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.
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.
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.
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 |
The height of a candle is measured every $15$15minutes.
Complete the table of values below:
Time (minutes) | $15$15 | $30$30 | $45$45 | $60$60 |
---|---|---|---|---|
Height (cm) |
$\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |
Consider the equation $y=5x+6$y=5x+6.
Complete the table of values below:
$x$x | $-10$−10 | $-5$−5 | $0$0 | $5$5 |
---|---|---|---|---|
$y$y | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |
A racing car starts the race with $140$140 litres of fuel. From there, it uses fuel at a rate of $2$2 litres per minute.
Complete the table of values:
Number of minutes passed ($x$x) | $0$0 | $5$5 | $10$10 | $15$15 | $20$20 | $70$70 |
---|---|---|---|---|---|---|
Amount of fuel left ($y$y) | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |