Skill: Recognize constant rate of change in a graph

A graph shows a constant rate of change when equal changes in $x$ always produce equal changes in $y$.

To check, choose points on the graph and compare the changes:

1. Find the change in $x$ between points.
2. Find the change in $y$ between the same points.
3. Check whether the ratio $\frac{\text{change in }y}{\text{change in }x}$ stays the same.

Example

Suppose a graph passes through these points:

$$(0,2),\ (1,5),\ (2,8),\ (3,11)$$

Check the changes as you move from left to right:

• From $(0,2)$ to $(1,5)$: $x$ increases by $1$, and $y$ increases by $3$.
• From $(1,5)$ to $(2,8)$: $x$ increases by $1$, and $y$ increases by $3$.
• From $(2,8)$ to $(3,11)$: $x$ increases by $1$, and $y$ increases by $3$.

Each equal horizontal change of $1$ gives the same vertical change of $3$. The rate of change is

$$\frac{\text{change in }y}{\text{change in }x}=\frac{3}{1}=3.$$

Therefore, the graph shows a constant rate of change. Its points form a straight line that rises from left to right. The line does not need to pass through $(0,0)$; here, it passes through $(0,2)$.