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B. The Graph of a Relation

B. The Graph of a Relation

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CHAPTER 1 Relations, Functions, and Graphs



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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CHAPTER 1 Relations, Functions, and Graphs



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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Section 1.1 Rectangular Coordinates; Graphing Circles and Other Relations



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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



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