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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. Domain and range in interval notation

Warm-up

Three quick questions from earlier lessons.

Discover

Which numbers can you put into $f(x) = \sqrt{x}$ ?

0 and the positive numbers
Every number
Only the positive numbers, not 0

Key idea

The domain is the set of inputs a function accepts. The range is the set of outputs it gives.

Leave out of the domain every

x

that makes a denominator

0

or puts a negative number under a square root. A square bracket includes the end:

[2, \infty)

contains 2; a parenthesis leaves it out:

(2, \infty)

does not.

Worked example

Find the domain of $f(x) = \sqrt{x + 2}$ .

x + 2 \ge 0

The inside of a square root must not be negative.

x \ge -2

Subtract 2 from both sides.

Domain:

[-2, \infty)

−2 is allowed (√0 = 0), so use a square bracket.

Your turn, guided

Your turn: $f(x) = \sqrt{x - 5}$ . Fill in the missing line.

x - 5 \ge 0

The inside must not be negative.

The domain starts at

x = 5

[5, \infty)

Add 5 to both sides: the domain starts at this number.

Practice ★

What is the domain of $f(x) = \sqrt{x - 9}$ ? Give it as an interval.
What is the domain of $f(x) = \sqrt{x - 5}$ ? Give it as an interval.
What is the domain of $f(x) = \sqrt{x - 6}$ ? Give it as an interval.
What is the domain of $f(x) = \sqrt{x - 4}$ ? Give it as an interval.

Watch out!

Practice ★★

What is the domain of $f(x) = \dfrac{ 9 }{x - 6}$ ? Give it as an interval or a union of intervals.
What is the domain of $f(x) = \dfrac{ 6 }{x - 5}$ ? Give it as an interval or a union of intervals.
What is the domain of $f(x) = \dfrac{ 4 }{x - 3}$ ? Give it as an interval or a union of intervals.

What is the range of $f(x) = (x - 5)^2 + 9$ ? Give it as an interval.
What is the range of $f(x) = (x - 3)^2 + 6$ ? Give it as an interval.
What is the range of $f(x) = (x - 5)^2 + 4$ ? Give it as an interval.

Exit ticket

Three questions to check you have got it.

What is the range of $f(x) = (x - 5)^2 + 9$ ? Give it as an interval.
What is the range of $f(x) = (x - 3)^2 + 6$ ? Give it as an interval.
What is the range of $f(x) = (x - 5)^2 + 4$ ? Give it as an interval.

Summary card

Domain and range

Domain: the inputs. Leave out zero denominators and negatives under a square root.
Range: the outputs.
$[a, \infty)$ includes $a$ ; $(a, \infty)$ does not.

Watch out: the range uses $y$ -values, not the $x$ of a vertex.

4. 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.

5. Transformations: shifts and reflections

Warm-up

Three quick questions from earlier lessons.

Discover

Move the sliders and watch the graph of $y = (x - h)^2 + k$ .

The graph of y = (x − h)² + k, a parabola whose vertex is at (h, k) and moves with the sliders.

When $h$ goes from 0 to 2, which way does the graph move?

To the right
To the left

Key idea

Changing the rule moves or flips the graph without changing its shape.

y = f(x - h) + k

moves the graph of

f

h

units to the right and

k

units up.

y = -f(x)

reflects it across the

x

-axis;

y = f(-x)

reflects it across the

y

-axis.

A change inside $f(\dots)$ acts on $x$ (horizontally); a change outside acts on $y$ (vertically).

Worked example

Describe $y = (x + 1)^2 - 3$ compared with $y = x^2$ .

1 unit to the left

x + 1 = x − (−1): h = −1.

3 units down

− 3 outside: k = −3.

Vertex:

(-1, -3)

The vertex (0, 0) moves with the graph.

Your turn, guided

Your turn: $y = (x - 4)^2 + 2$ .

4 units to the right

x − 4: h = 4.

2 units up

+ 2 outside.

The vertex is at

x = 4

The vertex moves from x = 0 to x = …

Practice ★

The graph of $y = x^2$ is moved 1 units to the right and 6 units down. Write the equation of the new graph.
The graph of $y = x^2$ is moved 3 units to the right and 1 units down. Write the equation of the new graph.
The graph of $y = x^2$ is moved 4 units to the right and 6 units down. Write the equation of the new graph.
The graph of $y = x^2$ is moved 7 units to the right and 1 units down. Write the equation of the new graph.

Watch out!

Practice ★★

Draw the graph of $y = (x - 1)^2 - 3$ : place the vertex and one more point.
Draw the graph of $y = (x - 1)^2 - 1$ : place the vertex and one more point.

