Print the chapter

Every lesson of the chapter, ready for paper. Exercise sets are lettered A, B, C… and answered at the end.

Include answers

To save a PDF, choose “Save as PDF” in the print dialog.

1. What is a function: one output for each input

Warm-up

Three quick questions from earlier lessons.

Discover

A vending machine: you press a button (input) and get a snack (output). Pressing B4 sometimes gives chips and sometimes gives a cookie. What is wrong?

One input gives two different outputs
Two buttons might give the same snack

Key idea

A function assigns exactly one output to each input.

A function is a rule that assigns to each input exactly one output. Different inputs may share an output. On a graph, no vertical line meets a function's graph twice. A function can be given by a table, a graph, an equation or words.

Worked example

$(1, 5), (2, 5), (3, 7)$ and $(1, 5), (1, 6), (2, 7)$ .

First: a function

Each input once.

Second: not a function

Input 1 has outputs 5 and 6.

Your turn, guided

Your turn: the rule “triple the input, then subtract 4”. The output for input 5.

11

15 − 4.

Practice ★

A table pairs inputs $x$ with outputs $y$ : $(x, y) = (5, 1), (5, 2), (9, 3)$ . Is $y$ a function of $x$ ?
1. No: the input 5 has two different outputs
2. Yes: every input has an output
3. Yes: the outputs are all different
A table pairs inputs $x$ with outputs $y$ : $(x, y) = (4, 1), (4, 2), (6, 3)$ . Is $y$ a function of $x$ ?
1. No: the input 4 has two different outputs
2. Yes: every input has an output
3. Yes: the outputs are all different

A table pairs inputs $x$ with outputs $y$ : $(x, y) = (1, 5), (5, 5), (9, 5)$ . Is $y$ a function of $x$ ?
1. Yes: each input has exactly one output
2. No: the output 5 repeats
3. No: a function needs a formula

Watch out!

Practice ★★

A function's rule: multiply the input by 6, then add 6. Which input gives the output 24?
A function's rule: multiply the input by 4, then add 9. Which input gives the output 21?

Exit ticket

Three questions to check you have got it.

A table pairs inputs $x$ with outputs $y$ : $(x, y) = (5, 1), (5, 2), (9, 3)$ . Is $y$ a function of $x$ ?
1. No: the input 5 has two different outputs
2. Yes: every input has an output
3. Yes: the outputs are all different
A table pairs inputs $x$ with outputs $y$ : $(x, y) = (4, 1), (4, 2), (6, 3)$ . Is $y$ a function of $x$ ?
1. No: the input 4 has two different outputs
2. Yes: every input has an output
3. Yes: the outputs are all different
A table pairs inputs $x$ with outputs $y$ : $(x, y) = (2, 1), (2, 2), (8, 3)$ . Is $y$ a function of $x$ ?
1. No: the input 2 has two different outputs
2. Yes: every input has an output
3. Yes: the outputs are all different

Summary card

Functions

Exactly one output for each input.

Watch out: shared outputs are fine; two outputs for one input are not.

2. Function notation: evaluate and read f(x)

Discover

$f(x) = x + 5$ is a rule: it takes a number and adds 5.

What does $f(2)$ mean?

The output of $f$ when the input is 2
$f$ times 2
The input that gives the output 2

Key idea

A function gives exactly one output for each input. We write the output as $f(x)$ .

f(a)

is the output of

f

for the input

a

. To find it, replace every

x

in the rule by

(a)

, with parentheses.

On a graph, $f(a)$ is the height of the graph above $x = a$ .

Worked example

$f(x) = 3x^2 - x$ . Find $f(-2)$ .

f(-2) = 3(-2)^2 - (-2)

Replace each x by (−2), with the parentheses.

= 3 \cdot 4 + 2

(−2)² = 4, and subtracting −2 adds 2.

= 14

Multiply, then add.

