Algebra and Trigonometry Real Mathematics, Real People

Ron Larson

Chapter 1

Functions and Their Graphs - all with Video Answers

Educators

+ 2 more educators

Section 1

Graphs of Equations

Problem 1

For an equation in $x$ and $y,$ if substitution of $a$ for $x$ and $b$ for $y$ satisfies the equation, then the point $(a, b)$ is a ________.

Jennifer Southers

Jennifer Southers

Numerade Educator

Problem 2

The set of all solution points of an equation is the ___________ of the equation.

Sirat Shah

Numerade Educator

Problem 3

Name three common approaches you can use to solve problems mathematically.

AG

Ankit Gupta

Numerade Educator

Problem 4

List the steps for sketching the graph of an equation by point plotting.

AG

Ankit Gupta

Numerade Educator

Problem 5

$$\begin{array}{cc}\text{Equation} && \text{Points} \\ y=\sqrt{x+4} && (a) (0,2)\quad(b) (12,4) \end{array}$$

Erika Bustos

Numerade Educator

Problem 6

Determine whether each point lies on the graph of the equation.
$$\begin{array}{cc}\text{Equation} && \text{Points} \\ y=x^{2}-3 x+2 && (a) (2,0)\quad (b) (-2,8) \end{array}$$

DM

Dominique Madrid

Numerade Educator

Problem 7

Determine whether each point lies on the graph of the equation.
$$\begin{array}{cc}\text{Equation} && \text{Points} \\ y=4-|x-2| && (a) (2,0)\quad (b) (1.2,3.2)\end{array}$$

Erika Bustos

Numerade Educator

Problem 8

Determine whether each point lies on the graph of the equation.
$$\begin{array}{cc}\text{Equation} && \text{Points} \\ 2 x-y-3=0 && (a) (1,2)\quad (b) (1,-1)\end{array}$$

AG

Ankit Gupta

Numerade Educator

Problem 9

Determine whether each point lies on the graph of the equation.
$$\begin{array}{cc}\text{Equation} && \text{Points} \\ x^{2}+y^{2}=20 && (a) (3,-2)\quad (b) (-4,2)\end{array}$$

AG

Ankit Gupta

Numerade Educator

Problem 10

Determine whether each point lies on the graph of the equation.
$$\begin{array}{cc}\text{Equation} && \text{Points} \\ y=\frac{1}{3} x^{3}-2 x^{2} && \text { (a) }\left(2,-\frac{16}{3}\right)\quad (b) (-3,9)\end{array}$$

Erika Bustos

Numerade Educator

Problem 11

Complete the table. Use the resulting solution points to sketch the graph of the equation. Use a graphing utility to verify the graph.
$$3 x-2 y=2$$ $$\begin{array}{|l|l|l|l|l|l|}\hline x & -2 & 0 & \frac{2}{3} & 1 & 2 \\\hline y & & & & & \\\hline \text { Solution point } & & & & & \\\hline
\end{array}$$

Erika Bustos

Numerade Educator

Problem 12

Complete the table. Use the resulting solution points to sketch the graph of the equation. Use a graphing utility to verify the graph.
$$8 x+4 y=24$$ $$\begin{array}{|l|l|l|l|l|l|}\hline x & -3 & -1 & 0 & 2 & 3 \\\hline y & & & & & \\\hline \text { Solution point } & & & & & \\\hline
\end{array}$$

Erika Bustos

Numerade Educator

Problem 13

Complete the table. Use the resulting solution points to sketch the graph of the equation. Use a graphing utility to verify the graph.
$$2 x+y=x^{2}$$ $$\begin{array}{|l|l|l|l|l|l|}\hline x & -1 & 0 & 1 & 2 & 3 \\\hline y & & & & & \\\hline \text { Solution point } & & & & & \\\hline
\end{array}$$

