SECTION 1.5 Circles
45
(c)
(d)
(e)
(f)
(g)
(h)
Move B to the point whose coordinates are (4, 5) . What is the slope of the line?
Move B to the point whose coordinates are (4, 4) . What is the slope of the line?
Move B to the point whose coordinates are (4, 1) . What is the slope of the line?
Move B to the point whose coordinates are (3, - 2) . What is the slope of the line?
Slowly move B to a point whose x-coordinate is 1. What happens to the value of the slope as the x-coordinate approaches 1?
What can be said about a line whose slope is positive? What can be said about a line whose slope is negative? What can be said
about a line whose slope is 0?
(i) Consider the results of parts (a)–(c). What can be said about the steepness of a line with positive slope as its slope
increases?
( j) Move B to the point whose coordinates are (3, 5) . What is the slope of the line? Move B to the point whose coordinates
are (5, 6) . What is the slope of the line? Move B to the point whose coordinates are (-1, 3) . What is the slope of the line?
TEKS 1.F
1.5 Circles
PREPARING FOR THIS SECTION Before getting started, review the following:
• Completing the Square
(Appendix A, Section A.3, pp. A28–A29)
• Square Root Method
(Appendix A, Section A.6, p. A47)
Now Work the ’Are You Prepared?’ problems on page 49.
OBJECTIVES 1 Write the Standard Form of the Equation of a Circle (p. 45)
2 Graph a Circle by Hand and by Using a Graphing Utility (p. 46)
3 Work with the General Form of the Equation of a Circle (p. 48)
1 Write the Standard Form of the Equation of a Circle
One advantage of a coordinate system is that it enables us to translate a geometric
statement into an algebraic statement, and vice versa. Consider, for example, the
following geometric statement that defines a circle.
DEFINITION
Figure 67
y
A circle is a set of points in the xy-plane that are a fixed distance r from a fixed
point (h, k) . The fixed distance r is called the radius, and the fixed point (h, k)
is called the center of the circle.
Figure 67 shows the graph of a circle. To find the equation, let (x, y) represent
the coordinates of any point on a circle with radius r and center (h, k) . Then the
distance between the points (x, y) and (h, k) must always equal r. That is, by the
distance formula
(x, y)
r
(h, k)
21x - h2 2 + 1y - k2 2 = r
x
or, equivalently,
1x - h2 2 + 1y - k2 2 = r 2
DEFINITION
The standard form of an equation of a circle with radius r and center (h, k) is
1x - h2 2 + 1y - k2 2 = r 2
(1)
46
CHAPTER 1 Graphs
THEOREM
The standard form of an equation of a circle of radius r with center at the
origin 10, 02 is
x2 + y2 = r 2
DEFINITION
If the radius r = 1 , the circle whose center is at the origin is called the unit
circle and has the equation
x2 + y2 = 1
See Figure 68. Notice that the graph of the unit circle is symmetric with respect to
the x-axis, the y-axis, and the origin.
Figure 68
Unit circle x 2 + y 2 = 1
y
1
⫺1
(0,0)
1
x
⫺1
E X AMP LE 1
Writing the Standard Form of the Equation of a Circle
Write the standard form of the equation of the circle with radius 5 and center 1 -3, 62 .
Solution
Using equation (1) and substituting the values r = 5, h = -3 , and k = 6 , we have
1x - h2 2 + 1y - k2 2 = r 2
1x - 1 -322 2 + 1y - 62 2 = 52
1x + 32 2 + 1y - 62 2 = 25
Now Work
PROBLEM
7
2 Graph a Circle by Hand and by Using a Graphing Utility
E X AMP LE 2
Graphing a Circle by Hand and by Using a Graphing Utility
Graph the equation: 1x + 32 2 + 1y - 22 2 = 16
Solution
Figure 69
(–3, 6)
1x + 32 2 + 1y - 22 2 = 16
y
6
1x - 1 -322 2 + 1y - 22 2 = 42
c
4
(–7, 2)
–10
(–3, 2)
(1, 2)
2 x
–5
(–3, –2)
Since the equation is in the form of equation (1), its graph is a circle. To graph the
equation by hand, compare the given equation to the standard form of the equation
of a circle. The comparison yields information about the circle.
c
c
1x - h2 2 + 1y - k2 2 = r 2
We see that h = -3, k = 2, and r = 4. The circle has center 1 -3, 22 and a
radius of 4 units. To graph this circle, first plot the center 1 -3, 22. Since the radius
is 4, locate four points on the circle by plotting points 4 units to the left, to the
right, up, and down from the center. These four points can then be used as guides
to obtain the graph. See Figure 69.
