dehollander

Adam Dehollander

Numerade Educator

Problem 30

Graph each quadratic function. Give the ( $a$ ) vertex, ( $b$ ) axis, ( $c$ ) domain, and ( $d$ ) range.
Then determine ( $e$ ) the largest open interval of the domain over which the function is increasing and ( $f$ ) the largest open interval over which the function is decreasing.
$$f(x)=x^{2}+6 x+5$$

Adam Dehollander

Adam Dehollander

Numerade Educator

Problem 31

Graph each quadratic function. Give the ( $a$ ) vertex, ( $b$ ) axis, ( $c$ ) domain, and ( $d$ ) range.
Then determine ( $e$ ) the largest open interval of the domain over which the function is increasing and ( $f$ ) the largest open interval over which the function is decreasing.
$$f(x)=x^{2}-10 x+21$$

Adam Dehollander

Adam Dehollander

Numerade Educator

Problem 32

Graph each quadratic function. Give the ( $a$ ) vertex, ( $b$ ) axis, ( $c$ ) domain, and ( $d$ ) range.
Then determine ( $e$ ) the largest open interval of the domain over which the function is increasing and ( $f$ ) the largest open interval over which the function is decreasing.
$$f(x)=2 x^{2}-4 x+5$$

Adam Dehollander

Adam Dehollander

Numerade Educator

Problem 33

Graph each quadratic function. Give the ( $a$ ) vertex, ( $b$ ) axis, ( $c$ ) domain, and ( $d$ ) range.
Then determine ( $e$ ) the largest open interval of the domain over which the function is increasing and ( $f$ ) the largest open interval over which the function is decreasing.
$$f(x)=-2 x^{2}-12 x-16$$

Adam Dehollander

Adam Dehollander

Numerade Educator

Problem 34

Graph each quadratic function. Give the ( $a$ ) vertex, ( $b$ ) axis, ( $c$ ) domain, and ( $d$ ) range.
Then determine ( $e$ ) the largest open interval of the domain over which the function is increasing and ( $f$ ) the largest open interval over which the function is decreasing.
$$f(x)=-3 x^{2}+24 x-46$$

Adam Dehollander

Adam Dehollander

Numerade Educator

Problem 35

Graph each quadratic function. Give the ( $a$ ) vertex, ( $b$ ) axis, ( $c$ ) domain, and ( $d$ ) range.
Then determine ( $e$ ) the largest open interval of the domain over which the function is increasing and ( $f$ ) the largest open interval over which the function is decreasing.
$$f(x)=-\frac{1}{2} x^{2}-3 x-\frac{1}{2}$$

Adam Dehollander

Adam Dehollander

Numerade Educator

Problem 36

Graph each quadratic function. Give the ( $a$ ) vertex, ( $b$ ) axis, ( $c$ ) domain, and ( $d$ ) range.
Then determine ( $e$ ) the largest open interval of the domain over which the function is increasing and ( $f$ ) the largest open interval over which the function is decreasing.
$$f(x)=\frac{2}{3} x^{2}-\frac{8}{3} x+\frac{5}{3}$$

Adam Dehollander

Adam Dehollander

Numerade Educator

Problem 37

The figure shows the graph of a quadratic function $y=f(x) .$ Use it to answer each question. (FIGURE CAN'T COPY)
What is the minimum value of $f(x) ?$

Swati Agarwal

Numerade Educator

Problem 38

The figure shows the graph of a quadratic function $y=f(x) .$ Use it to answer each question. (FIGURE CAN'T COPY)
For what value of $x$ is $f(x)$ as small as possible?

Swati Agarwal

Numerade Educator

Problem 39

The figure shows the graph of a quadratic function $y=f(x) .$ Use it to answer each question. (FIGURE CAN'T COPY)
How many real solutions are there to the equation $f(x)=1 ?$

Allison Knapp

Numerade Educator

Problem 40

The figure shows the graph of a quadratic function $y=f(x) .$ Use it to answer each question. (FIGURE CAN'T COPY)
How many real solutions are there to the equation $f(x)=4 ?$

Allison Knapp

Numerade Educator

Problem 41

Several graphs of the quadratic function $$f(x)=a x^{2}+b x+c$$ are shown below. For the given restrictions on $a,$ $b,$ and $c,$ select the corresponding graph from choices $A-F$. (GRAPH A-F CAN'T COPY)
$$a<0 ; b^{2}-4 a c=0$$