Erika Bustos

Numerade Educator

Problem 14

Complete the table. Use the resulting solution points to sketch the graph of the equation. Use a graphing utility to verify the graph.
$$6 x-2 y=-2 x^{2}$$ $$\begin{array}{|l|l|l|l|l|l|}\hline x & -4 & -3 & -2 & 0 & 1 \\\hline y & & & & & \\\hline \text { Solution point } & & & & & \\\hline
\end{array}$$

Jennifer Southers

Jennifer Southers

Numerade Educator

Problem 15

Match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]
$$y=2 \sqrt{x}$$

Erika Bustos

Numerade Educator

Problem 16

Match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]
$$y=4-x^{2}$$

Erika Bustos

Numerade Educator

Problem 17

Match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]
$$y=\sqrt{9-x^{2}}$$

JC

James Choi

Numerade Educator

Problem 18

Match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]
$$y=|x|-3$$

Erika Bustos

Numerade Educator

Problem 19

Sketch the graph of the equation.
$$y=-4 x+1$$

Erika Bustos

Numerade Educator

Problem 20

Sketch the graph of the equation.
$$y=2 x-3$$

Erika Bustos

Numerade Educator

Problem 21

Sketch the graph of the equation.
$$y=2-x^{2}$$

AG

Ankit Gupta

Numerade Educator

Problem 22

Sketch the graph of the equation.
$$y=x^{2}-1$$

AG

Ankit Gupta

Numerade Educator

Problem 23

Sketch the graph of the equation.
$$y=x^{2}-3 x$$

Erika Bustos

Numerade Educator

Problem 24

Sketch the graph of the equation.
$$y=-x^{2}-4 x$$

Erika Bustos

Numerade Educator

Problem 25

Sketch the graph of the equation.
$$y=x^{3}+2$$

Erika Bustos

Numerade Educator

Problem 26

Sketch the graph of the equation.
$$y=x^{3}-3$$

AG

Ankit Gupta

Numerade Educator

Problem 27

Sketch the graph of the equation.
$$y=\sqrt{x-3}$$

AG

Ankit Gupta

Numerade Educator

Problem 28

Sketch the graph of the equation.
$$y=\sqrt{1-x}$$

AG

Ankit Gupta

Numerade Educator

Problem 29

Sketch the graph of the equation.
$$y=|x-2|$$

AG

Ankit Gupta

Numerade Educator

Problem 30

Sketch the graph of the equation.
$$y=4-|x|$$

AG

Ankit Gupta

Numerade Educator

Problem 31

Sketch the graph of the equation.
$$x=y^{2}-1$$

AG

Ankit Gupta

Numerade Educator

Problem 32

Sketch the graph of the equation.
$$x=y^{2}+4$$

AG

Ankit Gupta

Numerade Educator

Problem 33

Use a graphing uti equation. Use a standard viewing window. Approximate any $x$ - or $y$ -intercepts of the graph.
$$y=x-7$$

Jennifer Southers

Jennifer Southers

Numerade Educator

Problem 34

Use a graphing uti equation. Use a standard viewing window. Approximate any $x$ - or $y$ -intercepts of the graph.
$$y=x+1$$

Jennifer Southers

Jennifer Southers

Numerade Educator

Problem 35

Use a graphing uti equation. Use a standard viewing window. Approximate any $x$ - or $y$ -intercepts of the graph.
$$y=3-\frac{1}{2} x$$

Jennifer Southers

Jennifer Southers

Numerade Educator

Problem 36

Use a graphing uti equation. Use a standard viewing window. Approximate any $x$ - or $y$ -intercepts of the graph.
$$y=\frac{2}{3} x-1$$

Jennifer Southers

Jennifer Southers

Numerade Educator

Problem 37

Use a graphing uti equation. Use a standard viewing window. Approximate any $x$ - or $y$ -intercepts of the graph.
$$y=\frac{2 x}{x-1}$$

Jennifer Southers

Jennifer Southers

Numerade Educator

Problem 38

Use a graphing uti equation. Use a standard viewing window. Approximate any $x$ - or $y$ -intercepts of the graph.
$$y=\frac{6}{x}$$

