Determine the rate of change from an equation

For a linear equation, the rate of change is the constant change in the dependent variable for each one-unit change in the independent variable, represented by the coefficient of $x$ when written as $y=mx+b$ , or found from $\Delta y/\Delta x$ . Learners interpret its sign, magnitude, and units in tables, graphs, and equations, distinguishing it from the initial value $b$ ; this scope excludes instantaneous rates of change and nonlinear functions.

Detailed Explanation: 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}}$.

Learn by doing: Determine the rate of change from an equation

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