Kian Manafi

Numerade Educator

Problem 42

Several graphs of the quadratic function $$f(x)=a x^{2}+b x+c$$ are shown below. For the given restrictions on $a,$ $b,$ and $c,$ select the corresponding graph from choices $A-F$. (GRAPH A-F CAN'T COPY)
$$a>0 ; b^{2}-4 a c<0$$

Kian Manafi

Numerade Educator

Problem 43

Several graphs of the quadratic function $$f(x)=a x^{2}+b x+c$$ are shown below. For the given restrictions on $a,$ $b,$ and $c,$ select the corresponding graph from choices $A-F$. (GRAPH A-F CAN'T COPY)
$$a<0 ; b^{2}-4 a c<0$$

Kian Manafi

Numerade Educator

Problem 44

Several graphs of the quadratic function $$f(x)=a x^{2}+b x+c$$ are shown below. For the given restrictions on $a,$ $b,$ and $c,$ select the corresponding graph from choices $A-F$. (GRAPH A-F CAN'T COPY)
$$a<0 ; b^{2}-4 a c>0$$

Kian Manafi

Numerade Educator

Problem 45

Several graphs of the quadratic function $$f(x)=a x^{2}+b x+c$$ are shown below. For the given restrictions on $a,$ $b,$ and $c,$ select the corresponding graph from choices $A-F$. (GRAPH A-F CAN'T COPY)
$$a>0 ; b^{2}-4 a c>0$$

Kian Manafi

Numerade Educator

Problem 46

Several graphs of the quadratic function $$f(x)=a x^{2}+b x+c$$ are shown below. For the given restrictions on $a,$ $b,$ and $c,$ select the corresponding graph from choices $A-F$. (GRAPH A-F CAN'T COPY)
$$a>0 ; b^{2}-4 a c=0$$

Kian Manafi

Numerade Educator

Problem 47

Find a quadratic function $f$ having the graph shown. (GRAPH CAN'T COPY)

Kian Manafi

Numerade Educator

Problem 48

Find a quadratic function $f$ having the graph shown. (GRAPH CAN'T COPY)

Kian Manafi

Numerade Educator

Problem 49

Find a quadratic function $f$ having the graph shown. (GRAPH CAN'T COPY)

Kian Manafi

Numerade Educator

Problem 50

Find a quadratic function $f$ having the graph shown. (GRAPH CAN'T COPY)

Kian Manafi

Numerade Educator

Problem 51

In each scatter diagram, tell whether a linear or a quadratic model is appropriate for the data. If linear, tell whether the slope should be positive or negative. If quadratic, tell whether the leading coefficient of $x^{2}$ should be positive or negative. (FIGURE CAN'T COPY).
number of shopping centers as a function of time

Kian Manafi

Numerade Educator

Problem 52

In each scatter diagram, tell whether a linear or a quadratic model is appropriate for the data. If linear, tell whether the slope should be positive or negative. If quadratic, tell whether the leading coefficient of $x^{2}$ should be positive or negative. (FIGURE CAN'T COPY)
growth in science centers/museums as a function of time

Kian Manafi

Numerade Educator

Problem 53

In each scatter diagram, tell whether a linear or a quadratic model is appropriate for the data. If linear, tell whether the slope should be positive or negative. If quadratic, tell whether the leading coefficient of $x^{2}$ should be positive or negative. (FIGURE CAN'T COPY)
value of U.S. salmon catch as a function of time

Kian Manafi

Numerade Educator

Problem 54

In each scatter diagram, tell whether a linear or a quadratic model is appropriate for the data. If linear, tell whether the slope should be positive or negative. If quadratic, tell whether the leading coefficient of $x^{2}$ should be positive or negative. (FIGURE CAN'T COPY).
height of an object projected upward as a function of time

Kian Manafi

Numerade Educator

Problem 55

In each scatter diagram, tell whether a linear or a quadratic model is appropriate for the data. If linear, tell whether the slope should be positive or negative. If quadratic, tell whether the leading coefficient of $x^{2}$ should be positive or negative. (FIGURE CAN'T COPY).
Social Security assets as a function of time

Kian Manafi

Numerade Educator

Problem 56

In each scatter diagram, tell whether a linear or a quadratic model is appropriate for the data. If linear, tell whether the slope should be positive or negative. If quadratic, tell whether the leading coefficient of $x^{2}$ should be positive or negative. (FIGURE CAN'T COPY).
newborns with AIDS as a function of time

Kian Manafi

Numerade Educator

Problem 57

Solve each problem. Give approximations to the nearest hundredth.
A toy rocket (not internally powered) is launched straight up from the top of a building 50 ft tall at an initial velocity of 200 ft per sec.
(a) Give the function that describes the height of the rocket in terms of time $t$.
(b) Determine the time at which the rocket reaches its maximum height and the maximum height in feet.
(c) For what time interval will the rocket be more than 300 ft above ground level?
(d) After how many seconds will it hit the ground?

