Skill: Determine rate of change from a graph

The rate of change tells how much the vertical value, $y$, changes for each 1-unit change in the horizontal value, $x$.

For a straight-line graph, use two points on the line:

$$\text{rate of change}=\frac{\text{change in }y}{\text{change in }x} =\frac{\Delta y}{\Delta x}$$

Example

A line on a graph passes through the points $(1,3)$ and $(5,11)$. Find its rate of change.

Step 1: Identify the two points.

Let

$$(x_1,y_1)=(1,3) \quad\text{and}\quad (x_2,y_2)=(5,11).$$

Step 2: Find the change in $y$.

$$\Delta y=y_2-y_1=11-3=8$$

The vertical value increases by $8$.

Step 3: Find the change in $x$.

$$\Delta x=x_2-x_1=5-1=4$$

The horizontal value increases by $4$.

Step 4: Divide the changes.

$$\text{rate of change}=\frac{\Delta y}{\Delta x} =\frac{8}{4}=2$$

The rate of change is $\boxed{2}$, or 2 vertical units for every 1 horizontal unit.

Because the answer is positive, the line increases from left to right. If the answer were negative, the line would decrease. A horizontal line has a rate of change of $0$ because its $y$-value does not change.