Table of values
Tables are used everywhere in mathematics, usually to show data for two or more related quantities (represented by variables).
If we have a table of values relating two quantities, it is often useful to figure out if there is a relationship between them, and what the relationship is. If we can find such a relationship, it can be used to predict future values and patterns.
We will start by looking at simple relationships, where we can get from one variable to the other by a single operation ($+$+, $-$−, $\times$× or $\div$÷).
Example 1
Use the following table of values to write an equation for $y$y in terms of $x$x.
| $x$x | $1$1 | $2$2 | $3$3 | $4$4 | $5$5 |
|---|---|---|---|---|---|
| $y$y | $5$5 | $6$6 | $7$7 | $8$8 | $9$9 |
Think: The $y$y values are larger than the $x$x values, so there must be addition or multiplication involved.
Do: To get from $x=1$x=1 to $y=5$y=5 we can either add $4$4 or multiply by $5$5. To get from $x=2$x=2 to $y=6$y=6 we can either add $4$4 or multiply by $3$3.
So the pattern is "add $4$4 to $x$x" (make sure to check that this works for the rest of the numbers). We can write this as an equation as $y=x+4$y=x+4.
- Are the numbers in the second row smaller or larger than the numbers in the first row?
- If they are larger, then addition or multiplication is involved.
- If they are smaller, then subtraction or division is involved.
- Addition and subtraction are opposites, as are multiplication and division. If you figure out the pattern in the wrong order, you can reverse it by using the opposite operation.
- For example, if we did the previous question in the wrong order, we would have found that "$x=y-4$x=y−4".
- Since addition is the opposite of subtraction, we could convert this to "$y=x+4$y=x+4" to get an equation for $y$y in terms of $x$x.
Example 2
Use the following table of values to write an equation for $q$q in terms of $p$p.
| $p$p | $10$10 | $15$15 | $20$20 | $25$25 | $30$30 |
|---|---|---|---|---|---|
| $q$q | $2$2 | $3$3 | $4$4 | $5$5 | $6$6 |
Think: The $q$q values are smaller than the $p$p values, so there must be subtraction or division involved.
Do: To get from $p=10$p=10 to $q=2$q=2 we can either subtract $8$8 or divide by $5$5. To get from $p=15$p=15 to $q=3$q=3 we can either subtract $12$12 or divide by $5$5.
So the pattern is "divide $p$p by $5$5" (make sure to check that this works for the rest of the numbers). We can write this as an equation as $q=\frac{p}{5}$q=p5.
Worked Examples
Question 1
Write an equation for $g$g in terms of $f$f.
Question 2
Write an equation for $g$g in terms of $f$f.