Skill: Determine the rate of change from an equation

For a linear equation written as

$$y=mx+b,$$

the rate of change is $m$, the coefficient of $x$. It tells how much $y$ changes when $x$ increases by 1.

• If $m$ is positive, $y$ increases.
• If $m$ is negative, $y$ decreases.
• The number $b$ is the initial value, not the rate of change.

Example

Suppose the equation for the cost of renting a bike is

$$C=4h+10,$$

where $h$ is the number of hours and $C$ is the cost in dollars.

Step 1: Identify the coefficient of the independent variable.

The coefficient of $h$ is $4$.

$$m=4$$

Step 2: State the rate of change with units.

The cost increases by $4 per hour. Therefore, the rate of change is

$$\boxed{\$4\text{ per hour}}.$$

The $10$ represents the starting fee when $h=0$; it is the initial value, not the rate of change.

You can check this using two values. At $h=1$,

$$C=4(1)+10=14,$$

and at $h=2$,

$$C=4(2)+10=18.$$

The change in cost is 18-$14=$4$for a change of$2-1=1$ hour:

$$\frac{\Delta C}{\Delta h}=\frac{4}{1}=4.$$

So the rate of change is still 4\text{ per hour}}$.