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College Algebra—
2
CHAPTER CONNECTIONS
Functions and
Graphs
CHAPTER OUTLINE
2.1 Rectangular Coordinates; Graphing Circles
and Other Relations 152
2.2 Graphs of Linear Equations 165
2.3 Linear Graphs and Rates of Change 178
2.4 Functions, Function Notation, and the Graph
of a Function 190
2.5 Analyzing the Graph of a Function 206
2.6 The Toolbox Functions and Transformations 225
2.7 Piecewise-Defined Functions 240
Viewing a function in terms of an equation, a
table of values, and the related graph, often
brings a clearer understanding of the relationships involved. For example, the power
generated by a wind turbine is often modeled
8v 3
by the function P1v2 , where P is
125
the power in watts and v is the wind velocity
in miles per hour. While the formula enables
us to predict the power generated for a given
wind speed, the graph offers a visual representation of this relationship, where we note
a rapid growth in power output as the wind
speed increases. This application appears as
Exercise 107 in Section 2.6.
Check out these other real-world connections:
2.8 The Algebra and Composition of Functions 254
Earthquake Area (Section 2.1, Exercise 84)
Height of an Arrow (Section 2.5, Exercise 61)
Garbage Collected per Number of Garbage
Trucks (Section 2.2, Exercise 42)
Number of People Connected to the Internet
(Section 2.3, Exercise 109)
151
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College Algebra—
2.1 Rectangular Coordinates; Graphing Circles and Other Relations
Learning Objectives
In everyday life, we encounter a large variety of relationships. For instance, the time
it takes us to get to work is related to our average speed; the monthly cost of heating
a home is related to the average outdoor temperature; and in many cases, the amount
of our charitable giving is related to changes in the cost of living. In each case we say
that a relation exists between the two quantities.
In Section 2.1 you will learn how to:
A. Express a relation in
mapping notation and
ordered pair form
B. Graph a relation
C. Develop the equation of
a circle using the
distance and midpoint
formulas
D. Graph circles
WORTHY OF NOTE
EXAMPLE 1
Figure 2.1
In the most general sense, a relation is simply a
P
B
correspondence between two sets. Relations can be
represented in many different ways and may even
Missy
April 12
Jeff
be very “unmathematical,” like the one shown in
Nov 11
Angie
Figure 2.1 between a set of people and the set of their
Sept 10
Megan
corresponding birthdays. If P represents the set of
Nov 28
people and B represents the set of birthdays, we say Mackenzie
May 7
Michael
that elements of P correspond to elements of B, or the
April
14
Mitchell
birthday relation maps elements of P to elements of B.
Using what is called mapping notation, we might
simply write P S B.
Figure 2.2
The bar graph in Figure 2.2 is also
155
($145)
145
an example of a relation. In the graph,
135
each year is related to average annual
($123)
125
consumer spending on Internet media
115
(music downloads, Internet radio, Web105
based news articles, etc.). As an alterna($98)
95
tive to mapping or a bar graph, the
($85)
85
relation could also be represented using
75
($69)
ordered pairs. For example, the
65
ordered pair (3, 98) would indicate that
in 2003, spending per person on Internet
2
3
5
1
7
media averaged $98 in the United
Year (1 → 2001)
States. Over a long period of time, we
Source: 2006 Statistical Abstract of the United States
could collect many ordered pairs of the
form (t, s), where consumer spending s depends on the time t. For this reason we often
call the second coordinate of an ordered pair (in this case s) the dependent variable,
with the first coordinate designated as the independent variable. In this form, the set
of all first coordinates is called the domain of the relation. The set of all second coordinates is called the range.
Consumer spending
(dollars per year)
From a purely practical
standpoint, we note that
while it is possible for two
different people to share the
same birthday, it is quite
impossible for the same
person to have two different
birthdays. Later, this observation will help us mark the
difference between a relation
and a function.
A. Relations, Mapping Notation, and Ordered Pairs
Expressing a Relation as a Mapping and in Ordered Pair Form
Represent the relation from Figure 2.2 in mapping notation
and ordered pair form, then state its domain and range.
Solution
A. You’ve just learned
how to express a relation in
mapping notation and ordered
pair form
152
Let t represent the year and s represent consumer spending.
The mapping t S s gives the diagram shown. In ordered pair
form we have (1, 69), (2, 85), (3, 98), (5, 123), and (7, 145).
The domain is {1, 2, 3, 5, 7}, the range is {69, 85, 98,
123, 145}.
t
s
1
2
3
5
7
69
85
98
123
145
Now try Exercises 7 through 12
For more on this relation, see Exercise 81.
2-2
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College Algebra—
2-3
153
Section 2.1 Rectangular Coordinates; Graphing Circles and Other Relations
Table 2.1 y x 1
x
y
4
5
2
3
0
1
2
1
4
3
B. The Graph of a Relation
Relations can also be stated in equation form. The equation y x 1 expresses a
relation where each y-value is one less than the corresponding x-value (see Table 2.1).
