Chapter 0

Preparation for Calculus

Educators

JC
JD
BP
JC
+ 18 more educators

Problem 1

Finding Intercepts Describe how to find the $x$-and $y$-intercepts of the graph of an equation.

JC
Jacob C.
Numerade Educator

Problem 2

Verifying Points of Intersection How can you check that an ordered pair is a point of intersection of two graphs?

JD
Jacob D.
Numerade Educator

Problem 3

match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).

$y=-\frac{3}{2} x+3$

BP
Barbara P.
Numerade Educator

Problem 4

match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).

$y=\sqrt{9-x^{2}}$

JC
James C.
Numerade Educator

Problem 5

match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).
$y=3-x^{2}$

Amy J.
Numerade Educator

Problem 6

match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).
$y=x^{3}-x$

JC
James C.
Numerade Educator

Problem 7

Sketching a Graph by Point Plotting In Exercises $7-16,$ sketch the graph of the equation by point plotting.

$y=\frac{1}{2} x+2$

A M.
Numerade Educator

Problem 8

Sketching a Graph by Point Plotting In Exercises $7-16,$ sketch the graph of the equation by point plotting.

$y=5-2 x$

JC
Jacob C.
Numerade Educator

Problem 9

Sketching a Graph by Point Plotting In Exercises $7-16,$ sketch the graph of the equation by point plotting.

$y=4-x^{2}$

LL
Linjun L.
Numerade Educator

Problem 10

Sketching a Graph by Point Plotting In Exercises $7-16,$ sketch the graph of the equation by point plotting.
$y=(x-3)^{2}$

JC
Jacob C.
Numerade Educator

Problem 11

Sketching a Graph by Point Plotting In Exercises $7-16,$ sketch the graph of the equation by point plotting.
$y=|x+1|$

JC
James C.
Numerade Educator

Problem 12

Sketching a Graph by Point Plotting In Exercises $7-16,$ sketch the graph of the equation by point plotting.
$y=|x|-1$

JC
Jacob C.
Numerade Educator

Problem 13

Sketching a Graph by Point Plotting In Exercises $7-16,$ sketch the graph of the equation by point plotting.
$y=\sqrt{x}-6$

Robert S.
Numerade Educator

Problem 14

Sketching a Graph by Point Plotting In Exercises $7-16,$ sketch the graph of the equation by point plotting.
$y=\sqrt{x+2}$

JC
Jacob C.
Numerade Educator

Problem 15

Sketching a Graph by Point Plotting In Exercises $7-16,$ sketch the graph of the equation by point plotting.
$y=\frac{3}{x}$

SG
Steven G.
Numerade Educator

Problem 16

Sketching a Graph by Point Plotting In Exercises $7-16,$ sketch the graph of the equation by point plotting.
$y=\frac{1}{x+2}$

JC
Jacob C.
Numerade Educator

Problem 17

Approximating Solution Points Using Technology In Exercises 17 and $18,$ use a graphing utility to graph the equation. Move the cursor along the curve to approximate the
unknown coordinate of each solution point accurate to two decimal places.

$$y=\sqrt{5-x}$$
(a) $(2, y)$
(b) $(x, 3)$

JD
Jacob D.
Numerade Educator

Problem 18

Approximating Solution Points Using Technology In Exercises 17 and $18,$ use a graphing utility to graph the equation. Move the cursor along the curve to approximate the
unknown coordinate of each solution point accurate to two decimal places.
$$y=x^{5}-5 x$$
(a) $(-0.5, y)$
(b) $(x,-4)$

JD
Jacob D.
Numerade Educator

Problem 19

Finding Intercepts In Exercises $19-28,$ find any intercepts.

$y=2 x-5$

TH
Thomas H.
Numerade Educator

Problem 20

Finding Intercepts In Exercises $19-28,$ find any intercepts.
$y=4 x^{2}+3$

JC
Jacob C.
Numerade Educator

Problem 21

Finding Intercepts In Exercises $19-28,$ find any intercepts.
$y=x^{2}+x-2$

ST
Shaurya T.
Numerade Educator

Problem 22

Finding Intercepts In Exercises $19-28,$ find any intercepts.
$y^{2}=x^{3}-4 x$

JC
Jacob C.
Numerade Educator

Problem 23

Finding Intercepts In Exercises $19-28,$ find any intercepts.
$y=x \sqrt{16-x^{2}}$

JR
Jack R.
Numerade Educator

Problem 24

Finding Intercepts In Exercises $19-28,$ find any intercepts.
$y=(x-1) \sqrt{x^{2}+1}$

JC
Jacob C.
Numerade Educator

Problem 25

Finding Intercepts In Exercises $19-28,$ find any intercepts.
$y=\frac{2-\sqrt{x}}{5 x+1}$

PG
Paul G.
Numerade Educator

Problem 26

Finding Intercepts In Exercises $19-28,$ find any intercepts.
$y=\frac{x^{2}+3 x}{(3 x+1)^{2}}$

MM
Maral M.
Numerade Educator

Problem 27

Finding Intercepts In Exercises $19-28,$ find any intercepts.
$x^{2} y-x^{2}+4 y=0$

JC
James C.
Numerade Educator

Problem 28

Finding Intercepts In Exercises $19-28,$ find any intercepts.
$y=2 x-\sqrt{x^{2}+1}$

JC
Jacob C.
Numerade Educator

Problem 29

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.

