Skill: Interpret the slope of a linear relation

Slope tells how much the dependent variable changes for each 1-unit change in the independent variable:

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

Example

A taxi’s total fare depends linearly on the distance traveled.

Distance, $x$ (km)	Fare, $y$ (dollars)
$2$	$7$
$5$	$13$

Step 1: Identify the variables.

• Independent variable: distance, $x$
• Dependent variable: fare, $y$

Step 2: Find the changes.

From $2$ km to $5$ km:

$$\Delta x=5-2=3\text{ km}$$

From 7$to$$13$:

$$\Delta y=13-7=\$6$$

Step 3: Calculate the slope.

$$m=\frac{\Delta y}{\Delta x}=\frac{\$6}{3\text{ km}}=\$2\text{ per km}$$

Step 4: Interpret the slope.

The slope is 2\text{ per km}}$. This means the fare increases by$$2$ for every additional kilometre traveled.

The slope is not the point $(2,7)$. That point means the fare is 7$when the distance is$2$ km. The slope describes the constant rate of change between points.

• A positive slope means $y$ increases as $x$ increases.
• A negative slope means $y$ decreases as $x$ increases.
• A zero slope means $y$ stays constant as $x$ increases.

Always include units when interpreting a slope: use the units of $y$ divided by the units of $x$.