The equation x y expresses a relation where each x-value corresponds to the
absolute value of y (see Table 2.2). In each case, the relation is the set of all ordered
pairs (x, y) that create a true statement when substituted, and a few ordered pair solutions are shown in the tables for each equation.
Relations can be expressed graphically using a rectangular coordinate
system. It consists of a horizontal number line (the x-axis) and a vertical number
line (the y-axis) intersecting at their zero marks. The
Figure 2.3
point of intersection is called the origin. The x- and
y
y-axes create a flat, two-dimensional surface called
5
the xy-plane and divide the plane into four regions
4
called quadrants. These are labeled using a capital
3
QII
QI
2
“Q” (for quadrant) and the Roman numerals I through
1
IV, beginning in the upper right and moving counterclockwise (Figure 2.3). The grid lines shown denote 5 4 3 2 11 1 2 3 4 5 x
the integer values on each axis and further divide the
2
QIII
QIV
plane into a coordinate grid, where every point in
3
4
the plane corresponds to an ordered pair. Since a
5
point at the origin has not moved along either axis, it
has coordinates (0, 0). To plot a point (x, y) means we
place a dot at its location in the xy-plane. A few of the
Figure 2.4
ordered pairs from y x 1 are plotted in Figure
y
5
2.4, where a noticeable pattern emerges—the points
seem to lie along a straight line.
(4, 3)
If a relation is defined by a set of ordered pairs, the
graph of the relation is simply the plotted points. The
(2, 1)
graph of a relation in equation form, such as y x 1,
5 x
is the set of all ordered pairs (x, y) that make the equa- 5
(0, 1)
tion true. We generally use only a few select points to
(2, 3)
determine the shape of a graph, then draw a straight line
(4, 5)
or smooth curve through these points, as indicated by
5
any patterns formed.
Table 2.2 x y
x
y
2
2
1
1
0
0
1
1
2
2
EXAMPLE 2
Graphing Relations
Graph the relations y x 1 and x y using the ordered pairs given earlier.
Solution
For y x 1, we plot the points then connect them with a straight line
(Figure 2.5). For x y, the plotted points form a V-shaped graph made up
of two half lines (Figure 2.6).
Figure 2.5
5
Figure 2.6
y yx1
5
(2, 2)
(2, 1)
5
(0, 1)
(2, 3)
x y
(3, 3)
(4, 3)
5
y
x
(0, 0)
5
5
(2, 2)
x
(3, 3)
5
5
Now try Exercises 13 through 16
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College Algebra—
154
2-4
CHAPTER 2 Functions and Graphs
While we used only a few points to graph the relations in Example 2, they are actually made up of an infinite number of ordered pairs that satisfy each equation, including those that might be rational or irrational. All of these points together make these
graphs continuous, which for our purposes means you can draw the entire graph
without lifting your pencil from the paper.
Actually, a majority of graphs cannot be drawn using only a straight line or
directed line segments. In these cases, we rely on a “sufficient number” of points
to outline the basic shape of the graph, then connect the points with a smooth curve.
As your experience with graphing increases, this “sufficient number of points” tends
to get smaller as you learn to anticipate what the graph of a given relation should
look like.
WORTHY OF NOTE
As the graphs in Example 2
indicate, arrowheads are
used where appropriate to
indicate the infinite extension
of a graph.
EXAMPLE 3
Graphing Relations
Graph the following relations by completing the tables given.
a. y x2 2x
b. y 29 x2
c. x y2
Solution
For each relation, we use each x-input in turn to determine the related y-output(s),
if they exist. Results can be entered in a table and the ordered pairs used to draw
the graph.
a.
y x2 2x
Figure 2.7
y
x
y
(x, y)
Ordered Pairs
4
24
(4, 24)
3
15
(3, 15)
2
8
(2, 8)
1
3
(1, 3)
0
1
0
1
2
0
(2, 0)
3
3
(3, 3)
4
8
(4, 8)
(0, 0)
(2, 8)
y x2 2x
(4, 8)
5
(1, 3)
(3, 3)
(2, 0)
(0, 0)
5
5
(1, 1)
(1, 1)
2
x
The result is a fairly common graph (Figure 2.7), called a vertical parabola.
Although (4, 24) and 13, 152 cannot be plotted here, the arrowheads
indicate an infinite extension of the graph, which will include these points.
y 29 x2
b.
Figure 2.8
(x, y)
Ordered Pairs
x
y
4
not real
—
3
0
(3, 0)
2
15
(2, 15)
1
212
(1, 212)
0
3
(0, 3)
1
212
(1, 212)
2
15
(2, 15)
3
0
(3, 0)
4
not real
—
y 9 x2
5
(1, 22)
(2, 5)
(3, 0)
y
(0, 3)
(1, 22)
(2, 5)
(3, 0)
5
5
5
x