Your turn, guided

Your turn: $f(x) = 2x^2 + 1$ . Fill in the missing line.

f(-3) = 2(-3)^2 + 1

Replace x by (−3).

f(-3) = 19

(−3)² = 9, then 2 · 9 + 1.

Practice ★

$f(x) = 2x^2 + 3$ . Find $f(-3)$ .
$f(x) = 4x^2 + 8$ . Find $f(-2)$ .
$f(x) = 4x^2 + 2$ . Find $f(-5)$ .
$f(x) = 5x^2 + 5$ . Find $f(-4)$ .
$f(x) = 3x^2 + 3$ . Find $f(-4)$ .

Watch out!

Practice ★★

$f(x) = x^2 + 3$ . Find $f(2 + 3)$ .
$f(x) = x^2 + 2$ . Find $f(4 + 4)$ .
$f(x) = x^2 + 7$ . Find $f(4 + 1)$ .
$f(x) = x^2 + 3$ . Find $f(3 + 3)$ .

Here is the graph of $y = f(x)$ . Tap the point of the graph that shows $f(2)$ .
Here is the graph of $y = f(x)$ . Tap the point of the graph that shows $f(2)$ .

Exit ticket

Three questions to check you have got it.

$f(x) = x^2 + 3$ . Find $f(2 + 3)$ .
$f(x) = x^2 + 2$ . Find $f(4 + 4)$ .
$f(x) = x^2 + 7$ . Find $f(4 + 1)$ .

Summary card

Function notation

$f(a)$ is the output for the input $a$ .
Replace every $x$ by $(a)$ , with parentheses: $f(-2)$ for $x^2$ is $(-2)^2 = 4$ .
On a graph: the height above $x = a$ .

Watch out: $f(a + b)$ is not $f(a) + f(b)$ .

3. Average rate of change; where a function increases or decreases

Warm-up

Three quick questions from earlier lessons.

Discover

A car drives 120 miles in 2 hours. What is its average speed?

60 miles per hour
$\tfrac{1}{60}$ hour per mile
120 miles per hour

Key idea

The average rate of change tells how fast the output changes, per unit of input.

The average rate of change of

f

from

x = a

x = b

\dfrac{f(b) - f(a)}{b - a}

, the slope of the line through the two points of the graph.

$f$ is increasing on an interval if its graph goes up from left to right there, and decreasing if it goes down.

Worked example

Find the average rate of change of $f(x) = x^2$ from $x = 1$ to $x = 4$ .

f(4) - f(1) = 16 - 1 = 15

Change in the output.

4 - 1 = 3

Change in the input.

Rate

= \dfrac{15}{3} = 5

Divide: output change per unit of input.

Your turn, guided

Your turn: $f(x) = 2x^2$ , from $x = 0$ to $x = 3$ .

f(3) - f(0) = 18

f(3) = 18, f(0) = 0.

Change in

x

3

3 − 0.

Rate

= 6

18 ÷ 3.

Practice ★

Find the average rate of change of $f(x) = 3x^2$ from $x = 0$ to $x = 6$ .
Find the average rate of change of $f(x) = 4x^2$ from $x = 1$ to $x = 5$ .
Find the average rate of change of $f(x) = 1x^2$ from $x = 1$ to $x = 6$ .
Find the average rate of change of $f(x) = 1x^2$ from $x = 1$ to $x = 4$ .

Watch out!

Practice ★★

A tank holds 16 liters at time 0 and $61$ liters after 5 minutes. What is the average rate of change of the volume, in liters per minute?
A tank holds 29 liters at time 0 and $49$ liters after 4 minutes. What is the average rate of change of the volume, in liters per minute?
A tank holds 12 liters at time 0 and $36$ liters after 6 minutes. What is the average rate of change of the volume, in liters per minute?