$y=x^{2}-6$

JC
James C.
Numerade Educator

Problem 30

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.
$y=9 x-x^{2}$

JC
James C.
Numerade Educator

Problem 31

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.
$y^{2}=x^{3}-8 x$

Diego R.
Numerade Educator

Problem 32

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.
$y=x^{3}+x$

JC
James C.
Numerade Educator

Problem 33

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.
$x y=4$

Amy J.
Numerade Educator

Problem 34

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.
$x y^{2}=-10$

JC
James C.
Numerade Educator

Problem 35

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.
$y=4-\sqrt{x+3}$

JC
James C.
Numerade Educator

Problem 36

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.
$x y-\sqrt{4-x^{2}}=0$

JC
James C.
Numerade Educator

Problem 37

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.
$y=\frac{x}{x^{2}+1}$

JC
James C.
Numerade Educator

Problem 38

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.
$y=\frac{x^{5}}{4-x^{2}}$

JC
James C.
Numerade Educator

Problem 39

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.
$y=\left|x^{3}+x\right|$

JD
Jacob D.
Numerade Educator

Problem 40

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.
$|y|-x=3$

JD
Jacob D.
Numerade Educator

Problem 41

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.

$y=2-3 x$

JD
Jacob D.
Numerade Educator

Problem 42

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.

$y=\frac{2}{3} x+1$

JD
Jacob D.
Numerade Educator

Problem 43

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.
$y=9-x^{2}$

PD
Paul-Yvann D.
Numerade Educator

Problem 44

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.
$y=2 x^{2}+x$

JD
Jacob D.
Numerade Educator

Problem 45

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.
$y=x^{3}+2$

JD
Jacob D.
Numerade Educator

Problem 46

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.
$y=x^{3}-4 x$

JD
Jacob D.
Numerade Educator

Problem 47

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.
$y=x \sqrt{x+5}$

TW
Timothy W.
Numerade Educator

Problem 48

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.
$y=\sqrt{25-x^{2}}$

JD
Jacob D.
Numerade Educator

Problem 49

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.
$x=y^{3}$

JD
Jacob D.
Numerade Educator

Problem 50

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.
$x=y^{4}-16$

JD
Jacob D.
Numerade Educator

Problem 51

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.
$y=\frac{8}{x}$

Amy J.
Numerade Educator

Problem 52

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.
$y=\frac{10}{x^{2}+1}$

JD
Jacob D.
Numerade Educator

Problem 53

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.
$y=6-|x|$

WC
William C.
Numerade Educator

Problem 54

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.
$y=|6-x|$

JD
Jacob D.
Numerade Educator

Problem 55

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.
$3 y^{2}-x=9$

LT
Linxiao T.
Numerade Educator

Problem 56

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.
$x^{2}+4 y^{2}=4$

JD
Jacob D.
Numerade Educator

Problem 57

Finding Points of Intersection In Exercises $57-62,$ find the points of intersection of the graphs of the equations.

$x+y=8$
$4 x-y=7$

Georgiann A.
Numerade Educator

Problem 58

Finding Points of Intersection In Exercises $57-62,$ find the points of intersection of the graphs of the equations.

$3 x-2 y=-4$
$4 x+2 y=-10$

JD
Jacob D.
Numerade Educator

Problem 59

Finding Points of Intersection In Exercises $57-62,$ find the points of intersection of the graphs of the equations.

$\begin{aligned} x^{2}+y &=15 \\-3 x+y &=11 \end{aligned}$

Catherine R.
Numerade Educator

Problem 60

Finding Points of Intersection In Exercises $57-62,$ find the points of intersection of the graphs of the equations.

$x=3-y^{2}$
$y=x-1$

JD
Jacob D.
Numerade Educator

Problem 61

Finding Points of Intersection In Exercises $57-62,$ find the points of intersection of the graphs of the equations.

$x^{2}+y^{2}=5$
$x-y=1$

Amy J.
Numerade Educator

Problem 62

Finding Points of Intersection In Exercises $57-62,$ find the points of intersection of the graphs of the equations.

$x^{2}+y^{2}=16$
$x+2 y=4$

JD
Jacob D.
Numerade Educator

Problem 63

Finding Points of Intersection Using Technology In Exercises $63-66$ , use a graphing utility to find the points of intersection of the graphs of the equations. Check your results
analytically.

$y=x^{3}-2 x^{2}+x-1$
$y=-x^{2}+3 x-1$

Amy J.
Numerade Educator

Problem 64

Finding Points of Intersection Using Technology In Exercises $63-66$ , use a graphing utility to find the points of intersection of the graphs of the equations. Check your results
analytically.