Swati Agarwal

Numerade Educator

Problem 58

Solve each problem. Give approximations to the nearest hundredth.
A rock is projected directly upward from ground level with an initial velocity of 90 ft per sec.
(a) Give the function that describes the height of the rock in terms of time $t$.
(b) Determine the time at which the rock reaches its maximum height and the maximum height in feet.
(c) For what time interval will the rock be more than $120 \mathrm{ft}$ above ground level?
(d) After how many seconds will it hit the ground?

Allison Knapp

Numerade Educator

Problem 59

Solve each problem. Give approximations to the nearest hundredth.
One campus of Houston Community College has plans to construct a rectangular parking lot on land bordered on one side by a highway. There are 640 ft of fencing available to fence the other three sides. Let $x$ represent the length of each of the two parallel sides of fencing.
(a) Express the length of the remaining side to be fenced in terms of $x .$
(b) What are the restrictions on $x ?$
(c) Determine a function $\mathscr{A}$ that represents the area of the parking lot in terms of $x .$
(d) Determine the values of $x$ that will give an area between $30,000$ and $40,000 \mathrm{ft}^{2}$.
(e) What dimensions will give a maximum area, and what will this area be?

Swati Agarwal

Numerade Educator

Problem 60

Solve each problem. Give approximations to the nearest hundredth.
A farmer wishes to enclose a rectangular region bordering a river with fencing, as shown in the diagram. Suppose that $x$ represents the length of each of the three parallel pieces of fencing. She has $600 \mathrm{ft}$ of fencing available. (FIGURE CAN'T COPY)
(a) What is the length of the remaining piece of fencing in terms of $x ?$
(b) Determine a function $\mathscr{A}$ that represents the total area of the enclosed region. Give any restrictions on $x$.
(c) What dimensions for the total enclosed region would give an area of $22,500 \mathrm{ft}^{2} ?$
(d) What is the maximum area that can be enclosed?

Stanley Enemuo

Numerade Educator

Problem 61

Solve each problem. Give approximations to the nearest hundredth.
A piece of cardboard is twice as long as it is wide. It is to be made into a box with an open top by cutting 2 -in. squares from each corner and folding up the sides. Let $x$ represent the width (in inches) of the original piece of cardboard.
(a) Represent the length of the original piece of cardboard in terms of $x .$
(b) What will be the dimensions of the bottom rectangular base of the box? Give the restrictions on $x$.
(c) Determine a function $V$ that represents the volume of the box in terms of $x .$
(d) For what dimensions of the bottom of the box will the volume be 320 in. $^{3} ?$
(e) Find the values of $x$ if such a box is to have a volume between 400 and 500 in. $^{3}$.

David Mccaslin

Numerade Educator

Problem 62

Solve each problem. Give approximations to the nearest hundredth.
A piece of sheet metal is 2.5 times as long as it is wide. It is to be made into a box with an open top by cutting 3 -in. squares from each corner and folding up the sides. Let $x$ represent the width (in inches) of the original piece.
(a) Represent the length of the original piece of sheet metal in terms of $x .$
(b) What are the restrictions on $x ?$
(c) Determine a function $V$ that represents the volume of the box in terms of $x .$
(d) For what values of $x$ (that is, original widths) will the volume of the box be between 600 and 800 in. $^{3} ?$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 63

Solve each problem. Give approximations to the nearest hundredth.
If a person shoots a free throw from a position 8 ft above the floor, then the path of the ball may be modeled by the parabola $$y=\frac{-16 x^{2}}{0.434 v^{2}}+1.15 x+8,$$ where $v$ is the initial velocity of the ball in feet per second, as illustrated in the figure. (FIGURE CAN'T COPY)
(a) If the basketball hoop is $10 \mathrm{ft}$ high and located $15 \mathrm{ft}$ away, what initial velocity $v$ should the basketball have?
(b) What is the maximum height of the basketball?

