4.8.4 Rate of Change of Linear Functions

Rate of Change of Linear Functions.

 The graph of a linear function is identical to the graph of the linear equation that describes it. The rate of change of a linear equation is constant. The rate of change is the slope, $m=\frac{y_2-y_1}{x_2-x_1}$

1. Find the rate of change of a linear function from a graph.

The rate of change of a linear function is constant, so you can choose any 2 points to find the slope.

Use $(16,1)$ and $(24,1.5)$.

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{1.5-1}{24-16}=\frac{0.5}{8}=\frac{1}{16}$

The rate of change is $\frac{1}{16}$

2. Find the rate of change of a linear function from a table.

The rate of change of a linear function is constant, so you can choose any 2 points to find the slope.

Use $(-2,3)$ and $(8,-2)$.

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{-2-3}{8-(-2)}=\frac{-5}{10}=-\frac{1}{2}$

The rate of change is $-\frac{1}{2}$.

3. Find the rate of change of a linear function from an equation.

$F(x)=7x+6$

Use the slope intercept form, $y=mx+b$, where $m$ is the slope and $b$ is the $y$-intercepts.

$y=7x+6$

The slope is 7. The rate of change is 7.

Try it

Try It: Rate of Change of Linear Functions

The function $V(x)$ gives the volume of water that flows from a faucet in gallons during $𝒙$ minutes. $V(x)$ is a linear function with the graph shown.What is the rate of change of the volume?

Compare your answer:

Here is how to find the rate of change of the linear function:

Identify two points on the graph.

Use $(0,0)$ and $(7,4)$.

Find the slope:

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{4-0}{7-0}=\frac{4}{7}$

The slope is $\frac{4}{7}$, so, the rate of change is $\frac{4}{7}$ gallons per minute.