Skill: Represent linear relationships using tables

A table represents a linear relationship by matching each input, $x$, with exactly one output, $y$. In a linear relationship, equal changes in $x$ always cause equal changes in $y$.

Example: A video game score starts at $12$ points and decreases by $3$ points each round. Make a table for rounds $0$ through $4$.

1. Let the number of rounds be the input, $x$.
2. Let the score be the output, $y$.
3. Start with $12$ points when $x=0$.
4. Subtract $3$ points for each additional round.

Rounds, $x$	Score, $y$
$0$	$12$
$1$	$9$
$2$	$6$
$3$	$3$
$4$	$0$

Check the changes:

• The input increases by $1$ each time.
• The output decreases by $3$ each time.

Because the output changes by the same amount for equal changes in the input, the relationship is linear. Its constant rate of change is $-3$ points per round.

The table gives ordered pairs such as $(0,12)$, $(1,9)$, and $(2,6)$. These pairs can be plotted on a graph. The relationship can also be written as an equation:

$$y=-3x+12$$

Here, $-3$ is the rate of change, and $12$ is the starting value.