SECTION 1.5 Circles
47
To graph a circle on a graphing utility, we must write the equation in the form
y = 5 expression involving x 6 . * We must solve for y in the equation
In Words
The symbol { is read “plus or
minus.” It means to add and
subtract the quantity following the
{ symbol. For example, 5 { 2
means 5 - 2 = 3 or 5 + 2 = 7.
1x + 32 2 + 1y - 22 2 = 16
Subtract (x + 3)2 from both sides.
1y - 22 2 = 16 - 1x + 32 2
y - 2 = { 216 - 1x + 32 2
Use the Square Root Method.
y = 2 { 216 - 1x + 32 2 Add 2 to both sides.
Figure 70
To graph the circle, we graph the top half
6
Y1 = 2 + 216 - 1x + 32 2
Y1
and the bottom half
Y2 = 2 - 216 - 1x + 32 2
–9
3
Y2
–2
Also, be sure to use a square screen. Otherwise, the circle will appear distorted.
Figure 70 shows the graph on a TI-84 Plus. The graph is “disconnected” due to the
resolution of the calculator.
Now Work
EX AMP L E 3
PROBLEMS
23(a)
AND
(b)
Finding the Intercepts of a Circle
For the circle 1x + 32 2 + 1y - 22 2 = 16, find the intercepts, if any, of its graph.
Solution
This is the equation discussed and graphed in Example 2. To find the x-intercepts, if
any, let y = 0 and solve for x. Then
1x + 32 2 + 1y - 22 2 = 16
1x + 32 2 + 10 - 22 2 = 16
1x + 32 2 + 4 = 16
1x + 32 = 12
2
x + 3 = { 212
y=0
Simplify.
Subtract 4 from both sides.
Apply the Square Root Method.
x = -3 { 223 Solve for x.
The x-intercepts are -3 - 223 ⬇ -6.46 and -3 + 223 ⬇ 0.46.
To find the y-intercepts, if any, let x = 0 and solve for y. Then
1x + 32 2 + 1y - 22 2 = 16
10 + 32 2 + 1y - 22 2 = 16
x=0
9 + 1y - 22 = 16
2
1y - 22 2 = 7
y - 2 = { 27 Apply the Square Root Method.
y = 2 { 27 Solve for y.
The y-intercepts are 2 - 27 ⬇ -0.65 and 2 + 27 ⬇ 4.65.
Look back at Figure 69 to verify the approximate locations of the intercepts.
Now Work
PROBLEM
23 (c)
*Some graphing utilities (e.g., TI-83, TI-84, and TI-86) have a CIRCLE function that allows the user to
enter only the coordinates of the center of the circle and its radius to graph the circle.
48
CHAPTER 1 Graphs
3 Work with the General Form of the Equation of a Circle
If we eliminate the parentheses from the standard form of the equation of the circle
given in Example 3, we get
1x + 32 2 + 1y - 22 2 = 16
x2 + 6x + 9 + y 2 - 4y + 4 = 16
which, upon simplifying, is equivalent to
x2 + y 2 + 6x - 4y - 3 = 0
It can be shown that any equation of the form
x 2 + y 2 + ax + by + c = 0
has a graph that is a circle or a point, or has no graph at all. For example, the graph of
the equation x2 + y 2 = 0 is the single point 10, 02. The equation x2 + y 2 + 5 = 0,
or x2 + y 2 = -5, has no graph, because sums of squares of real numbers are never
negative.
DEFINITION
ELPS
ELPS
ELP
When its graph is a circle, the equation
x2 + y 2 + ax + by + c = 0
ELPS 1.A.2
You may recall using the method of completing
the square in prior courses, such as solving
quadratic equations in an algebra course.