Swati Agarwal

Numerade Educator

Problem 64

Solve each problem. Give approximations to the nearest hundredth.
See Exercise $63 .$ If a person shoots a free throw from an underhand position 3 ft above the floor, then the path of the ball may be modeled by $$y=\frac{-16 x^{2}}{0.117 v^{2}}+2.75 x+3.$$ Repeat parts (a) and (b) from Exercise $63 .$ Then compare the paths for the overhand shot and the underhand shot.

Allison Knapp

Numerade Educator

Problem 65

Solve each problem. Give approximations to the nearest hundredth.
Sum and Product of Two Numbers Find two numbers whose sum is 20 and whose product is the maximum possible value.

Kian Manafi

Numerade Educator

Problem 66

Solve each problem. Give approximations to the nearest hundredth.
Find two numbers whose sum is 32 and whose product is the maximum possible value.

Kian Manafi

Numerade Educator

Problem 67

Solve each problem. Give approximations to the nearest hundredth.
If an object is projected upward from ground level with an initial velocity of 32 ft per sec, then its height in feet after $t$ seconds is given by $$s(t)=-16 t^{2}+32 t.$$ Find the number of seconds it will take the object to reach its maximum height. What is this maximum height?

Kian Manafi

Numerade Educator

Problem 68

Solve each problem. Give approximations to the nearest hundredth.
If an object is projected upward from an initial height of $100 \mathrm{ft}$ with an initial velocity of $64 \mathrm{ft}$ per sec, then its height in feet after $t$ seconds is given by $$s(t)=-16 t^{2}+64 t+100.$$ Find the number of seconds it will take the object to reach its maximum height. What is this maximum height?

Kian Manafi

Numerade Educator

Problem 69

Solve each problem.
The average price in dollars of a pound of chocolate chip cookies from 2002 to 2013 is shown in the table.
$$\begin{array}{|c|c|}\hline\text { Year } & \text { Price per Pound } \\\hline 2002 & 2.59 \\\hline 2003 & 2.81 \\\hline 2004 & 2.65 \\\hline 2005 & 2.67 \\\hline 2006 & 2.88 \\\hline 2007 & 2.70 \\\hline\end{array}$$ $$\begin{array}{|c|c|}\hline \text { Year } & \text { Price per Pound } \\\hline 2008 & 2.88 \\\hline 2009 & 3.17 \\\hline 2010 & 3.25 \\\hline 2011 & 3.35 \\\hline 2012 & 3.62 \\\hline 2013 & 3.64 \\\hline\end{array}$$
The data are modeled by the quadratic function $$f(x)=0.0095 x^{2}-0.0076 x+2.660,$$ where $x=0$ corresponds to 2002 and $f(x)$ is the price in dollars. If this model continues to apply, what will it predict for the price of a pound of chocolate chip cookies
in $2018 ?$

Kian Manafi

Numerade Educator

Problem 70

Solve each problem.
The quadratic function $$f(x)=0.0118 x^{2}+0.8633 x+317$$ models the worldwide atmospheric concentration of carbon dioxide in parts per million (ppm) over the period $1960-2013$, where $x=0$ represents the year 1960 . If this model continues to apply, what will be the atmospheric $\mathrm{CO}_{2}$ concentration in $2020 ?$ Round to the nearest unit.

Kian Manafi

Numerade Educator

Problem 71

Solve each problem.
The total amount spent by Americans on shoes and clothing from 2000 to 2013 can be modeled by $$f(x)=0.7714 x^{2}-3.693 x+297.9,$$ where $x=0$ represents 2000 and $f(x)$ is in billions of dollars. Based on this model, in what year did spending on shoes and clothing reach a minimum?

Kian Manafi

Numerade Educator

Problem 72

Solve each problem.
According to data from the National Highway Traffic Safety Administration, the accident rate as a function of the age of the driver in years $x$ can be approximated by the function $$f(x)=0.0232 x^{2}-2.28 x+60.0, \quad \text { for } 16 \leq x \leq 85.$$ Find both the age at which the accident rate is a minimum and the minimum rate to the nearest hundredth.