Match each equation with what it does to the graph of $y = f(x)$ .
1. $y = f(x) + 3$
2. $y = f(x - 3)$
3. $y = -f(x)$
4. $y = f(-x)$
1. moves it up 3 units
2. moves it 3 units to the right
3. reflects it across the x-axis
4. reflects it across the y-axis

Exit ticket

Three questions to check you have got it.

The graph of $y = x^2$ is moved 1 units to the right and 6 units down. Write the equation of the new graph.
The graph of $y = x^2$ is moved 3 units to the right and 1 units down. Write the equation of the new graph.
The graph of $y = x^2$ is moved 4 units to the right and 6 units down. Write the equation of the new graph.

Summary card

Shifts and reflections

$f(x - h)$ : $h$ units right. $f(x) + k$ : $k$ units up.
$-f(x)$ : flip across the $x$ -axis. $f(-x)$ : flip across the $y$ -axis.
Inside $f(\dots)$ acts on $x$ ; outside acts on $y$ .

Watch out: $f(x + 3)$ moves the graph left.

6. Sequences: explicit and recursive formulas

Warm-up

Three quick questions from earlier lessons.

Discover

A sequence starts $5, 8, 11, 14, \dots$ Which rule gives the next term from the one before?

Add 3
Multiply by 3

Key idea

A sequence is a function whose inputs are the whole numbers $1, 2, 3, \dots$ : $a_n$ is its $n$ th term.

An explicit formula gives

a_n

from

n

directly (for example

a_n = 3n + 2

). A recursive formula gives a first term and a rule from one term to the next (for example

a_1 = 5

a_{n+1} = a_n + 3

Worked example

$a_1 = 2$ and $a_{n+1} = 3a_n - 1$ . Find $a_4$ .

a_2 = 3 \cdot 2 - 1 = 5

Apply the rule to a₁.

a_3 = 3 \cdot 5 - 1 = 14

Then to a₂.

a_4 = 3 \cdot 14 - 1 = 41

Then to a₃.

Your turn, guided

Your turn: $a_n = n^2 - 1$ . Find $a_5$ .

a_5 = 5^2 - 1

Replace n by 5.

a_5 = 24

25 − 1.

Practice ★

$a_n = n^2 + 5$ . Complete the table.
$a_n = n^2 + 9$ . Complete the table.

Watch out!

Practice ★★

$a_1 = 3$ and $a_{n+1} = 2a_n - 7$ . Find $a_3$ .
$a_1 = 6$ and $a_{n+1} = 2a_n - 3$ . Find $a_3$ .
$a_1 = 4$ and $a_{n+1} = 2a_n - 6$ . Find $a_3$ .

Write a formula for $a_n$ , starting at $n = 1$ : $1,6, 11, 16, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $3,10, 17, 24, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $8,14, 20, 26, \dots$

Exit ticket

Three questions to check you have got it.

Write a formula for $a_n$ , starting at $n = 1$ : $1,6, 11, 16, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $3,10, 17, 24, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $8,14, 20, 26, \dots$

Summary card

Sequences

Explicit: $a_n$ straight from $n$ .
Recursive: a first term and a rule from $a_n$ to $a_{n+1}$ .

Watch out: check your formula with $n = 1$ .

7. Arithmetic sequences

Warm-up

Three quick questions from earlier lessons.

Discover

A taxi charges 3 dollars, plus 2 dollars per mile. Mile by mile, the fare is $5, 7, 9, \dots$ What does the 10th mile cost in total?

23 dollars
25 dollars

Key idea

An arithmetic sequence adds the same number $d$ (the common difference) each time.

a_n = a_1 + (n - 1)d

. From

a_p

a_q

there are

q - p

steps, so

d = \dfrac{a_q - a_p}{q - p}

Worked example

An arithmetic sequence has $a_3 = 11$ and $a_8 = 31$ . Find $d$ and $a_{20}$ .

d = \dfrac{31 - 11}{5} = 4

5 steps from a₃ to a₈.

a_1 = 11 - 2 \cdot 4 = 3

Back 2 steps from a₃.

a_{20} = 3 + 19 \cdot 4 = 79

19 steps from a₁.

Your turn, guided

Your turn: $a_1 = 7$ , $d = 5$ . Find $a_{12}$ .

a_{12} = 7 + 11 \cdot 5

11 steps.

a_{12} = 62

7 + 55.

Practice ★

An arithmetic sequence has $a_1 = 14$ and common difference 2. Find $a_{ 43 }$ .
An arithmetic sequence has $a_1 = 17$ and common difference 3. Find $a_{ 24 }$ .
An arithmetic sequence has $a_1 = 8$ and common difference 5. Find $a_{ 26 }$ .
An arithmetic sequence has $a_1 = 20$ and common difference 5. Find $a_{ 28 }$ .

Watch out!