Jennifer Southers

Jennifer Southers

Numerade Educator

Problem 39

Use a graphing uti equation. Use a standard viewing window. Approximate any $x$ - or $y$ -intercepts of the graph.
$$y=x \sqrt{x+3}$$

Jennifer Southers

Jennifer Southers

Numerade Educator

Problem 40

Use a graphing uti equation. Use a standard viewing window. Approximate any $x$ - or $y$ -intercepts of the graph.
$$y=(6-x) \sqrt{x}$$

Jennifer Southers

Jennifer Southers

Numerade Educator

Problem 41

Use a graphing uti equation. Use a standard viewing window. Approximate any $x$ - or $y$ -intercepts of the graph.
$$y=\sqrt[3]{x-8}$$

Jennifer Southers

Jennifer Southers

Numerade Educator

Problem 42

Use a graphing uti equation. Use a standard viewing window. Approximate any $x$ - or $y$ -intercepts of the graph.
$$y=\sqrt[3]{x+1}$$

Jennifer Southers

Jennifer Southers

Numerade Educator

Problem 43

Use a graphing uti equation. Use a standard viewing window. Approximate any $x$ - or $y$ -intercepts of the graph.
$$x^{2}-y=4 x-3$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 44

Use a graphing uti equation. Use a standard viewing window. Approximate any $x$ - or $y$ -intercepts of the graph.
$$2 y-x^{2}+8=2 x$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 45

Use a graphing uti equation. Use a standard viewing window. Approximate any $x$ - or $y$ -intercepts of the graph.
$$y-4 x=x^{2}(x-4)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 46

Use a graphing uti equation. Use a standard viewing window. Approximate any $x$ - or $y$ -intercepts of the graph.
$$x^{3}+y=1$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 47

Describe the viewing window of the graph shown.
$$y=-10 x+50$$

AG

Ankit Gupta

Numerade Educator

Problem 48

Describe the viewing window of the graph shown.
$$y=\sqrt{x+2}-1$$

AG

Ankit Gupta

Numerade Educator

Problem 49

Explain how to use a graphing utility to verify that $y_{1}=y_{2}$ Identify the rule of algebra that is illustrated.
$$\begin{aligned}&y_{1}=\frac{1}{4}\left(x^{2}-8\right)\\&y_{2}=\frac{1}{4} x^{2}-2\end{aligned}$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 50

Explain how to use a graphing utility to verify that $y_{1}=y_{2}$ Identify the rule of algebra that is illustrated.
$$\begin{aligned}&y_{1}=\frac{1}{2} x+(x+1)\\&y_{2}=\frac{3}{2} x+1\end{aligned}$$

AG

Ankit Gupta

Numerade Educator

Problem 51

Explain how to use a graphing utility to verify that $y_{1}=y_{2}$ Identify the rule of algebra that is illustrated.
$$\begin{aligned}&y_{1}=\frac{1}{5}\left[10\left(x^{2}-1\right)\right]\\&y_{2}=2\left(x^{2}-1\right)\end{aligned}$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 52

Explain how to use a graphing utility to verify that $y_{1}=y_{2}$ Identify the rule of algebra that is illustrated.
$$\begin{aligned}&y_{1}=(x-3) \cdot \frac{1}{x-3}\\&y_{2}=1\end{aligned}$$

Numerade Educator

Numerade Educator

Numerade Educator

Problem 53

Use a graphing utility to graph the equation. Use the trace feature of the graphing utility to approximate the unknown coordinate of each solution point accurate to two decimal places. (Hint: You may need to use the zoom feature of the graphing utility to obtain the required accuracy.)
$y=\sqrt{5-x}$
(a) $(3, y)$
(b) $(x, 3)$

AG

Ankit Gupta

Numerade Educator

Problem 54

Use a graphing utility to graph the equation. Use the trace feature of the graphing utility to approximate the unknown coordinate of each solution point accurate to two decimal places. (Hint: You may need to use the zoom feature of the graphing utility to obtain the required accuracy.)
$y=x^{3}(x-3)$
(a) $(2.25, y)$
(b) $(x, 20)$