Before working on Example 4, with a partner,
discuss any prior experiences you have had
with the method of completing the square and
how this may be useful when graphing the
equation of a circle in general form.
E X AMP LE 4
is referred to as the general form of the equation of a circle.
If an equation of a circle is in the general form, we use the method of completing
the square to put the equation in standard form so that we can identify its center
and radius.
Graphing a Circle Whose Equation Is in General Form
Graph the equation x2 + y 2 + 4x - 6y + 12 = 0 .
Solution
Group the expression involving x, group the expression involving y, and put the
constant on the right side of the equation. The result is
1x2 + 4x2 + 1y 2 - 6y2 = -12
Next, complete the square of each expression in parentheses. Remember that any
number added on the left side of the equation must be added on the right.
(x2 ⫹ 4x ⫹ 4) ⫹ (y2 ⫺ 6y ⫹ 9) ⫽ ⫺12 ⫹ 4 ⫹ 9
( 42 )
2
(⫺6
)
2
2
⫽4
⫽9
(x ⫹ 2) ⫹ (y ⫺ 3)2 ⫽ 1
2
Factor
This equation is the standard form of the equation of a circle with radius 1 and center
1 -2, 32. To graph the equation by hand, use the center 1 -2, 32 and the radius 1. See
Figure 71(a).
To graph the equation using a graphing utility, solve for y.
1y - 32 2 = 1 - 1x + 22 2
y - 3 = {21 - 1x + 22 2
y = 3 {21 - 1x + 22
Use the Square Root Method.
2
Add 3 to both sides.
SECTION 1.5 Circles
49
Figure 71(b) illustrates the graph on a TI-84 Plus graphing calculator.
Figure 71
y
(⫺2, 4)
4
1
(⫺3, 3)
Y1 ⫽ 3 ⫹ 1 ⫺ (x ⫹ 2)2 5
(⫺1, 3)
(⫺2, 3)
(⫺2, 2)
⫺3
⫺4.5
1 x
Y2 ⫽ 3 ⫺ 1 ⫺ (x ⫹
Now Work
1
(b)
(a)
EX AMP L E 5
1.5
2)2
PROBLEM
27
Finding the General Equation of a Circle
Find the general equation of the circle whose center is 11, -22 and whose graph
contains the point 14, -22.
Solution
Figure 72
y
3
To find the equation of a circle, we need to know its center and its radius. Here,
the center is 11, -22. Since the point 14, -22 is on the graph, the radius r will
equal the distance from 14, -22 to the center 11, -22. See Figure 72. Thus,
r = 2 14 - 12 2 + 3 -2 - 1 -22 4 2
r
(1, ⫺2)
5
= 29 = 3
x
(4, ⫺2)
The standard form of the equation of the circle is
1x - 12 2 + 1y + 22 2 = 9
⫺5
Eliminate the parentheses and rearrange the terms to get the general equation
x2 + y 2 - 2x + 4y - 4 = 0
Now Work
PROBLEM
13
Overview
The discussion in Sections 1.4 and 1.5 about lines and circles dealt with two main
types of problems that can be generalized as follows:
1. Given an equation, classify it and graph it.
2. Given a graph, or information about a graph, find its equation.
This text deals with both types of problems. We shall study various equations,
classify them, and graph them. The second type of problem is usually more
difficult to solve than the first. In many instances a graphing utility can be used
to solve problems when information about the problem (such as data) is given.
1.5 Assess Your Understanding
‘Are You Prepared?’
Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.
1. To complete the square of x 2 + 10x , you would (add/
subtract) the number ______. (p. A48)
2. Use the Square Root Method to solve the equation
1x - 22 2 = 9 . (p. A47)
50
CHAPTER 1 Graphs
Concepts and Vocabulary
3. True or False
Every equation of the form
5. True or False The radius of the circle x 2 + y 2 = 9 is 3.
x 2 + y 2 + ax + by + c = 0
6. True or False The center of the circle
1x + 32 2 + 1y - 22 2 = 13
has a circle as its graph.
is (3, - 2) .
4. For a circle, the ______ is the distance from the center to any
point on the circle.
