Interpret slope as a rate of change

Slope is the constant rate at which one quantity changes relative to another in a linear relationship, calculated as $\Delta y/\Delta x$ and interpreted with appropriate units, such as dollars per hour. Its value and sign indicate the direction and steepness of change—positive, negative, or zero—and can be determined from a graph, table, equation $y=mx+b$ , or two points; slope is not generally $y/x$ or merely a visual angle. This treatment concerns linear rates, not instantaneous rates of nonlinear functions.

Detailed Explanation: Interpret slope as a rate of change

Slope tells you how much one quantity changes for each 1-unit increase in another quantity.

For a linear relationship,

\text{slope}=\frac{\Delta y}{\Delta x} =\frac{\text{change in }y}{\text{change in }x}.

The symbol $\Delta$ means “change in,” so:

\Delta y=y_2-y_1 \quad\text{and}\quad \Delta x=x_2-x_1.

Example

A taxi fare is 8 $after 2 miles and$ $14$ after 5 miles. Find and interpret the slope.

Step 1: Identify two points.

Let $x$ be the number of miles and $y$ be the fare:

(x_1,y_1)=(2,8), \qquad (x_2,y_2)=(5,14).

Step 2: Find the change in the fare.

\Delta y=14-8=6

The fare increases by 6$.

Step 3: Find the change in miles.

\Delta x=5-2=3

The distance increases by 3 miles.

Step 4: Divide the changes.

m=\frac{\Delta y}{\Delta x} =\frac{6}{3}=2

The slope is $2$ .

Step 5: Include the units and interpret.

The slope is

\boxed{\$2\text{ per mile}}.

This means the fare increases by 2$ for every additional mile. Because the slope is positive, the fare increases as the number of miles increases.

Remember, slope is the change in $y$ divided by the change in $x$ , not usually $y/x$ . For a linear relationship, the slope stays constant between any two points.

Learn by doing: Interpret slope as a rate of change

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Slope - Find Equivalent - X,Y Chart to Decimal Slope

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