Graphing functions

High School Math 1 · Unit 11 · function-features · Teacher edition

function-features

Name ______________________________Date ______________

1.
A piecewise function is defined by

$f(x) = \begin{cases} x + 3 & \text{if } x < 0 \\ 2x & \text{if } x \ge 0 \end{cases}$

Find $f(-2)$ .
Answer: ______________
2.
Let $f(x) = x^{2} - 2x$ . Find the average rate of change of $f$ from $x = 1$ to $x = 4$ .
Answer: ______________
3.
A graph starts at $(-4, 2)$ , falls to a low point at $(-1, -3)$ , rises to a high point at $(2, 5)$ , then falls to its end at $(5, 0)$ . On which interval is the function increasing?
1. (A)
  $(-1, 2)$
2. (B)
  $(-4, -1)$
3. (C)
  $(2, 5)$
4. (D)
  $(-3, 5)$
4.
The table gives the height $h(t)$ , in metres, of a drone $t$ seconds after launch.

$t$ $0$ $2$ $5$
$h(t)$ $12$ $20$ $44$

What is the average rate of change of $h$ from $t = 2$ to $t = 5$ , in metres per second?
1. (A)
  $24$
2. (B)
  $8$
3. (C)
  $\dfrac{32}{3}$
4. (D)
  $6.4$
5.
What is the maximum value of the function $f(x) = -x^{2} + 6x - 5$ ?
Answer: ______________

$t$	$0$	$2$	$5$
$h(t)$	$12$	$20$	$44$

Answer key — Graphing functions

1.
1function-features-01
$-2 < 0$ , so $f(-2) = -2 + 3 = \mathbf{1}$ . (Using $2x$ would give $-4$ , but that rule is only for $x \ge 0$ .)
2.
3function-features-02
$\dfrac{f(4) - f(1)}{4 - 1} = \dfrac{8 - (-1)}{3} = \dfrac{9}{3} = \mathbf{3}$ .
3.
(A)
$(-1, 2)$
function-features-03
The graph rises between the low point and the high point, so $f$ is increasing on $\mathbf{(-1, 2)}$ . ( $(-3, 5)$ lists the $y$ -values of those two points, not an $x$ -interval; on $(-4, -1)$ and $(2, 5)$ the graph is falling.)
4.
(B)
$8$
function-features-04
$\dfrac{h(5) - h(2)}{5 - 2} = \dfrac{44 - 20}{3} = \mathbf{8}$ m/s. ( $24$ is $h(5) - h(2)$ without dividing by the time; $\frac{32}{3}$ uses the height change from $t = 0$ ; $6.4$ is $\frac{32}{5}$ , the rate over the whole table.)
5.
4function-features-05
Vertex at $x = 3$ ; $f(3) = -9 + 18 - 5 = \mathbf{4}$ . The maximum value is $4$ , reached at $x = 3$ . (Reporting $3$ answers “where,” not “what.”)