Skill Building
In Problems 7–10, find the center and radius of each circle. Write the standard form of the equation.
7. y
8.
9. y
y
10. y
(4, 2)
(2, 3)
(1, 2)
(0, 1)
(2, 1)
(0, 1)
(1, 2)
x
x
x
(1, 0)
x
In Problems 11–20, write the standard form of the equation and the general form of the equation of each circle of radius r and center 1h, k2 .
Graph each circle.
11. r = 2;
1h, k2 = 10, 02
12. r = 3;
1h, k2 = 10, 02
13. r = 2;
1h, k2 = 10, 22
14. r = 3;
15. r = 5;
1h, k2 = 14, -32
16. r = 4;
1h, k2 = 12, -32
17. r = 4;
1h, k2 = 1 -2, 12
18. r = 7; 1h, k2 = 1- 5, - 22
1
;
2
1
1h, k2 = a , 0 b
2
19. r =
20. r =
1
;
2
1h, k2 = 11, 02
1
1h, k2 = a0, - b
2
In Problems 21–34, (a) find the center 1h, k2 and radius r of each circle; (b) graph each circle; (c) find the intercepts, if any.
21. x 2 + y 2 = 4
22. x 2 + 1y - 12 2 = 1
23. 2 1x - 32 2 + 2y 2 = 8
24. 3 1x + 12 2 + 3 1y - 12 2 = 6
25. x 2 + y 2 - 2x - 4y - 4 = 0
26. x 2 + y 2 + 4x + 2y - 20 = 0
27. x 2 + y 2 + 4x - 4y - 1 = 0
1
30. x 2 + y 2 + x + y - = 0
2
33. 2x 2 + 8x + 2y 2 = 0
28. x 2 + y 2 - 6x + 2y + 9 = 0
29. x 2 + y 2 - x + 2y + 1 = 0
31. 2x 2 + 2y 2 - 12x + 8y - 24 = 0
32. 2x 2 + 2y 2 + 8x + 7 = 0
34. 3x 2 + 3y 2 - 12y = 0
In Problems 35–42, find the standard form of the equation of each circle.
35. Center at the origin and containing the point 1 - 2, 32
36. Center 11, 02 and containing the point 1 - 3, 22
37. Center 12, 32 and tangent to the x-axis
38. Center 1 -3, 12 and tangent to the y-axis
39. With endpoints of a diameter at 11, 42 and 1 - 3, 22
40. With endpoints of a diameter at 14, 32 and 10, 12
41. Center 1 - 1, 32 and tangent to the line y = 2
42. Center 14, -22 and tangent to the line x = 1
In Problems 43–46, match each graph with the correct equation.
(a) 1x - 32 2 + 1y + 32 2 = 9
(b) 1x + 12 2 + 1y - 22 2 = 4
43.
44.
⫺4
46.
⫺9
9
⫺6
6
4
6
6
(d) 1x + 32 2 + 1y - 32 2 = 9
45.
4
⫺6
(c) 1x - 12 2 + 1y + 22 2 = 4
⫺6
6
⫺4
⫺9
9
⫺6
SECTION 1.5 Circles
51
Applications and Extensions
47. Find the area of the square in the figure.
y
51. Weather Satellites Earth is represented on a map of a
portion of the solar system so that its surface is the circle
with equation x 2 + y 2 + 2x + 4y - 4091 = 0 . A weather
satellite circles 0.6 unit above Earth with the center of its
circular orbit at the center of Earth. Find the equation for
the orbit of the satellite on this map.
x2 y2 9
x
r
48. Find the area of the blue shaded region in the figure, assuming
the quadrilateral inside the circle is a square.
y
52. The tangent line to a circle may be defined as the line that
intersects the circle in a single point, called the point of
tangency. See the figure.
x 2 y 2 36
y
x
r
x
49. Ferris Wheel The original Ferris wheel was built in 1893 by
Pittsburgh, Pennsylvania, bridge builder George W. Ferris.
The Ferris wheel was originally built for the 1893 World’s
Fair in Chicago and was later reconstructed for the 1904
World’s Fair in St. Louis. It had a maximum height of
264 feet and a wheel diameter of 250 feet. Find an equation
for the wheel if the center of the wheel is on the y-axis.
