Given two variables, say $x$x and $y$y, that are connected in some way, we can describe the relationship between the two variable using a formula, a graph or a table of values.
A table of values shows certain values of $y$y that occur for certain values of $x$x - this gives us a snapshot of the relationship. This can often be the first step in finding a formula or finding pairs to graph the relationship.
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 |
A table of values is a table used to present the quantities of two variables that are related in some way.
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 and asked to create a table that shows a snapshot of the relationship for certain values. 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. You may also be able to complete the table by noticing a pattern.
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$22 |
Practice questions
Question 1
Complete the table for the figures in the given pattern.
Step number ($x$x) $1$1 $2$2 $3$3 $4$4 $5$5 $10$10 Number of matches ($y$y) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
Question 2
Complete the table of values using the formula $q=2p-3$q=2p−3.
$p$p $0$0 $1$1 $2$2 $3$3 $4$4 $q$q $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
QUESTION 3
There are $20$20 litres of water in a rainwater tank. It rains for a period of $24$24 hours and during this time the tank fills up at a rate of $8$8 litres per hour.
Complete the table of values:
Number of hours passed ($x$x) $0$0 $4$4 $6$6 $7$7 $9$9 $11$11 $12$12 Amount of water in tank ($y$y) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$