B. The Graph of a Relation
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We generally use only a few select points to determine the shape of a graph, then draw a
straight line or smooth curve through these points, as indicated by any patterns formed.
EXAMPLE 2
ᮣ
Graphing Relations
Graph the relations y ϭ x Ϫ 1 and x ϭ ͿyͿ using the ordered pairs given in Tables 1.1
and 1.2.
Solution
ᮣ
For y ϭ x Ϫ 1, we plot the points then connect them with a straight line (Figure 1.5).
For x ϭ ͿyͿ, the plotted points form a V-shaped graph made up of two half lines
(Figure 1.6).
Figure 1.5
5
Figure 1.6
y
y yϭxϪ1
5
x ϭ ԽyԽ
(4, 3)
(2, 2)
(2, 1)
(0, 0)
Ϫ5
5
x
Ϫ5
5
(Ϫ2, Ϫ3)
x
(2, Ϫ2)
(0, Ϫ1)
Ϫ5
Ϫ5
(Ϫ4, Ϫ5)
Now try Exercises 13 through 16
WORTHY OF NOTE
As the graphs in Example 2
indicate, arrowheads are used
where appropriate to indicate the
infinite extension of a graph.
ᮣ
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. This understanding is an important part of
reading and interpreting graphs, and is illustrated for you in Figures 1.7 through 1.10.
Figure 1.7
Figure 1.8
y ϭ x Ϫ 1: selected integer values
y ϭ x Ϫ 1: selected rational values
y
y
5
5
Ϫ5
5
x
Ϫ5
5
Ϫ5
Ϫ5
Figure 1.9
Figure 1.10
y ϭ x Ϫ 1: selected real number values
y ϭ x Ϫ 1: all real number values
y
y
5
Ϫ5
5
5
Ϫ5
x
x
Ϫ5
5
Ϫ5
x
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Section 1.1 Rectangular Coordinates; Graphing Circles and Other Relations
Since there are an infinite number of ordered pairs forming the graph of y ϭ x Ϫ 1, the
domain cannot be given in list form. Here we note x can be any real number and write
D: x ʦ ޒ. Likewise, y can be any real number and for the range we have R: y ʦ ޒ. 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. In particular,
for the linear graph in Figure 1.5 we notice that both the x- and y-variables have an
implied exponent of 1. This is in fact a characteristic of linear equations and graphs. In
Example 3 we’ll notice that if the exponent on one of the variables is 2 (either x or y is
squared ) while the other exponent is 1, the result is a graph called a parabola. If the
x-term is squared (Example 3a) the parabola is oriented vertically, as in Figure 1.11,
and its highest or lowest point is called the vertex. If the y-term is squared (Example 3c),
the parabola is oriented horizontally, as in Figure 1.13, and the leftmost or rightmost
point is the vertex. The graphs and equations of other relations likewise have certain
identifying characteristics. See Exercises 85 through 92.
EXAMPLE 3
ᮣ
Graphing Relations
Graph the following relations by completing the tables given. Then use the graph
to state the domain and range of the relation.
a. y ϭ x2 Ϫ 2x
b. y ϭ 29 Ϫ x2
c. x ϭ 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 assist
in drawing a complete graph.
Figure 1.11
a.
y ؍x2 Ϫ 2x
y
x
y
(x, y)
Ordered Pairs
Ϫ4
24
(Ϫ4, 24)
Ϫ3
15
(Ϫ3, 15)
Ϫ2
8
(Ϫ2, 8)
Ϫ1
3
(Ϫ1, 3)
0
(0, 0)
1
0
Ϫ1
(1, Ϫ1)
2
0
(2, 0)
3
3
(3, 3)
4
8
(4, 8)
(4, 8)
(Ϫ2, 8)
y ϭ x2 Ϫ 2x
5
(Ϫ1, 3)
(3, 3)
(0, 0)
(2, 0)
Ϫ5
5
Ϫ2
x
(1, Ϫ1)
The resulting vertical parabola is shown in Figure 1.11. Although (Ϫ4, 24) and
(Ϫ3, 15) cannot be plotted here, the arrowheads indicate an infinite extension of
the graph, which will include these points. This “infinite extension” in the upward
direction shows there is no largest y-value (the graph becomes infinitely “tall”).
Since the smallest possible y-value is Ϫ1 [from the vertex (1, Ϫ1)], the range is
y Ն Ϫ1. However, this extension also continues forever in the outward direction
as well (the graph gets wider and wider). This means the x-value of all possible
ordered pairs could vary from negative to positive infinity, and the domain is all
real numbers. We then have D: x ʦ ޒand R: y Ն Ϫ1.
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y ؍29 ؊ x2
b.