$f(x) = -(x - 5)^2 + 8$ . On which interval is $f$ increasing?
1. $(-\infty, 5)$
2. $(5, \infty)$
3. $(-\infty, 8)$
$f(x) = -(x - 1)^2 + 7$ . On which interval is $f$ increasing?
1. $(-\infty, 1)$
2. $(1, \infty)$
3. $(-\infty, 7)$
$f(x) = -(x - 3)^2 + 2$ . On which interval is $f$ increasing?
1. $(-\infty, 3)$
2. $(3, \infty)$
3. $(-\infty, 2)$

Exit ticket

Three questions to check you have got it.

Find the average rate of change of $f(x) = 3x^2$ from $x = 0$ to $x = 6$ .
Find the average rate of change of $f(x) = 4x^2$ from $x = 1$ to $x = 5$ .
Find the average rate of change of $f(x) = 1x^2$ from $x = 1$ to $x = 6$ .

Summary card

Average rate of change

From $a$ to $b$ : $\dfrac{f(b) - f(a)}{b - a}$ , output change over input change.
Increasing: the graph rises from left to right.
Give intervals of increase with $x$ -values.

Watch out: change in $f(x)$ on top, change in $x$ below.

4. Volume of prisms and cylinders

Warm-up

Three quick questions from earlier lessons.

Discover

A stack of identical coins forms a cylinder. If each coin has area $A$ and the stack is $h$ high, what is its volume?

$A \times h$
$A + h$

Key idea

Base area times height.

For any prism or cylinder:

V = B \cdot h

, with

B

the area of the base. Cylinder:

V = \pi r^2 h

Worked example

A cylinder with $r = 3$ , $h = 10$ .

Not to scale

B = 9\pi

Base area.

V = 90\pi \approx 282.7

Times height.

Your turn, guided

Your turn: a rectangular prism 4 by 5 by 6.

120

4 × 5 × 6.

Practice ★

Find the volume of a cylinder with radius 3 cm and height 4 cm (exact, in terms of π).Not to scale
Find the volume of a cylinder with radius 8 cm and height 11 cm (exact, in terms of π).Not to scale

A prism has a triangular base with base 2 cm and height 7 cm, and the prism is 7 cm long. Find its volume (cm³).Not to scale

Watch out!

Practice ★★

A cylinder of radius 5 cm has volume $175\pi$ cm³. Find its height (cm).Not to scale
A cylinder of radius 6 cm has volume $360\pi$ cm³. Find its height (cm).Not to scale

Exit ticket

Three questions to check you have got it.

Find the volume of a cylinder with radius 3 cm and height 4 cm (exact, in terms of π).Not to scale
Find the volume of a cylinder with radius 8 cm and height 11 cm (exact, in terms of π).Not to scale
Find the volume of a cylinder with radius 5 cm and height 5 cm (exact, in terms of π).Not to scale

Summary card

Prisms and cylinders

$V = Bh$ ; cylinder $V = \pi r^2 h$ .

Watch out: base area, not circumference.

5. Volume of pyramids and cones

Warm-up

Three quick questions from earlier lessons.

Discover

A cone and a cylinder have the same base and height. How many cones of water fill the cylinder?

Not to scale

Tilt them to compare. How many full cones of water fill the cylinder?

Key idea

Pointed solids hold one third.

For a pyramid or cone:

V = \tfrac{1}{3}Bh

. Cone:

V = \tfrac{1}{3}\pi r^2 h

Worked example

A square pyramid, base 6 m, height 10 m.

B = 36

Base area.

V = \tfrac{1}{3} \cdot 36 \cdot 10 = 120

m³

One third of Bh.

Your turn, guided

Your turn: a cone with $r = 3$ , $h = 4$ (in terms of π).

12\pi

⅓ · 9π · 4.

Practice ★

Find the volume of a cone with radius 6 cm and height 6 cm (exact, in terms of π).Not to scale
Find the volume of a cone with radius 6 cm and height 12 cm (exact, in terms of π).Not to scale

A pyramid has a square base of side 4 m and height 15 m. Find its volume (m³).Not to scale

Watch out!

Practice ★★

A cone and a cylinder have the same radius 5 and the same height. What fraction of the cylinder's volume is the cone's?