JL

Jennifer L

Numerade Educator

Problem 55

Use a graphing utility to graph the equation. Use the trace feature of the graphing utility to approximate the unknown coordinate of each solution point accurate to two decimal places. (Hint: You may need to use the zoom feature of the graphing utility to obtain the required accuracy.)
$y=x^{5}-5 x$
(a) $(-0.5, y)$
(b) $(x,-4)$

JL

Jennifer L

Numerade Educator

Problem 56

Use a graphing utility to graph the equation. Use the trace feature of the graphing utility to approximate the unknown coordinate of each solution point accurate to two decimal places. (Hint: You may need to use the zoom feature of the graphing utility to obtain the required accuracy.)
$y=\left|x^{2}-6 x+5\right|$
(a) $(2, y)$
(b) $(x, 1.5)$

AG

Ankit Gupta

Numerade Educator

Problem 57

Solve for y and use a graphing utility to graph each of the resulting equations in the same viewing window. (Adjust the viewing window so that the circle appears circular.)
$$x^{2}+y^{2}=16$$

AG

Ankit Gupta

Numerade Educator

Problem 58

Solve for y and use a graphing utility to graph each of the resulting equations in the same viewing window. (Adjust the viewing window so that the circle appears circular.)
$$x^{2}+y^{2}=36$$

AG

Ankit Gupta

Numerade Educator

Problem 59

Solve for y and use a graphing utility to graph each of the resulting equations in the same viewing window. (Adjust the viewing window so that the circle appears circular.)
$$(x-1)^{2}+(y-2)^{2}=9$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 60

Solve for y and use a graphing utility to graph each of the resulting equations in the same viewing window. (Adjust the viewing window so that the circle appears circular.)
$$(x-3)^{2}+(y-1)^{2}=25$$

AG

Ankit Gupta

Numerade Educator

Problem 61

Determine which point lies on the graph of the circle. (There may be more than one correct answer.)
$(x-1)^{2}+(y-2)^{2}=25$
(a) (1,3)
(b) (-2,6)
(c) (5,-1)
(d) $(0,2+2 \sqrt{6})$

Erika Bustos

Numerade Educator

Problem 62

Determine which point lies on the graph of the circle. (There may be more than one correct answer.)
$(x+2)^{2}+(y-3)^{2}=25$
(a) (-2,3)
(b) (0,0)
(c) (1,-1)
(d) $(-1,3-2 \sqrt{6})$

Erika Bustos

Numerade Educator

Problem 63

A manufacturing plant purchases a new molding machine for 225,000 dollar. The depreciated value (decreased value) $y$ after $t$ years is $y=225,000-20,000 t,$ for $0 \leq t \leq 8.$
(a) Use the constraints of the model and a graphing utility to graph the equation using an appropriate viewing window.
(b) Use the value feature or the zoom and trace features of the graphing utility to determine the value of $y$ when $t=5.8 .$ Verify your answer algebraically.
(c) Use the value feature or the zoom and trace features of the graphing utility to determine the value of $y$ when $t=2.35 .$ Verify your answer algebraically.

James Kiss

Numerade Educator

Problem 64

You buy a personal watercraft for 8100 dollar. The depreciated value $y$ after $t$ years is $y=8100-929 t$ for $0 \leq t \leq 6.$
(a) Use the constraints of the model and a graphing utility to graph the equation using an appropriate viewing window.
(b) Use the zoom and trace features of the graphing utility to determine the value of $t$ when $y=5545.25 .$ Verify your answer algebraically.
(c) Use the value feature or the zoom and trace features of the graphing utility to determine the value of $y$ when $t=5.5 .$ Verify your answer algebraically.