Figure 1.12
x
y
(x, y)
Ordered Pairs
Ϫ4
not real
—
Ϫ3
0
(Ϫ3, 0)
Ϫ2
15
(Ϫ2, 15)
Ϫ1
212
(Ϫ1, 212)
0
3
(0, 3)
1
212
(1, 212)
2
15
(2, 15)
3
0
(3, 0)
4
not real
—
y ϭ ͙9 Ϫ x2
y
5
(Ϫ1, 2͙2)
(Ϫ2, ͙5)
(0, 3)
(1, 2͙2)
(2, ͙5)
(Ϫ3, 0)
(3, 0)
Ϫ5
5
x
Ϫ5
The result is the graph of a semicircle (Figure 1.12). The points with irrational
coordinates were graphed by estimating their location. Note that when x 6 Ϫ3
or x 7 3, the relation y ϭ 29 Ϫ x2 does not represent a real number and no
points can be graphed. Also note that no arrowheads are used since the graph
terminates at (Ϫ3, 0) and (3, 0). These observations and the graph itself show
that for this relation, D: Ϫ3 Յ x Յ 3, and R: 0 Յ x Յ 3.
c. Similar to x ϭ ͿyͿ, the relation x ϭ y2 is defined only for x Ն 0 since y2 is
always nonnegative (Ϫ1 ϭ y2 has no real solutions). In addition, we reason
that each positive x-value will correspond to two y-values. For example, given
x ϭ 4, (4, Ϫ2) and (4, 2) are both solutions to x ϭ y2.
x ؍y2
Figure 1.13
y
x
y
(x, y)
Ordered Pairs
Ϫ2
not real
—
Ϫ1
not real
—
0
(0, 0)
0
1
Ϫ1, 1
(1, Ϫ1) and (1, 1)
2
Ϫ12, 12
(2, Ϫ12) and (2, 12)
3
Ϫ13, 13
(3, Ϫ13) and (3, 13)
4
Ϫ2, 2
(4, Ϫ2) and (4, 2)
5
x ϭ y2
(4, 2)
(2, ͙2)
(0, 0)
Ϫ5
5
Ϫ5
x
(2, Ϫ͙2)
(4, Ϫ2)
This relation is a horizontal parabola, with a vertex at (0, 0) (Figure 1.13). The
graph begins at x ϭ 0 and extends infinitely to the right, showing the domain
is x Ն 0. Similar to Example 3a, this “infinite extension” also extends in both
the upward and downward directions and the y-value of all possible ordered
pairs could vary from negative to positive infinity. We then have D: x Ն 0 and
R: y ʦ ޒ.
B. You’ve just seen how
we can graph relations
Now try Exercises 17 through 24
ᮣ
C. Graphing Relations on a Calculator
For relations given in equation form, the TABLE feature of a graphing calculator can
be used to compute ordered pairs, and the GRAPH feature to draw the related graph. To use
these features, we first solve the equation for the variable y (write y in terms of x),
then enter the right-hand expression on the calculator’s Y= (equation editor) screen.
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We can then select either the GRAPH feature, or set-up, create, and use the TABLE
feature. We’ll illustrate here using the relation Ϫ2x ϩ y ϭ 3.
1. Solve for y in terms of x.
Ϫ2x ϩ y ϭ 3
given equation
y ϭ 2x ϩ 3 add 2x to each side
2. Enter the equation.
Press the Y= key to access the equation editor,
then enter 2x ϩ 3 as Y1 (see Figure 1.14). The
calculator automatically highlights the equal sign,
showing that equation Y1 is now active. If there
are other equations on the screen, you can either
them or deactivate them by moving the cursor to overlay the equal sign and pressing .
3. Use the TABLE or GRAPH .
To set up the table, we use the keystrokes 2nd
(TBLSET). For this exercise, we’ll put the
table in the “Indpnt: Auto Ask” mode, which
will have the calculator automatically generate the
input and output values. In this mode, we can tell
the calculator where to start the inputs (we chose
TblStart ϭ Ϫ3), and have the calculator produce the input values using any increment desired
(we choose ⌬Tbl ϭ 1). See Figure 1.15A. Access
the table using 2nd GRAPH (TABLE), and the table
resulting from this setup is shown in Figure 1.15B.
Notice that all ordered pairs satisfy the equation
y ϭ 2x ϩ 3, or “y is twice x increased by 3.”
Figure 1.14
CLEAR
ENTER
Figure 1.15A
WINDOW
Figure 1.15B
Since much of our graphical work is centered at (0, 0)
on the coordinate grid, the calculator’s default settings for the standard viewing
are [Ϫ10, 10] for
both x and y (Figure 1.16). The Xscl and Yscl values
give the scale used on each axis, and indicate here that each “tick mark” will be 1 unit
apart. To graph the line in this window, we can use the ZOOM key and select 6:ZStandard (Figure 1.17), which resets the window to these default settings and automatically graphs the line (Figure 1.18).
WINDOW
Figure 1.16
Figure 1.18
Figure 1.17
10
Ϫ10
10
Ϫ10
In addition to using the calculator’s TABLE feature to find ordered pairs for a
given graph, we can also use the calculator’s TRACE feature. As the name implies, this
feature allows us to “trace” along the graph by moving a cursor to the left and right
using the arrow keys. The calculator displays the coordinates of the cursor’s location each time it moves. After pressing the TRACE key, the marker appears automatically
and as you move it to the left or right, the current coordinates are shown at the bottom