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Calculus of a Single Variable

Ron Larson, Bruce Edwards

Chapter 0

Preparation for Calculus

Educators


Problem 1

In Exercises $1-4,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]
$$
y=-\frac{3}{2} x+3
$$

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Problem 2

In Exercises $1-4,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]
$$
y=\sqrt{9-x^{2}}
$$

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Problem 3

In Exercises $1-4,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]
$$
y=3-x^{2}
$$

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Problem 4

In Exercises $1-4,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]
$$
y=x^{3}-x
$$

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Problem 5

In Exercises $9-14,$ sketch the graph of the equation by point plotting.
$$
y=\frac{1}{2} x+2
$$

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Problem 6

In Exercises $9-14,$ sketch the graph of the equation by point plotting.
$$
y=5-2 x
$$

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Problem 7

In Exercises $9-14,$ sketch the graph of the equation by point plotting.
$$
y=4-x^{2}
$$

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Problem 8

In Exercises $9-14,$ sketch the graph of the equation by point plotting.
$$
y=(x-3)^{2}
$$

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Problem 9

In Exercises $9-14,$ sketch the graph of the equation by point plotting.
$$
y=|x|-1
$$

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Problem 10

In Exercises $9-14,$ sketch the graph of the equation by point plotting.
$$
y=|x|-1
$$

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Problem 11

In Exercises $9-14,$ sketch the graph of the equation by point plotting.
$$
y=\sqrt{x}-6
$$

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Problem 12

In Exercises $9-14,$ sketch the graph of the equation by point plotting.
$$
y=\sqrt{x+2}
$$

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Problem 13

In Exercises $9-14,$ sketch the graph of the equation by point plotting.
$$
y=\frac{3}{x}
$$

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Problem 14

In Exercises $9-14,$ sketch the graph of the equation by point plotting.
$$
y=\frac{1}{x+2}
$$

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Problem 15

In Exercises 15 and $16,$ describe the viewing window that yields the figure.
$$
y=x^{3}+4 x^{2}-3
$$

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Problem 16

In Exercises 15 and $16,$ describe the viewing window that yields the figure.
$$
y=|x|+|x-16|
$$

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Problem 17

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

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Problem 18

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.
$$
y=x^{5}-5 x \quad \text { (a) }(-0.5, y) \quad \text { (b) }(x,-4)
$$

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Problem 19

In Exercises $19-28$ , find any intercepts.
$$
y=2 x-5
$$

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Problem 20

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

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Problem 21

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

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Problem 22

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

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Problem 23

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

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Problem 24

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

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Problem 25

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

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Problem 26

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

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Problem 27

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

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Problem 28

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

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Problem 29

In Exercises $29-40,$ test for symmetry with respect to each axis and to the origin.
$$
y=x^{2}-6
$$

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Problem 30

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

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Problem 31

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

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Problem 32

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

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Problem 33

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

$$
x y=4
$$

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Problem 34

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

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Problem 35

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

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Problem 36

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

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Problem 37

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

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Problem 38

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

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Problem 39

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

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Problem 40

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

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Problem 41

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=2-3 x
$$

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Problem 42

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=-\frac{3}{2} x+6
$$

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Problem 43

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=\frac{1}{2} x-4
$$

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Problem 44

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=\frac{2}{3} x+1
$$

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Problem 45

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=9-x^{2}
$$

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Problem 46

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=x^{2}+3
$$

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Problem 47

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=(x+3)^{2}
$$

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Problem 48

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=2 x^{2}+x
$$

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Problem 49

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=x^{3}+2
$$

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Problem 50

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=x^{3}-4 x
$$

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Problem 51

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=x \sqrt{x+5}
$$

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Problem 52

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=\sqrt{25-x^{2}}
$$

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Problem 53

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
x=y^{3}
$$

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Problem 54

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
x=y^{2}-4
$$

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Problem 55

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=\frac{8}{x}
$$

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Problem 56

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=\frac{10}{x^{2}+1}
$$

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Problem 57

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=6-|x|
$$

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Problem 58

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=|6-x|
$$

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Problem 59

In Exercises $59-62$ , use a graphing utility to graph the equation. Identify any intercepts and test for symmetry.
$$
y^{2}-x=9
$$

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Problem 60

In Exercises $59-62$ , use a graphing utility to graph the equation. Identify any intercepts and test for symmetry.
$$
x^{2}+4 y^{2}=4
$$

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Problem 61

In Exercises $59-62$ , use a graphing utility to graph the equation. Identify any intercepts and test for symmetry.
$$
x+3 y^{2}=6
$$

Himanshu G.
Numerade Educator

Problem 62

In Exercises $59-62$ , use a graphing utility to graph the equation. Identify any intercepts and test for symmetry.
$$
3 x-4 y^{2}=8
$$

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Problem 63

In Exercises $63-70,$ find the points of intersection of the graphs of the equations.
$$
\begin{array}{c}{x+y=8} \\ {4 x-y=7}\end{array}
$$

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Problem 64

In Exercises $63-70,$ find the points of intersection of the graphs of the equations.
$$
\begin{array}{l}{3 x-2 y=-4} \\ {4 x+2 y=-10}\end{array}
$$

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Problem 65

In Exercises $63-70,$ find the points of intersection of the graphs of the equations.
$$
\begin{aligned} x^{2}+y &=6 \\ x+y &=4 \end{aligned}
$$

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Problem 66

In Exercises $63-70,$ find the points of intersection of the graphs of the equations.
$$
\begin{aligned} x^{2}+y &=6 \\ x+y &=4 \end{aligned}
$$

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Problem 67

In Exercises $63-70,$ find the points of intersection of the graphs of the equations.
$$
\begin{aligned} x^{2}+y^{2} &=5 \\ x-y &=1 \end{aligned}
$$