Source: inventors.about.com
50. Ferris Wheel In 2008, the Singapore Flyer opened as
the world’s largest Ferris wheel. It has a maximum height
of 165 meters and a diameter of 150 meters, with one full
rotation taking approximately 30 minutes. Find an equation
for the wheel if the center of the wheel is on the y-axis.
Source: Wikipedia
If the equation of the circle is x 2 + y 2 = r 2 and the equation
of the tangent line is y = mx + b , show that:
(a) r 2 11 + m 2 2 = b2
[Hint: The quadratic equation x 2 + 1mx + b2 2 = r 2
has exactly one solution.]
- r2 m r2
(b) The point of tangency is ¢
, ≤.
b
b
(c) The tangent line is perpendicular to the line containing
the center of the circle and the point of tangency.
53. The Greek Method The Greek method for finding the
equation of the tangent line to a circle uses the fact that at
any point on a circle the lines containing the center and the
tangent line are perpendicular (see Problem 52). Use this
method to find an equation of the tangent line to the circle
x 2 + y 2 = 9 at the point 11, 2 122 .
54. Use the Greek method described in Problem 53
to find an equation of the tangent line to the circle
x 2 + y 2 - 4x + 6y + 4 = 0 at the point 13, 2 12 - 32.
55. Refer to Problem 52. The line x - 2y + 4 = 0 is tangent to
a circle at 10, 22 . The line y = 2x - 7 is tangent to the same
circle at 13, - 12 . Find the center of the circle.
56. Find an equation of the line containing the centers of the
two circles
x 2 + y 2 - 4x + 6y + 4 = 0
and
x 2 + y 2 + 6x + 4y + 9 = 0
57. If a circle of radius 2 is made to roll along the x-axis, what is
an equation for the path of the center of the circle?
58. If the circumference of a circle is 6p , what is its radius?
52
CHAPTER 1
Graphs
Explaining Concepts: Discussion and Writing
59. Which of the following equations might have the graph
shown? (More than one answer is possible.)
y
(a) 1x - 22 2 + 1y + 32 2 = 13
(b) 1x - 22 2 + 1y - 22 2 = 8
(c) 1x - 22 2 + 1y - 32 2 = 13
(d) 1x + 22 2 + 1y - 22 2 = 8
(e) x 2 + y 2 - 4x - 9y = 0
x
(f) x 2 + y 2 + 4x - 2y = 0
2
2
(g) x + y - 9x - 4y = 0
(h) x 2 + y 2 - 4x - 4y = 4
60. Which of the following equations might have the graph
shown? (More than one answer is possible.)
(a) 1x - 22 2 + y 2 = 3
y
(b) 1x + 22 2 + y 2 = 3
(c) x 2 + 1y - 22 2 = 3
(d) 1x + 22 2 + y 2 = 4
(e) x 2 + y 2 + 10x + 16 = 0
x
(f) x 2 + y 2 + 10x - 2y = 1
(g) x 2 + y 2 + 9x + 10 = 0
(h) x 2 + y 2 - 9x - 10 = 0
61. Explain how the center and radius of a circle can be used to graph the circle.
62. What Went Wrong? A student stated that the center and radius of the graph whose equation is (x + 3)2 + (y - 2)2 = 16 are
(3, -2) and 4, respectively. Why is this incorrect?
Interactive Exercises
Ask your instructor if the applets below are of interest to you.
63. Center of a Circle Open the “Circle: the role of the center”
applet. Place the cursor on the center of the circle and hold
the mouse button. Drag the center around the Cartesian
plane and note how the equation of the circle changes.
(a) What is the radius of the circle?
(b) Draw a circle whose center is at 11, 32 . What is the
equation of the circle?
(c) Draw a circle whose center is at 1 - 1, 32 . What is the
equation of the circle?
(d) Draw a circle whose center is at 1 - 1, - 32 . What is the
equation of the circle?
(e) Draw a circle whose center is at 11, - 32 . What is the
equation of the circle?
(f) Write a few sentences explaining the role the center of
the circle plays in the equation of the circle.
