5.3.2: Graphs and Tables

Learning Outcomes

Graph coordinate pairs from a table of values
Create a table of values from a graph
Write a relation as a set of ordered pairs
Identify the domain and range of a relation
Identify local maxima and minima
Identify lines of symmetry

KEY words

A relation is a set of ordered pairs.
The domain of a relation is the set of all $x-$ values.
The range of a relationship is the set of all $y-$ values.
A local maximum is the $y-$ value of a point where the graph turns to go back up after moving downwards.
A local minimum is the $y-$ value of a point where the graph turns to go back down after moving upwards.
A maximum is the highest $y-$ value on the graph.
A minimum is the lowest $y-$ value on the graph.

Graphs and Tables

Graphs and tables have a symbiotic relationship. Ordered pairs can be written in the rows and columns of a table or graphed as points using the rectangular coordinate system. The same data can be displayed as a table or a graph.

Figure 1 shows a table of data points that when converted to $(x,y)$ coordinates can be plotted on a rectangular coordinate system to become a graph.

Points on a graph

$x$	$y$
$-4$	$7$
$-2$	$6$
$0$	$5$
$2$	$4$
$4$	$3$
$6$	$2$
$8$	$0$

Figure 1

A set of ordered pairs is called a relation. From figure 1, the relation is $\{(-4,7),(-2,6),(0,5),(2,4),(4,3),(6,2),(8,0)\}$ . The set of all values of the independent variable $x$ is the domain of the relation, while the set of all the values of the dependent variable $y$ is the range of the relation. For the data in figure 1, domain = $\{-4,-2,0,2,4,6,8\}$ and range = $\{0,2,3,4,5,6,7\}$ .

Relation

A relation is a set of ordered pairs $(x,y)$ .

The domain of a relation is the set of all $x$ -values.

The range of a relation is the set of all $y$ -values.

We can also convert from coordinates on a graph to data points in a table.

example

Use the graph to complete the table. Then state the domain and range of the relation.

Points on a graph

$x$	$y$
$0$
$7$
	$0$
	$2$
$-4$
	$4$

Solution

$x$	$y$
$0$	$4$
$7$	$-3$
$-1$	$0$
$3$	$2$
$-4$	$-5$
$5$	$4$

domain = $\{-4,-1,0,3,5,7\}$ and range = $\{-5,-3,0,2,4\}$

We only listed $4$ once because it is not necessary to list it every time it appears in the range.

TRY IT

List the domain and range for the following table of values, then graph the data points on a coordinate system.

x	y
$−3$	$4$
$−2$	$4$
$−1$	$4$
$2$	$4$
$3$	$4$

Show Solution

The domain describes all the $x$ -values (independent variable values), and we can use set notation with braces { } to make the list.

$\text{Domain}:\{-3,-2,-1,2,3\}$

The range describes all the $y$ -values (dependent variable values).

$\text{Range}:\{4\}$

We only listed $4$ once because it is not necessary to list it every time it appears in the range.

points with y = 4 in a horizontal line

The graphs we have seen so far in this section have been of relations with a finite number of points. Some relations have an infinite number of points, with the points lying so close together on a graph they form a curve or a line. Since the relation has an infinite number of points, it is impossible to write every ordered pair in a table. However, a few points can be tabulated to show a pattern.