Task Sets: Calculate average rate of change - Common Core Math - HS Functions

A

Recall / Warm-Up

1.

A line passes through $(1, 3)$ and $(4, 12)$ . What is the slope of the line?

A.

3

B.

9

C.

$\dfrac{1}{3}$

D.

4

2.

If $f(x) = x^2 + 3x$ , what is $f(2)$ ?

A.

8

B.

10

C.

7

D.

11

3.

For a linear function $f(x) = 5x - 2$ , which statement about slope is true?

A.

The slope depends on which two points you pick.

B.

The slope is the same no matter which two points you pick.

C.

The slope equals the $y$ -intercept.

D.

The slope is always positive for linear functions.

B

Fluency Practice

1.

Let $f(x) = 3x + 7$ . Find the average rate of change of $f$ over the interval $[1, 5]$ .

2.

Let $g(x) = x^2$ . Find the average rate of change of $g$ over the interval $[2, 5]$ .

3.

The table below shows values of a function $h$ .

$x$	0	2	4	6
$h(x)$	5	11	21	35

Average rate of change from $x = 0$ to $x = 6$ : $\dfrac{h(6) - h(0)}{6 - 0} = \dfrac{\underline{\hspace{5em}} - \underline{\hspace{5em}}}{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}} = \underline{\hspace{5em}}$ .

h(6):

h(0):

denominator:

result:

4.

Let $p(t) = -16t^2 + 64t + 4$ . Find the average rate of change of $p$ from $t = 1$ to $t = 3$ .

5.

Let $f(x) = \sqrt{x}$ . Find the average rate of change of $f$ over the interval $[4, 16]$ .

C

Varied Practice

D

Word Problems

1.

A plumber charges $C(h) = 40h + 75$ dollars for a job lasting $h$ hours. Compute the average rate of change of $C$ over $[1, 3]$ and $[0, 5]$ . Then explain what you notice.

Show your computation for both intervals and describe what the results tell you about this function.

2.

The table shows the height of a sunflower plant over several weeks.

Week	0	2	4	6	8	10
Height (in)	2	8	18	30	38	42

1.

What is the average rate of change in height from week 0 to week 10? Give your answer in inches per week.

2.

Compare the average rate from week 4 to week 8 with the overall rate from part (a). During which period did the plant grow faster? What does this tell you about the plant's growth?

3.

A town's population (in thousands) is recorded every two years.

Year	0	2	4	6	8	10
Pop. (thousands)	25.0	27.4	30.2	33.8	38.0	43.2

Compute the average rate of change for each 2-year interval. Based on the results, is this population growing linearly? Explain.

4.

A cooling cup of coffee has its temperature recorded at various times.

Time (min)	0	5	15	30	60
Temp (F)	200	172	130	98	76

Find the average rate of change in temperature from $t = 5$ to $t = 30$ . Include units and interpret your answer in context.

E

Error Analysis

1.

A student finds the average rate of change of $f(x) = x^2$ from $x = 1$ to $x = 4$ :
$\frac{f(1) - f(4)}{4 - 1} = \frac{1 - 16}{3} = \frac{-15}{3} = -5$

What error did the student make?

A.

The student subtracted in the wrong order in the numerator — should be $f(4) - f(1)$ , giving $+5$ .

B.

The student computed $f(1)$ and $f(4)$ incorrectly.

C.

The denominator should be $1 - 4 = -3$ , not $4 - 1 = 3$ .

D.

The student should have computed $f(4)/f(1)$ instead of the difference.

2.

A student is asked for the average rate of change of $g(x) = x^2 + 4$ from $x = 0$ to $x = 6$ . The student computes:
$\text{Average rate} = \frac{g(0) + g(6)}{2} = \frac{4 + 40}{2} = 22$

What mistake did the student make, and what is the correct answer?

A.

The student found the average of the two output values instead of the average rate of change. The correct formula is $\dfrac{g(6)-g(0)}{6-0}$ , giving $\dfrac{36}{6} = 6$ .

B.

The student computed $g(0)$ and $g(6)$ incorrectly.

C.

The student should have divided by the number of $x$ -values, which is 3 (for $x = 0, 3, 6$ ).

D.

The student needed to use the slope formula: $g(6)/g(0) = 40/4 = 10$ .

F

Challenge / Extension

1.

For $f(x) = x^2$ , find an interval of the form $[a, a+2]$ where the average rate of change equals 10. Find the value of $a$ and show your work. Is this the only such interval of width 2? Explain.

2.