64. Radius of a Circle Open the “Circle: the role of the radius”
applet. Place the cursor on point B, press and hold the mouse
button. Drag B around the Cartesian plane.
(a) What is the center of the circle?
(b) Move B to a point in the Cartesian plane directly above
the center such that the radius of the circle is 5.
(c) Move B to a point in the Cartesian plane such that the
radius of the circle is 4.
(d) Move B to a point in the Cartesian plane such that the
radius of the circle is 3.
(e) Find the coordinates of two points with integer
coordinates in the fourth quadrant on the circle that
result in a circle of radius 5 with center equal to that
found in part (a).
(f) Use the concept of symmetry about the center, vertical
line through the center of the circle, and horizontal line
through the center of the circle to find three other points
with integer coordinates in the other three quadrants
that lie on the circle of radius 5 with center equal to that
found in part (a).
‘Are You Prepared?’ Answers
1. add; 25
2. 5 - 1, 5 6
CHAPTER REVIEW
Things to Know
Formulas
Distance formula (p. 5)
Midpoint formula (p. 7)
Slope (p. 29)
Parallel lines (p. 37)
Perpendicular lines (38)
d = 21x2 - x1 2 2 + 1y2 - y1 2 2
x1 + x2 y 1 + y 2
,
≤
2
2
y2 - y1
if x1 ⬆ x2; undefined if x1 = x2
m=
x2 - x1
Equal slopes 1m 1 = m 2 2 and different y-intercepts 1b1 ⬆ b2 2
1x, y2 = ¢
Product of slopes is - 1 1m 1 # m 2 = -12 ; slopes are negative reciprocals of
1
each other am 1 = b
m2
Chapter Review
53
Equations of Lines and Circles
Vertical line (p. 33)
x = a ; a is the x-intercept
Horizontal line (p. 34)
y = b ; b is the y-intercept
Point–slope form of the equation of a line (p. 33)
y - y1 = m(x - x1); m is the slope of the line, (x1, y1) is a point on the line
Slope–intercept form of the equation of a line (p. 35)
y = mx + b; m is the slope of the line, b is the y-intercept
General form of the equation of a line (p. 36)
Ax + By = C; A, B not both 0
Standard form of the equation of a circle (p. 45)
(x - h)2 + (y - k)2 = r 2; r is the radius of the circle, (h, k) is the center of
the circle
Equation of the unit circle (p. 46)
x2 + y2 = 1
General form of the equation of a circle (p. 48)
x 2 + y 2 + ax + by + c = 0 , with restrictions on a, b , and c
Objectives
Section
1.1
You should be able to . . .
Example(s)
Review Exercises
Use the distance formula (p. 4)
Use the midpoint formula (p. 7)
Graph equations by hand by plotting points (p. 7)
Graph equations using a graphing utility (p. 10)
Use a graphing utility to create tables (p. 12)
Find intercepts from a graph (p. 12)
Use a graphing utility to approximate intercepts (p. 13)
2, 3, 4
5
6, 7, 8
9, 10
11
12
13
3
Find intercepts algebraically from an equation (p. 18)
Test an equation for symmetry (p.19)
Know how to graph key equations (p. 21)
1
2, 3
4–6
7–9
10–14
15
1.3
1
Solve equations using a graphing utility (p. 26)
1–3
16, 17
1.4
1
1
2
3
4, 5
6
1(c)–4(c), 1(d)–4(d), 32(b), 33
37
20
18, 19
21–23
18, 19, 21–25
26–27
7
24
25
1
2
3
4
5
6
7
1.2
1
2
Calculate and interpret the slope of a line (p. 29)
Graph lines given a point and the slope (p. 32)
3 Find the equation of a vertical line (p. 32)
4 Use the point–slope form of a line; identify horizontal lines (p. 33)
5 Find the equation of a line given two points (p. 34)
6 Write the equation of a line in slope–intercept form (p. 34)
7 Identify the slope and y-intercept of a line from its equation (p. 35)
8 Graph lines written in general form using intercepts (p. 36)
9 Find equations of parallel lines (p. 37)
10 Find equations of perpendicular lines (p. 38)
2
1.5
1
2
3
Write the standard form of the equation of a circle (p. 45)
Graph a circle by hand and by using a graphing utility (p. 46)
Work with the general form of the equation of a circle (p. 48)
7
8
9, 10
11, 12
1
2, 3
4, 5
Review Exercises
In Problems 1– 4, find the following for each pair of points:
(a) The distance between the points.
