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 $\div$÷).
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.
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.
Write an equation for $g$g in terms of $f$f.
Write an equation for $g$g in terms of $f$f.