Kian Manafi

Numerade Educator

Problem 73

Solve each problem.
The table lists total fall enrollments in degree-granting postsecondary colleges in the United States for selected years. $$\begin{array}{|c|c|}\hline \text {Year} & \begin{array}{c}\text { Enrollment } \\\text { (in millions) }\end{array} \\\hline 2008 & 19.1 \\\hline 2009 & 20.4 \\\hline 2010 & 21.0 \\\hline 2011 & 21.0 \\\hline 2012 & 20.6 \\\hline\end{array}$$
(a) Plot the data. Let $x=0$ correspond to the year 2008 .
(b) Find a quadratic function $f(x)=a x^{2}+b x+c$ that models the data.
(c) Plot the data together with $f$ in the same window. How well does $f$ model enrollment?
(d) Use $f$ to estimate total enrollment in 2013 to the nearest tenth of a million.

Joseph Palsic

Numerade Educator

Problem 74

Solve each problem.
The table lists total fall enrollments in degree-granting two-year colleges in the United States for selected years.
Solve each problem.$$\begin{array}{|c|c|}\hline\hline \text { Year } & \begin{array}{c}\text { Enrollment } \\\text { (in millions) }\end{array} \\\hline 2008 & 6.9 \\\hline 2009 & 7.5 \\\hline 2010 & 7.7 \\\hline 2011 & 7.5 \\\hline 2012 & 7.2\end{array}$$
(a) Plot the data. Let $x=0$ correspond to the year 2008 .
(b) Find a quadratic function $g(x)=a x^{2}+b x+c$ that models the data.
(c) Plot the data together with $g$ in the same window. How well does $g$ model enrollment?
(d) Use $g$ to estimate total enrollment in 2013 to the nearest tenth of a million.

Stanley Enemuo

Numerade Educator

Problem 75

Solve each problem.
The table lists the percent of the U.S. population that was foreign-born for selected years. $$\begin{array}{|c|c|}\hline \text { Year } & \text { Percent } \\\hline 1930 & 11.6 \\\hline 1940 & 8.8 \\\hline 1950 & 6.9 \\\hline 1960 & 5.4 \\\hline 1970 & 4.7 \\\hline 1980 & 6.2 \\\hline 1990 & 7.9 \\\hline 2000 & 11.1 \\\hline 2010 & 12.4 \\\hline 2012 & 12.9 \\\hline\end{array}$$
(a) Plot the data. Let $x=0$ correspond to the year 1930 , $x=10$ correspond to $1940,$ and so on.
(b) Find a quadratic function $f(x)=a(x-h)^{2}+k$ that models the data. Use $(40,4.7)$ as the vertex and $(20,6.9)$ as the other point to determine $a$.
(c) Plot the data together with $f$ in the same window. How well does $f$ model the percent of the U.S. population that is foreign-born?
(d) Use the quadratic regression feature of a graphing calculator to determine the quadratic function $g$ that provides the best fit for the data.
(e) Use functions $f$ and $g$ to predict the percent, to the nearest tenth, of the U.S. population in 2019 that will be foreign-born.

Stanley Enemuo

Numerade Educator

Problem 76

Solve each problem.
Selected values of the stopping distance $y,$ in feet, of a car traveling $x$ miles per hour are given in the table.
$$\begin{array}{|c|c|}\hline\hline\begin{array}{c}\text { Speed } \\\text { (in mph) }\end{array} & \begin{array}{c}\text { Stopping Distance } \\\text { (in feet) }\end{array} \\\hline 20 & 46 \\\hline 30 & 87 \\\hline 40 & 140 \\\hline 50 & 240 \\\hline 60 & 282 \\\hline 70 & 371\end{array}$$
(a) Plot the data.
(b) The quadratic function $$f(x)=0.056057 x^{2}+1.06657 x$$ is one model that has been used to approximate stopping distances. Find $f(45)$ to the nearest foot, and interpret this result.
(c) How well does $f$ model the car's stopping distance?

Stanley Enemuo

Numerade Educator

Problem 77

Work each problem.
Find a value of $c$ so that $y=x^{2}-10 x+c$ has exactly one $x$ -intercept.

Kian Manafi

Numerade Educator

Problem 78

Work each problem.
For what values of $a$ does $y=a x^{2}-8 x+4$ have no $x$ -intercepts?

Kian Manafi

Numerade Educator

Problem 79

Work each problem.
Define the quadratic function $f$ having $x$ -intercepts $(2,0)$ and $(5,0)$ and $y$ -intercept $(0,5)$.