(b) The midpoint of the line segment connecting the points.
(c) The slope of the line containing the points.
(d) Then interpret the slope found in part (c).
1. 10, 02; 14, 22
3. 14, -42; 14, 82
2. 11, -12; 1 - 2, 32
4. 1 -2, -12; 13, -12
1(a)–4(a), 32(a), 34–36
1(b)–4(b), 35
7–9
6, 7–9
6
5
6
28, 29
30, 31
30, 31, 35
54
CHAPTER 1
Graphs
6. Graph y = -x 2 + 15 using a graphing utility. Create a table
of values to determine a good initial viewing window. Use a
graphing utility to approximate the intercepts.
5. List the intercepts of the following graph.
y
2
⫺4
4
x
⫺2
In Problems 7–9, determine the intercepts and graph each equation by hand by plotting points. Verify your results using a graphing utility.
Label the intercepts on the graph.
8. y = x 2 - 9
7. 2x - 3y = 6
9. x 2 + 2y = 16
In Problems 10–14, test each equation for symmetry with respect to the x-axis, the y-axis, and the origin.
10. 2x = 3y 2
11. x 2 + 4y 2 = 16
13. y = x 3 - x
12. y = x 4 - 3x 2 - 4
14. x 2 + x + y 2 + 2y = 0
15. Sketch a graph of y = x .
3
In Problems 16 and 17, use a graphing utility to approximate the solutions of each equation rounded to two decimal places. All solutions
lie between - 10 and 10.
16. x 3 - 5x + 3 = 0
17. x 4 - 3 = 2x + 1
In Problems 18–25, find an equation of the line having the given characteristics. Express your answer using either the general form or the
slope–intercept form of the equation of a line, whichever you prefer. Graph the line.
18. Slope = -2; containing the point 13, - 12
19. Slope = 0; containing the point 1 -5, 42
20. Slope undefined; containing the point 1 - 3, 42
21. x@intercept = 2; containing the point 14, -52
22. y@intercept = -2; containing the point 15, - 32
23. Containing the points 13, -42 and 12, 12
24. Parallel to the line 2x - 3y = -4; containing the point 1 - 5, 32
25. Perpendicular to the line 3x - y = -4; containing the point 1 -2, 42
In Problems 26 and 27, find the slope and y-intercept of each line.
26. 4x + 6y = 36
27.
5
1
x + y = 10
2
2
In Problems 28 and 29, find the standard form of the equation of the circle whose center and radius are given.
28. 1h, k2 = 1 -2, 32; r = 4
29. 1h, k2 = 1 - 1, - 22; r = 1
In Problems 30 and 31, find the center and radius of each circle. Graph each circle by hand. Determine the intercepts of the graph of
each circle.
30. x 2 + y 2 - 2x + 4y - 4 = 0
31. 3x 2 + 3y 2 - 6x + 12y = 0
32. Show that the points A = 1 - 2, 02, B = 1 - 4, 42, and
C = 18, 52 are the vertices of a right triangle in two ways:
(a) By using the converse of the Pythagorean Theorem
(b) By using the slopes of the lines joining the vertices
36. Find two numbers y such that the distance from 1 - 3, 22 to
15, y2 is 10.
2
37. Graph the line with slope containing the point 11, 22.
3
38. Make up four problems that you might be asked to do given
the two points 1 - 3, 42 and 16, 12. Each problem should
involve a different concept. Be sure that your directions are
clearly stated.
39. Describe each of the following graphs in the xy-plane. Give
justification.
(a) x = 0
(b) y = 0
(c) x + y = 0
(d) xy = 0
33. Show that the points A = 12, 52, B = 16, 12,
C = 18, - 12 lie on a straight line by using slopes.
and
34. Show that the points A = 11, 52, B = 12, 42, and
C = 1 - 3, 52 lie on a circle with center 1 - 1, 22. What is the
radius of this circle?
