Summary
3
DEFINITION /PROCEDURE
EXAMPLE
REFERENCE
Ordered Pairs and Relations
Section 3.1
Ordered Pair
Given two related values, x and y, we write the pair of values as
(x, y).
p. 126
Relation
A set of ordered pairs.
The set (1, 4), (2, 5), (1, 6) is a
relation.
p. 126
Domain
The set of all first elements of a relation.
The domain is 1, 2.
p. 127
Range
The set of all second elements of a relation.
The range is 4, 5, 6.
p. 127
The Rectangular Coordinate System
Section 3.2
The rectangular coordinate system allows us to establish a
one-to-one correspondence between points in the plane and
ordered pairs of real numbers.
y
(x, y)
y axis
x coordinate
y coordinate
(x, y)
y
To graph (or plot) a point (x, y) in the plane:
1. Start at the origin.
2. Move to the right or left along the x axis according to the
value of the x coordinate.
x
x
Origin
x axis
3. Move up or down and parallel to the y axis according to the
value of the y coordinate.
It is not always desirable to use the same scale on both the x
and y axes. In these situations, we use a different marked scale.
This is called scaling the axes.
p. 131
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An Introduction to Functions
A function is set of ordered pairs (a relation) in which no two
first coordinates are equal.
The set of points in the plane that correspond to ordered
pairs in a relation or function is called the graph of that relation
or function.
Section 3.3
(1, 2), (2, 3), (3, 4) is a function.
(1, 2), (2, 3), (2, 4) is not a
function.
p. 142
Continued
173
174
CHAPTER 3
THE COORDINATE PLANE AND FUNCTIONS
DEFINITION /PROCEDURE
EXAMPLE
REFERENCE
An Introduction to Functions
Section 3.3
A useful means of determining whether a graph represents a
function is called the vertical line test.
If a vertical line meets the graph of a relation in two or
more points, the relation is not a function.
y
x
A function
y
x
p. 147
A relation—not a function
Reading Values from a Graph
For a specific value of x, let’s call it a, we can find f(a) with the
following algorithm.
Section 3.4
Find f(3)
y
1. Draw a vertical line through a on the x axis.
2. Find the point of intersection of that line with the graph.
3. Draw a horizontal line through the graph at that point.
4. Find the intersection of the horizontal line with the y axis.
5. f(a) is that y value.
x
If given the function value, you find the x values associated
with it as follows.
1. Find the given function value on the y axis.
2. Draw a horizontal line through that point.
line.
4. Draw a vertical line through each of those points of
intersection.
5. Each point of intersection of the vertical lines and the x axis
gives an x value.
Draw a line through x 3
Draw a horizontal line through the
intersection.
The horizontal line passes through
(0, 5)
f(3) 5
p. 162
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3. Find every point on the graph that intersects the horizontal
Summary Exercises
This summary exercise set is provided to give you practice with each of the objectives in the chapter. Each exercise is
keyed to the appropriate chapter section. The answers are provided in the Instructor’s Manual.
[3.1]
In exercises 1 and 2, identify which are ordered pairs.
1. (a) (2, 1)
(b) 3, 4
(c) 1, 4
(d) (4, 3)
(e) ((3, 2), 5)
2. (a) 1, 4
(b) 6, 8
(c) (3, 4)
(d) (3, 1), 4
(e) (2, 5)
In exercises 3 to 10, find the domain and range of each relation.
3. R (Maine, 5), (Massachusetts, 13), (Vermont, 7), (Connecticut, 11)
4. R (John Wayne, 1969), (Art Carney, 1974), (Peter Finch, 1976), (Marlon Brando, 1972)
5. R (Dean Smith, 65), (John Wooden, 47), (Denny Crum, 42), (Bob Knight, 41)
© 2001 McGraw-Hill Companies
6. R (Don Shula, 328), (George Halas, 318), (Tom Landry, 250), (Chuck Noll, 193)
7. (3, 5), (4, 6), (1, 2), (8, 1), (7, 3)
8. (1, 3), (2, 5), (3, 7), (1, 4), (2, 2)
9. (1, 3), (1, 5), (1, 7), (1, 9), (1, 10)
10. (2, 4), (1, 4), (3, 4), (1, 4), (6, 4)
175
176
[3.2]
THE COORDINATE PLANE AND FUNCTIONS
CHAPTER 3
In exercises 11 to 18, graph the following points in the Cartesian coordinate system.
y
x
11. A(2, 3)
12. B(3, 5)
13. C(0, 5)
14. D(2, 6)
15. E(4, 1)
16. F(6, 0)
17. G(4, 5)
18. H(0, 2)
In exercises 19 to 26, give the quadrant in which each of the following points is located or the axis on which the point lies.
19. (6, 15)
20. (8, 11)
21. (9, 0)
22. (15, 8)
23. (7, 2)
24. (0, 5)
25. (2.5, 5.6)
26. (7.1, 4.3)
In exercises 27 to 34, give the coordinates associated with the points indicated in the figure.
y
W
27. P
28. Q
29. R
30. S
31. T
32. V
33. W
34. X
P
T
x
R
V
S
X
[3.3]
In exercises 35 to 40, evaluate each function for the value specified.
35. f(x) x2 3x 5; find (a) f(0), (b) f(1), and (c) f(1).
36. f(x) 2x2 x 7; find (a) f(0), (b) f(2), and (c) f(2).
37. f(x) x3 x2 2x 5; find (a) f(1), (b) f(0), and (c) f(2).
38. f(x) x2 7x 9; find (a) f(3), (b) f(0), and (c) f(1).
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Q
SUMMARY EXERCISES
177
39. f(x) 3x2 5x 1; find (a) f(1), (b) f(0), and (c) f(2).
40. f(x) x3 3x 5; find (a) f(2), (b) f(0), and (c) f(1).
In exercises 41 to 46, determine which relations are also functions.
41. (1, 3), (2, 4), (5, 1), (1, 3)
42. (2, 4), (3, 6), (1, 5), (0, 1)
43. (1, 2), (0, 4), (1, 3), (2, 5)
44. (1, 3), (2, 3), (3, 3), (4, 3)
45.
46.
x
y
3
1
0
1
3
2
1
3
4
5
x
y
1
0
1
2
3
3
2
3
4
5
In exercises 47 to 50, use the vertical line test to determine whether the given graph represents a function. Find the domain
and range of the relation.
47.
48.
y
y
x
x
© 2001 McGraw-Hill Companies
49.
50.
y
x
y
x
178
[3.4]
51.
CHAPTER 3
THE COORDINATE PLANE AND FUNCTIONS
In exercises 51 to 54, find the coordinates of the labeled points.
y
52.
y
A
A
x
x
B
B
53.
y
54.
y
A
A
B
x
x
B
In exercises 55 to 58, use the graph to estimate (a) f(2), (b) f(0), and (c) f(2).
56.
y
y
x
x
57.
58.
y
x
y
x
© 2001 McGraw-Hill Companies
55.
SUMMARY EXERCISES
179
In exercises 59 to 62, use the graph of f(x) to find all values of x such that (a) f(x) 1, (b) f(x) 0, and (c) f(x) 1.
59.
60.
y
y
x
x
61.
62.
y
y
x
x
In exercises 63 to 70, find the x and y intercepts from the graph.
63.
64.
y
y
x
x
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65.
66.
y
x
y
x
67.
CHAPTER 3
THE COORDINATE PLANE AND FUNCTIONS
68.
y
y
x
x
69.
70.
y
x
y
x
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180