Table of values

Given two variables, $x$x and $y$y, is there a way to show how these two variables are related? At the very least, we may be able to see certain values of $y$y that occur at certain values of $x$x. We can collect this information into a table of values.

Exploration

Imagine we started with a triangle made out of matchsticks. We can form a pattern by adding two additional matchsticks each time as shown below.

The table of values for this pattern connects the number of triangles made ($x$x) with the number of matches needed ($y$y).

Number of triangles ($x$x)	$1$1	$2$2	$3$3	$4$4
Number of matches ($y$y)	$3$3	$5$5	$7$7	$9$9

As we saw before, a table of values may be used to describe a pattern. However, we may also be given an equation or a rule to describe the relationship between two variables. Let's take a look below.

Worked example

Consider the equation $y=3x-5$y=3x−5. Using this rule, we want to complete the following table of values.

$x$x	$1$1	$2$2	$3$3	$4$4
$y$y	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$

Think: We wish to find the value of $y$y at each value of $x$x in the table of values.

Do: First we find the value of $y$y when $x=1$x=1 by substitution.

Substituting $x=1$x=1 into $y=3x-5$y=3x−5 we end up with:

$y=3\times\left(1\right)-5$y=3×(1)−5

Which simplifies to give:

$y=-2$y=−2

So after finding the value of $y$y when $x=1$x=1, we have:

$x$x	$1$1	$2$2	$3$3	$4$4
$y$y	$-2$−2	$\editable{}$	$\editable{}$	$\editable{}$

Reflect: In general, we can complete a table of values by repeating this process of substitution for each variable given in the table.

Completing the rest of the table of values gives us:

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

For a table of values, the values of $x$x do not need to increase by one each time. We could obtain the following table of values repeating the same procedure as before:

$x$x	$1$1	$3$3	$5$5	$9$9
$y$y	$-2$−2	$4$4	$10$10	$22$22

Practice Examples

Question 1

Question 2

QUESTION 3