Kian Manafi

Numerade Educator

Problem 80

Work each problem.
Define the quadratic function $f$ having $x$ -intercepts $(1,0)$ and $(-2,0)$ and $y$ -intercept $(0,4)$.

Kian Manafi

Numerade Educator

Problem 81

Work each problem.
The distance between the two points $P\left(x_{1}, y_{1}\right)$ and $R\left(x_{2}, y_{2}\right)$ is $$d(P, R)=\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}} . \text { Distance formula }$$ Find the closest point on the line $y=2 x$ to the point $(1,7)$.

Allison Knapp

Numerade Educator

Problem 82

Work each problem.
A quadratic equation $f(x)=0$ has a solution $x=2 .$ Its graph has vertex $(5,3)$. What is the other solution of the equation?

Kian Manafi

Numerade Educator

Problem 83

A quadratic inequality such as $$x^{2}+2 x-8<0$$ can be solved by first solving the related quadratic equation $$x^{2}+2 x-8=0,$$ identifying intervals determined by the solutions of this equation, and then using a test value from each interval to determine which intervals form the solution set. Work Exercises in order to learn a graphical method of solving inequalities.
$$\text { Graph } f(x)=x^{2}+2 x-8.$$

Allison Knapp

Numerade Educator

Problem 84

A quadratic inequality such as $$x^{2}+2 x-8<0$$ can be solved by first solving the related quadratic equation $$x^{2}+2 x-8=0,$$ identifying intervals determined by the solutions of this equation, and then using a test value from each interval to determine which intervals form the solution set. Work Exercises in order to learn a graphical method of solving inequalities.
The real solutions of $x^{2}+2 x-8=0$ are the $x$ -values of the $x$ -intercepts of the graph in Exercise $83 .$ These are values of $x$ for which $f(x)=0 .$ What are these values? What is the solution set of this equation?

Allison Knapp

Numerade Educator

Problem 85

A quadratic inequality such as $$x^{2}+2 x-8<0$$ can be solved by first solving the related quadratic equation $$x^{2}+2 x-8=0,$$ identifying intervals determined by the solutions of this equation, and then using a test value from each interval to determine which intervals form the solution set. Work Exercises in order to learn a graphical method of solving inequalities.
The real solutions of $x^{2}+2 x-8<0$ are the $x$ -values for which the graph in Exercise 83 lies below the $x$ -axis. These are values of $x$ for which $f(x)<0$ is true. What interval of $x$ -values represents the solution set of this inequality?

Allison Knapp

Numerade Educator

Problem 86

A quadratic inequality such as $$x^{2}+2 x-8<0$$ can be solved by first solving the related quadratic equation $$x^{2}+2 x-8=0,$$ identifying intervals determined by the solutions of this equation, and then using a test value from each interval to determine which intervals form the solution set. Work Exercises in order to learn a graphical method of solving inequalities.
The real solutions of $x^{2}+2 x-8>0$ are the $x$ -values for which the graph in Exercise 83 lies above the $x$ -axis. These are values of $x$ for which $f(x)>0$ is true. What intervals of $x$ -values represent the solution set of this inequality?

Allison Knapp

Numerade Educator

Problem 87

Use the technique described in Exercises $83-86$ to solve each inequality. Write the solution set in interval notation.
$$x^{2}-x-6<0$$

Kian Manafi

Numerade Educator

Problem 88

Use the technique described in Exercises $83-86$ to solve each inequality. Write the solution set in interval notation.
$$x^{2}-9 x+20<0$$

Kian Manafi

Numerade Educator

Problem 89

Use the technique described in Exercises $83-86$ to solve each inequality. Write the solution set in interval notation.
$$2 x^{2}-9 x \geq 18$$

Kian Manafi

Numerade Educator

Problem 90

Use the technique described in Exercises $83-86$ to solve each inequality. Write the solution set in interval notation.
$$3 x^{2}+x \geq 4$$

Kian Manafi

Numerade Educator

Problem 91

Use the technique described in Exercises $83-86$ to solve each inequality. Write the solution set in interval notation.
$$-x^{2}+4 x+1 \geq 0$$

Allison Knapp

Numerade Educator

Problem 92

Use the technique described in Exercises $83-86$ to solve each inequality. Write the solution set in interval notation.
$$-x^{2}+2 x+6>0$$

Allison Knapp

Numerade Educator