Rate of Change of Linear Functions

4.8.4 Rate of Change of Linear Functions

4.8.4 • Rate of Change of Linear Functions

Activity

Three people, Adam, Bianca, and Carmen, are competing to see who can save the most money in one month. Use the table, graph, and equation below to answer the questions.

Three functions representing savings in dollars as a function of number of days for three people: Adam, Bianca, and Carmen the total amount Adam has saved is represented by the graph of a line that passes through the points (0, 3) and (1, 6) .The total amount Bianca has saved is represented by a table with the points (5, 17), (8, 26), (12, 38), and (20, 62). The total amount Carmen has saved is represented by the equation f of x equals 2x plus 30.

1. Use the graph to find the average rate of change of the function that represents Adam’s savings for the following intervals:

a. between 0 and 1 days

Compare your answer:

3 dollars per day

b. between 1 and 2 days

Compare your answer:

3 dollars per day

c. between 1 and 3 days

Compare your answer:

3 dollars per day

2. Use the table to find the average rate of change for the function that represents Bianca’s savings for the following intervals:

a. between 5 and 8 days

Compare your answer:

3 dollars per day

b. between 12 and 20 days

Compare your answer:

3 dollars per day

c. between 5 and 20 days

Compare your answer:

3 dollars per day

3. Use the equation to find the average rate of change for the function that represents Carmen’s savings for the following intervals:

a. between 1 and 2 days

Compare your answer:

2 dollars per day

b. between 2 and 4 days

Compare your answer:

2 dollars per day

c. between 1 and 4 days

Compare your answer:

2 dollars per day

4. Which statements are true?

All three functions are linear functions.
All three functions have the same rate of change.
All three functions have a constant rate of change.

Compare your answer:

a, c

5. Who has the most money after 30 days? Explain how you know.

Compare your answers:

Adam; Adam’s function is $A (30) = 3 (30) + 3 = 93$ ; Bianca’s function is $B (30) = 3 (30) + 2 = 92$ ; Carmen’s function is $C (30) = 2 (30) + 30 = 90$ .

Self Check

What is the average rate of change for the linear function?

Input	Output
1	2
2	-1
4	-7
6	-13

$\frac13$
$-\frac13$
$-3$
$3$

Additional Resources

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, $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

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

Graph of a line passing through the points (16, 1) and (24, 1.5).

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.

Input	Output
-2	3
8	-2
10	-3
20	-8

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.

$F (x) = 7 x + 6$

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

$y = 7 x + 6$

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?

A line graph shows volume in gallons on the y-axis and time in minutes on the x-axis. The orange line rises steadily, indicating a constant increase in volume over time.

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.