Boss problem. An ice cream cone has radius 3 cm and height 8 cm, topped with a hemisphere of ice cream of the same radius. Give exact answers in terms of π.

Exit ticket

Three questions to check you have got it.

Find the volume of a cone with radius 6 cm and height 6 cm (exact, in terms of π).Not to scale
Find the volume of a cone with radius 6 cm and height 12 cm (exact, in terms of π).Not to scale
Find the volume of a cone with radius 4 cm and height 9 cm (exact, in terms of π).Not to scale

Summary card

Pyramids and cones

$V = \tfrac{1}{3}Bh$ .

Watch out: do not forget the ⅓.

6. Volume of spheres; density

Warm-up

Three quick questions from earlier lessons.

Discover

A ball of radius $r$ fits exactly in a cylinder of radius $r$ and height $2r$ . The ball fills $\tfrac{2}{3}$ of the cylinder. What is the ball's volume?

Not to scale

$\tfrac{4}{3}\pi r^3$
$4\pi r^2$

Key idea

Spheres, and how heavy a solid is.

Sphere:

V = \tfrac{4}{3}\pi r^3

(surface area

4\pi r^2

). Density = mass ÷ volume, so mass = density × volume.

Worked example

A steel ball of radius 3 cm, density 7.8 g/cm³.

V = 36\pi \approx 113.1

cm³

(4/3)π · 27.

m \approx 7.8 \times 113.1 \approx 882

Mass = density × volume.

Your turn, guided

Your turn: mass 200 g, volume 50 cm³. The density.

4

g/cm³

200 ÷ 50.

Practice ★

Find the volume of a sphere of radius 5 cm (exact, in terms of π).Not to scale
Find the volume of a sphere of radius 9 cm (exact, in terms of π).Not to scale

A block has mass 526 g and volume 27 cm³. Find its density (g/cm³).

Watch out!

Practice ★★

Find the volume of a hemisphere (half a ball) of radius 3 cm (exact, in terms of π).
Find the volume of a hemisphere (half a ball) of radius 5 cm (exact, in terms of π).

Exit ticket

Three questions to check you have got it.

Find the volume of a sphere of radius 5 cm (exact, in terms of π).Not to scale
Find the volume of a sphere of radius 9 cm (exact, in terms of π).Not to scale
Find the volume of a sphere of radius 8 cm (exact, in terms of π).Not to scale

Summary card

Spheres and density

$V = \tfrac{4}{3}\pi r^3$ ; density = mass/volume.

Watch out: $r^3$ for volume, $r^2$ for area.

Answers: Functions and volume

1. What is a function: one output for each input

No: the input 5 has two different outputs
No: the input 4 has two different outputs

Yes: each input has exactly one output

No: the input 5 has two different outputs
No: the input 4 has two different outputs
No: the input 2 has two different outputs

2. Function notation: evaluate and read f(x)

$21$
$24$
$102$
$85$
$51$

$28$
$66$
$32$
$39$

$(2, -1)$
$(2, 3)$

$28$
$66$
$32$

3. Average rate of change; where a function increases or decreases

$18$
$24$
$7$
$5$

$(-\infty, 5)$
$(-\infty, 1)$
$(-\infty, 3)$

$18$
$24$
$7$

4. Volume of prisms and cylinders

$36 \pi$
$704 \pi$

$49$

$7$
$10$

$36 \pi$
$704 \pi$
$125 \pi$

5. Volume of pyramids and cones

$72 \pi$
$144 \pi$

$80$

$\frac{1}{3}$

$42 \pi$

$72 \pi$
$144 \pi$
$48 \pi$

6. Volume of spheres; density

$\frac{500 \pi}{3}$
$972 \pi$

$\frac{526}{27}$

$18 \pi$
$\frac{250 \pi}{3}$

$\frac{500 \pi}{3}$
$972 \pi$
$\frac{2048 \pi}{3}$