Recognize constant rate of change in a graph

A graph shows a constant rate of change when the ratio of the change in $y$ to the change in $x$ is the same across intervals; equal horizontal changes produce equal vertical changes. This relationship appears as a straight, nonvertical line and may have a positive, negative, or zero rate, whether or not it passes through the origin, supporting interpretation of slope and linear equations without requiring advanced function analysis.

Detailed Explanation: 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:

Find the change in $x$ between points.
Find the change in $y$ between the same points.
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 $(0,2)$ .

Learn by doing: Recognize constant rate of change in a graph

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