3.01 Graphs from tables

Remember that a table of values can be constructed by substituting multiple values into a linear equation. Once a table of values has been constructed, the table can then be used to create a graph.

Plotting points from a table of values

Each column in a table of values may be grouped together in the form $\left(x,y\right)$(x,y). This pairing of numbers is known as an ordered pair.

 

Exploration

Let's plot the ordered pairs from the following table of values:

$x$x	$1$1	$2$2	$3$3	$4$4
$y$y	$-2$−2	$1$1	$4$4	$7$7

The table of values has the following ordered pairs:

$\left(1,-2\right),\left(2,1\right),\left(3,4\right),\left(4,7\right)$(1,−2),(2,1),(3,4),(4,7)

Each ordered pair becomes a point on the $xy$xy-plane.

The ordered pair $\left(a,b\right)$(a,b) is plotted on the number plane by first identifying where $x=a$x=a is along the $x$x-axis and where $y=b$y=b is along the $y$y-axis.

Take $\left(3,4\right)$(3,4) as an example. Identify $x=3$x=3 along the $x$x-axis and draw a vertical line through this point. Then identify $y=4$y=4 along the $y$y-axis and draw a horizontal line through that point. Finally, plot a point where the two lines meet, and this represents the ordered pair $\left(3,4\right)$(3,4).

 

Drawing a straight line from points on the plane

By plotting points on the number plane that correspond to ordered pairs from the table of values, a straight line can then be drawn that passes through each of these points.

 

Exploration

In the example above, the line that passes through these points is given by:

This straight line is the graph of $y=-5+3x$y=−5+3x which was used to complete the table of values.

 

Drawing a straight line from a table of values

To draw a line from a table of values, it is useful to plot the significant points and draw the line that passes through them.

 

Exploration

Consider the following linear equation:

$y=-6+3x$y=−6+3x

And the following table of values:

$x$x	$0$0	$1$1	$2$2	$3$3
$y$y	$-6$−6	$-3$−3	$0$0	$3$3

There are two significant ordered pairs, namely the $x$x-intercept and the $y$y-intercept.

• The $x$x-intercept has the form $\left(a,0\right)$(a,0) which is a point that lies on the $x$x-axis.
• The $y$y-intercept has the form $\left(0,b\right)$(0,b) which is a point that lies on the $y$y-axis.

The $x$x-intercept in our example is $\left(2,0\right)$(2,0) and the $y$y-intercept is $\left(0,-6\right)$(0,−6).

The line represented by equation $y=-6+3x$y=−6+3x can be graphed by drawing a line which passes through these two points.

 

Practice questions

Question 1