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Problem 68

In Exercises $63-70,$ find the points of intersection of the graphs of the equations.
$$
\begin{array}{c}{x^{2}+y^{2}=25} \\ {-3 x+y=15}\end{array}
$$

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Problem 69

In Exercises $63-70,$ find the points of intersection of the graphs of the equations.
$$
\begin{array}{l}{y=x^{3}} \\ {y=x}\end{array}
$$

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Problem 70

In Exercises $63-70,$ find the points of intersection of the graphs of the equations.
$$
\begin{array}{l}{y=x^{3}-4 x} \\ {y=-(x+2)}\end{array}
$$

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Problem 71

In Exercises $71-74,$ use a graphing utility to find the points of intersection of the graphs. Check your results analytically.
$$
\begin{array}{l}{y=x^{3}-2 x^{2}+x-1} \\ {y=-x^{2}+3 x-1}\end{array}
$$

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Problem 72

In Exercises $71-74,$ use a graphing utility to find the points of intersection of the graphs. Check your results analytically.
$$
\begin{array}{l}{y=x^{4}-2 x^{2}+1} \\ {y=1-x^{2}}\end{array}
$$

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Problem 73

In Exercises $71-74,$ use a graphing utility to find the points of intersection of the graphs. Check your results analytically.
$$
\begin{array}{l}{y=\sqrt{x+6}} \\ {y=\sqrt{-x^{2}-4 x}}\end{array}
$$

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Problem 74

In Exercises $71-74,$ use a graphing utility to find the points of intersection of the graphs. Check your results analytically.
$$
\begin{array}{l}{y=-|2 x-3|+6} \\ {y=6-x}\end{array}
$$

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Problem 75

Modeling Data The table shows the Consumer Price Index (CPI) for selected years. (Source: Bureau of Labor Statistics)
$$
\begin{array}{|c|c|c|c|c|c|c|}\hline \text { Year } & {1975} & {1980} & {1985} & {1990} & {1995} & {2000} & {2005} \\ \hline \mathrm{CPI} & {53.8} & {82.4} & {107.6} & {130.7} & {152.4} & {172.2} & {195.3} \\ \hline\end{array}
$$
(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 CPI and $t$ represents the year, with $t=5$ corresponding to $1975 .$
(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 CPI for the year 2010.

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Problem 76

Modeling Data The table shows the numbers of cellular phone subscribers (in millions) in the United States for selected years. (Source: Cellular Telecommunications and Intemet Association)
$$
\begin{array}{|c|c|c|c|c|c|c|}\hline \text { Year } & {1990} & {1993} & {1996} & {1999} & {2002} & {2005} \\ \hline \text { Number } & {5} & {16} & {44} & {86} & {141} & {208} \\ \hline\end{array}
$$
(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 and $t$ represents the year, with $t=0$ corresponding to $1990 .$
(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 cellular phone subscribers in the United States in the year 2015 .

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Problem 77

Break-Even Point Find the sales necessary to break even $(R=C)$ if the cost $C$ of producing $x$ units is $C=5.5 \sqrt{x}+10,000$
and the revenue $R$ from selling $x$ units is
$R=3.29 x$

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Problem 78

Copper Wire The resistance $y$ in ohms of 1000 feet of solid copper wire at $77^{\circ} \mathrm{F}$ can be approximated by the model
$$y=\frac{10,770}{x^{2}}-0.37, \quad 5 \leq x \leq 100$$

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Problem 79

In Exercises 79 and $80,$ write an equation whose graph has the indicated property. (There may be more than one correct answer.)
The graph has intercepts at $x=-4, x=3,$ and $x=8$

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Problem 80

In Exercises 79 and $80,$ write an equation whose graph has the indicated property. (There may be more than one correct answer.)
The graph has intercepts at $x=-\frac{3}{2}, x=4,$ and $x=\frac{5}{2}$

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Problem 81

(a) Prove that if a graph is symmetric with respect to the $x$ -axis and to the $y$ -axis, then it is symmetric with respect to the origin. Give an example to show that the converse is not true.
(b) Prove that if a graph is symmetric with respect to one axis and to the origin, then it is symmetric with respect to the other axis.

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Problem 82

Match the equation or equations with the given characteristic.
$\begin{array}{ll}{\text { (i) } y=3 x^{3}-3 x} & {\text { (ii) } y=(x+3)^{2} \quad \text { (iii) } y=3 x-3} \\ {\text { (iv) } y=\sqrt[3]{x}} & {\text { (v) } y=3 x^{2}+3 \quad \text { (vi) } y=\sqrt{x+3}}\end{array}$
(a) Symmetric with respect to the $y$ -axis
(b) Three $x$ -intercepts
(c) Symmetric with respect to the $x$ -axis
(d) $(-2,1)$ is a point on the graph
(e) Symmetric with respect to the origin
(f) Graph passes through the origin

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Problem 83

True or False? In Exercises $83-86$ , 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.

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Problem 84

True or False? In Exercises $83-86$ , 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.

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Problem 85

True or False? In Exercises $83-86$ , 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 two $x$ -intercepts.

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Problem 86

True or False? In Exercises $83-86$ , 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.

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Problem 87

In Exercises 87 and $88,$ find an equation of the graph that consists of all points $(x, y)$ having the given distance from the origin. (For a review of the Distance Formula, see Appendix C.)

The distance from the origin is twice the distance from $(0,3)$

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Problem 88

In Exercises 87 and $88,$ find an equation of the graph that consists of all points $(x, y)$ having the given distance from the origin. (For a review of the Distance Formula, see Appendix C.)

The distance from the origin is $K(K \neq 1)$ times the distance from $(2,0) .$

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