$y=x^{4}-2 x^{2}+1$
$y=1-x^{2}$

JD
Jacob D.
Numerade Educator

Problem 65

Finding Points of Intersection Using Technology In Exercises $63-66$ , use a graphing utility to find the points of intersection of the graphs of the equations. Check your results
analytically.

$y=\sqrt{x+6}$
$y=\sqrt{-x^{2}-4 x}$

Amy J.
Numerade Educator

Problem 66

Finding Points of Intersection Using Technology In Exercises $63-66$ , use a graphing utility to find the points of intersection of the graphs of the equations. Check your results
analytically.

$y=-|2 x-3|+6$
$y=6-x$

JD
Jacob D.
Numerade Educator

Problem 67

Modeling Data The table shows the Gross Domestic Product, or GDP (in trillions of dollars), for 2009 through $2014 . \quad$ Source: U.S. Bureau of Economic Analysis)

$$\begin{array}{|c|c|c|c|c|c|c|}\hline \text { Year } & {2009} & {2010} & {2011} & {2012} & {2013} & {2014} \\ \hline \text { GDP } & {14.4} & {15.0} & {15.5} & {16.2} & {16.7} & {17.3} \\ \hline\end{array}$$

(a) Use the regression capabilities of a graphing utility to find a mathematical model of the form $y=a t+b$ for the data. In the model, $y$ represents the GDP (in trillions of dollars)
and $t$ represents the year, with $t=9$ corresponding to $2009 .$
(b) Use a graphing utility to plot the data and graph the model. Compare the data with the model.
(c) Use the model to predict the GDP in the year 2024 .

Amy J.
Numerade Educator

Problem 68

Modeling Data The table shows the numbers of cell phone subscribers (in millions) in the United States for selected years. (Source: CTIA-The Wireless Association)

$$\begin{array}{|c|c|c|c|c|}\hline \text { Year } & {2000} & {2002} & {2004} & {2006} \\ \hline \text { Number } & {109} & {141} & {182} & {233} \\ \hline \text { Year } & {2008} & {2010} & {2012} & {2014} \\ \hline \text { Number } & {270} & {296} & {326} & {355} \\ \hline\end{array}$$

(a) Use the regression capabilities of a graphing utility to find a mathematical model of the form $y=a t^{2}+b t+c$ for the data. In the model, $y$ represents the number of subscribers (in millions) and $t$ represents the year, with $t=0$ corresponding to $2000 .$
(b) Use a graphing utility to plot the data and graph the model. Compare the data with the model.
(c) Use the model to predict the number of cell phone subscribers in the United States in the year 2024 .

JD
Jacob D.
Numerade Educator

Problem 69

Break-Even Point Find the sales necessary to break even $(R=C)$ when the cost $C$ of producing $x$ units is $C=2.04 x+5600$ and the revenue $R$ from selling $x$ units is $R=3.29 x .$

Jason K.
Numerade Educator

Problem 70

Using Solution Points For what values of $k$ does the graph of $y^{2}=4 k x$ pass through the point?
$$\begin{array}{ll}{\text { (a) }(1,1)} & {\text { (b) }(2,4)} \\ {\text { (c) }(0,0)} & {\text { (d) }(3,3)}\end{array}$$

JD
Jacob D.
Numerade Educator

Problem 71

Using Intercepts Write an equation whose graph has intercepts at $x=-\frac{3}{2}, x=4,$ and $x=\frac{5}{2} .$ (There is more than one correct answer.)

Amy J.
Numerade Educator

Problem 72

Symmetry A graph is symmetric with respect to the $x$ -axis and to the $y$ -axis. Is the graph also symmetric with respect to the origin? Explain.

JD
Jacob D.
Numerade Educator

Problem 73

Symmetry A graph is symmetric with respect to one axis and to the origin. Is the graph also symmetric with respect to the other axis? Explain.

Amy J.
Numerade Educator

Problem 74

Use the graphs of the two equations to answer the questions below.

(a) What are the intercepts for each equation?
(b) Determine the symmetry for each equation.
(c) Determine the point of intersection of the two equations.

JD
Jacob D.
Numerade Educator

Problem 75

True or False? In Exercises $75-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $(-4,-5)$ is a point on a graph that is symmetric with respect to the $x$ -axis, then $(4,-5)$ is also a point on the graph.

Amy J.
Numerade Educator

Problem 76

True or False? In Exercises $75-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $(-4,-5)$ is a point on a graph that is symmetric with respect to the $y$ -axis, then $(4,-5)$ is also a point on the graph.

JD
Jacob D.
Numerade Educator

Problem 77

True or False? In Exercises $75-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $b^{2}-4 a c>0$ and $a \neq 0,$ then the graph of
$$y=a x^{2}+b x+c$$

Amy J.
Numerade Educator

Problem 78

True or False? In Exercises $75-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $b^{2}-4 a c=0$ and $a \neq 0,$ then the graph of
$$y=a x^{2}+b x+c$$
has only one $x$ -intercept.

JC
Jacob C.
Numerade Educator