Before you read

Answer first — see what you already know.

A tank has 50 gallons of water and is being filled at 8 gallons per minute. Which function gives the amount of water $W$ after $t$ minutes?

A
$W = 50 t + 8$
B
$W = 8 t + 50$
C
$W = 58 t$
D
$W = 50 - 8 t$

Check your answer

Answer:

Before you read

Answer first — see what you already know.

The table shows a car's distance from home.

Time (hours)	Distance (miles)
1	95
3	215
5	335

What is the car's average speed?

A
$55$ mph
B
$60$ mph
C
$67$ mph
D
$95$ mph

Check your answer

Answer:

Theory

Rate of Change = Slope

In a linear function $f(x) = mx + b$ :

- Rate of change = $m$ = slope = how fast the output changes per unit of input
- Initial value = $b$ = y-intercept = the output when the input is $0$

Rate of change = rise/run = 10/4 = 2.5

The rate of change is constant for linear functions — that's what makes them linear! No matter which two points you pick, $\frac{\Delta y}{\Delta x}$ is always the same.

Theory

Average Rate of Change

The average rate of change of a function $f$ from $x = a$ to $x = b$ is:

$\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$

For linear functions, this equals the slope everywhere.
For nonlinear functions, it varies depending on the interval (we'll cover this later).

Example 1

A plant is 3 cm tall at week 0 and 15 cm tall at week 4. What is the average growth rate?

$\text{Rate} = \frac{f(4) - f(0)}{4 - 0} = \frac{15 - 3}{4} = \frac{12}{4} = 3 \text{ cm/week}$

The plant grows at an average rate of

3

cm per week.

Example 2

A table shows:

Hours	Distance (miles)
2	130
5	310

What is the average speed?

$\text{Speed} = \frac{310 - 130}{5 - 2} = \frac{180}{3} = 60 \text{ mph}$

60

mph

Continue this lesson in the app

3 more sections including examples, practice problems, and step-by-step solutions.

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