James Kiss

Numerade Educator

Problem 65

The table shows the median (middle) sales prices $y$ (in thousands of dollars) of new one-family homes in the southern United States from 2000 through 2008. $$\begin{array}{|l|c|}\hline \text { Year } & \text { Median sales price, $y$} \\\hline 2000 & 148.0 \\2001 & 155.4 \\2002 & 163.4 \\2003 & 168.1 \\2004 & 181.1 \\2005 & 197.3 \\2006 & 208.2 \\2007 & 217.7 \\ 2008 & 203.7 \\\hline\end{array}$$ A model that represents the data is given by the $y=-0.4221 t^{3}+4.690 t^{2}-3.47 t+150.9,0 \leq t \leq 8$where $t$ represents the year, with $t=0$ corresponding to 2000.
(a) Use the model and the table feature of a graphing utility to find the median sales prices from 2000 through 2008 . How well does the model fit the data? Explain.
(b) Use the graphing utility to graph the data from the table and the model in the same viewing window. How well does the model fit the data? Explain.
(c) Use the model to estimate the median sales prices in 2012 and $2014 .$ Do the values seem reasonable? Explain.
(d) Use the zoom and trace features of the graphing utility to determine during which year(s) the median sales price was approximately 150,000 dollar.

AS

Alana Schneider

Numerade Educator

Problem 66

The table shows the life expectancy $y$ of a child (at birth) in the United States for each of the selected years from 1930 through 2000. $$\begin{array}{|c|c|}\hline \text { Year } & \text { Life expectancy, $y$ } \\\hline 1930 & 59.7 \\1940 & 62.9 \\1950 & 68.2 \\1960 & 69.7 \\1970 & 70.8 \\1980 & 73.7 \\1990 & 75.4 \\2000 & 77.0 \\\hline \end{array}$$ A model that represents the data is given by $$y=\frac{59.617+1.18 t}{1+0.012 t}, \quad 0 \leq t \leq 70$$ where $t$ is the time in years, with $t=0$ corresponding to 1930.
(a) Use a graphing utility to graph the data from the table above and the model in the same viewing window. How well does the model fit the data? Explain.
(b) Find the $y$ -intercept of the graph of the model. What does it represent in the context of the problem?
(c) Use the zoom and trace features of the graphing utility to determine the year when the life expectancy was $73.2 .$ Verify your answer algebraically.
(d) Determine the life expectancy in 1948 both graphically and algebraically.
(e) Use the model to estimate the life expectancy of a child born in 2020

AS

Alana Schneider

Numerade Educator

Problem 67

Determine whether the statement is true or false. Justify your answer.
A parabola can have only one $x$ -intercept.

AG

Ankit Gupta

Numerade Educator

Problem 68

Determine whether the statement is true or false. Justify your answer.
The graph of a linear equation can have either no $x$ -intercepts or only one $x$ -intercept.

AG

Ankit Gupta

Numerade Educator

Problem 69

Determine whether the statement is true or false. Justify your answer.
Your employer offers you a choice of wage scales: a monthly salary of 3000 dollar plus commission of $7 \%$ of sales or a salary of 3400 dollar plus a $5 \%$ commission. Write a short paragraph discussing how you would choose your option. At what sales level would the options yield the same salary?

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 70

Determine whether the statement is true or false. Justify your answer.
You open a savings account and deposit 200 dollar. Every week you withdraw 50 dollar. The account balance $y$ after $t$ weeks is $y=-50 t+200$ for $0 \leq t \leq 4$
(a) Use point plotting and graph paper to sketch the graph of $y=-50 t+200$
(b) Use a graphing utility to graph $y=-50 t+200$
(c) Explain how to find an appropriate viewing window for the graph of the equation.
(d) Find the $y$ intercept of the graph of the model. What does it represent in the context of the problem?

AS

Alana Schneider

Numerade Educator

Problem 71

Perform the operation and write the result in standard form.
$$(9 x-4)+\left(2 x^{2}-x+15\right)$$

AG

Ankit Gupta

Numerade Educator

Problem 72

Perform the operation and write the result in standard form.
$$\left(3 x^{2}-5\right)\left(-x^{2}+1\right)$$

AG

Ankit Gupta

Numerade Educator