35. The endpoints of the diameter of a circle are 1 - 3, 22 and
15, -62. Find the center and radius of the circle. Write the
general equation of this circle.
(e) x 2 + y 2 = 0
SECTION
Chapter
1.5 Project
Circles
CHAPTER TEST
1. Suppose the points 1 -2, -32 and 14, 52 are the endpoints
of a line segment.
(a) Find the distance between the two points.
(b) Find the midpoint of the line segment connecting the
two points.
In Problems 2 and 3, graph each equation by hand by plotting
points. Use a graphing utility to approximate the intercepts and
label them on the graph.
2. 2x - 7y = 21
55
The Chapter Test Prep Videos are step-by-step test solutions available in the
Video Resources DVD, in
Channel. Flip
, or on this text’s
back to the Student Resources page to see the exact web address for this
text’s YouTube channel.
3. y = x 2 - 5
In Problems 4–6, use a graphing utility to approximate the real
solutions of each equation rounded to two decimal places. All
solutions lie between - 10 and 10.
4. 2x 3 - x 2 - 2x + 1 = 0
5. x 4 - 5x 2 - 8 = 0
7. Use P1 = 1 -1, 32 and P2 = 15, -12 .
(a) Find the slope of the line containing P1 and P2 .
(b) Interpret this slope.
8. Sketch the graph of y 2 = x .
9. List the intercepts and test for symmetry: x 2 + y = 9 .
10. Write the slope–intercept form of the line with slope - 2
containing the point 13, -42 . Graph the line.
11. Write the general form of the circle with center 14, -32 and
radius 5.
12. Find the center and radius of the circle
x 2 + y 2 + 4x - 2y - 4 = 0 . Graph this circle.
13. For the line 2x + 3y = 6 , find a line parallel to it containing
the point 11, - 12 . Also find a line perpendicular to it
containing the point (0, 3).
6. - x 3 + 7x - 2 = x 2 + 3x - 3
TEKS 1.D
Internet-based Project
Determining the Selling Price of a Home Determining how
much to pay for a home is one of the more difficult decisions
that must be made when purchasing a home. There are many
factors that play a role in a home’s value. Location, size, number
of bedrooms, number of bathrooms, lot size, and building
materials are just a few. Fortunately, the website Zillow.com
has developed its own formula for predicting the selling price
of a home. This information is a great tool for predicting
the actual sale price. For example, the data to the right show
the “zestimate”—the selling price of a home as predicted by
the folks at Zillow and the actual selling price of the home for
homes in Oak Park, Illinois.
Zestimate
(000s of dollars)
Sale Price
(000s of dollars)
291.5
268
320
305
371.5
375
303.5
283
351.5
350
314
275
332.5
356
295
300
313
285
368
385
The graph below, called a scatter diagram, shows the points
(291.5, 268), (320, 305), . . . , (368, 385) in a Cartesian plane. From
the graph, it appears that the data follow a linear relation.
Zestimate vs. Sale Price
in Oak Park, IL
Sale Price (thousands of dollars)
CHAPTER PROJECT
380
360
340
320
300
280
300
320
340
360
Zestimate (thousands of dollars)
56
CHAPTER 1
Graphs
1. Imagine drawing a line through the data that appears to fit
the data well. Do you believe the slope of the line would be
positive, negative, or close to zero? Why?
2. Pick two points from the scatter diagram. Treat the zestimate
as the value of x and treat the sale price as the corresponding
value of y. Find the equation of the line through the two
points you selected.
3. Interpret the slope of the line.
4. Use your equation to predict the selling price of a home
whose zestimate is $335,000.
5. Do you believe it would be a good idea to use the equation
you found in part 2 if the zestimate is $950,000? Why or
why not?
6. Choose a location in which you would like to live. Go to
www.zillow.com and randomly select at least ten homes that
have recently sold.
(a) Draw a scatter diagram of your data.
(b) Select two points from the scatter diagram and find the
equation of the line through the points.
(c) Interpret the slope.
(d) Find a home from the Zillow website that interests you
under the “Make Me Move” option for which a zestimate
is available. Use your equation to predict the sale price
based on the zestimate.