Determine rate of change from a graph

Rate of change from a linear graph is the ratio of the change in the vertical quantity to the corresponding change in the horizontal quantity, calculated as $\frac{\Delta y}{\Delta x}$ using two points on the line. The value indicates the rate and units of change, while its sign indicates whether the relationship increases or decreases; a horizontal line has rate zero, and visual steepness alone does not determine the rate. This treatment concerns constant rates on straight-line graphs, not instantaneous rates or nonlinear models.

Detailed Explanation: 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.

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Slope of a Line from Graph (without points) - Integer

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