Practice ★★

In an arithmetic sequence, $a_{ 4 } = 28$ and $a_{ 9 } = 68$ . Find the common difference.
In an arithmetic sequence, $a_{ 3 } = 18$ and $a_{ 7 } = 34$ . Find the common difference.
In an arithmetic sequence, $a_{ 3 } = 11$ and $a_{ 10 } = 25$ . Find the common difference.

What kind of sequence is $3, 6, 12, 24, \dots$ ?
1. Geometric, with ratio 2
2. Arithmetic
3. Neither

Exit ticket

Three questions to check you have got it.

An arithmetic sequence has $a_1 = 14$ and common difference 2. Find $a_{ 43 }$ .
An arithmetic sequence has $a_1 = 17$ and common difference 3. Find $a_{ 24 }$ .
An arithmetic sequence has $a_1 = 8$ and common difference 5. Find $a_{ 26 }$ .

Summary card

Arithmetic sequences

Add $d$ each time: $a_n = a_1 + (n - 1)d$ .
$d = \dfrac{a_q - a_p}{q - p}$ .

Watch out: $n - 1$ steps from $a_1$ to $a_n$ .

8. Geometric sequences

Warm-up

Three quick questions from earlier lessons.

Discover

A rumor spreads: 3 people know it on day 1, and each day the number triples. How many know it on day 4?

Key idea

A geometric sequence multiplies by the same number $r$ (the common ratio) each time.

a_n = a_1 \cdot r^{\,n-1}

, and

r = \dfrac{a_{n+1}}{a_n}

Worked example

Find $a_7$ for $2, 6, 18, \dots$

r = 3

6 ÷ 2.

a_7 = 2 \cdot 3^6

6 multiplications from a₁ to a₇.

a_7 = 1458

3⁶ = 729.

Your turn, guided

Your turn: $a_1 = 5$ , $r = 2$ . Find $a_6$ .

a_6 = 5 \cdot 2^5

5 multiplications.

a_6 = 160

5 · 32.

Practice ★

A geometric sequence has $a_1 = 3$ and ratio 3. Find $a_{ 4 }$ .
A geometric sequence has $a_1 = 1$ and ratio 2. Find $a_{ 8 }$ .
A geometric sequence has $a_1 = 2$ and ratio 3. Find $a_{ 8 }$ .
A geometric sequence has $a_1 = 4$ and ratio 2. Find $a_{ 5 }$ .

Watch out!

Practice ★★

Find the common ratio of the geometric sequence $8, 32, 128, \dots$
Find the common ratio of the geometric sequence $7, 21, 63, \dots$

Write a formula for $a_n$ , starting at $n = 1$ : $2, 8, 32, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $9, 36, 144, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $8, 40, 200, \dots$

Exit ticket

Three questions to check you have got it.

Write a formula for $a_n$ , starting at $n = 1$ : $2, 8, 32, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $9, 36, 144, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $8, 40, 200, \dots$

Summary card

Geometric sequences

Multiply by $r$ each time: $a_n = a_1 r^{\,n-1}$ .
$r$ = a term divided by the one before.

Watch out: the exponent is $n - 1$ .

Answers: Functions and sequences

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. Domain and range in interval notation

$[9, \infty)$
$[5, \infty)$
$[6, \infty)$
$[4, \infty)$

$(-\infty, 6) \cup (6, \infty)$
$(-\infty, 5) \cup (5, \infty)$
$(-\infty, 3) \cup (3, \infty)$

$[9, \infty)$
$[6, \infty)$
$[4, \infty)$

$[9, \infty)$
$[6, \infty)$
$[4, \infty)$

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

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

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

$18$
$24$
$7$

5. Transformations: shifts and reflections

$\left(x - 1\right)^{2} - 6$
$\left(x - 3\right)^{2} - 1$
$\left(x - 4\right)^{2} - 6$
$\left(x - 7\right)^{2} - 1$

$y = \left(x - 1\right)^{2} - 3$
$y = \left(x - 1\right)^{2} - 1$

$y = f(x) + 3$ → moves it up 3 units · $y = f(x - 3)$ → moves it 3 units to the right · $y = -f(x)$ → reflects it across the x-axis · $y = f(-x)$ → reflects it across the y-axis

$\left(x - 1\right)^{2} - 6$
$\left(x - 3\right)^{2} - 1$
$\left(x - 4\right)^{2} - 6$

6. Sequences: explicit and recursive formulas

$6, 9, 14, 21$
$10, 13, 18, 25$

$-9$
$15$
$-2$

$5 n - 4$
$7 n - 4$
$6 n + 2$

$5 n - 4$
$7 n - 4$
$6 n + 2$

7. Arithmetic sequences

$98$
$86$
$133$
$155$

Geometric, with ratio 2

$98$
$86$
$133$

8. Geometric sequences

$81$
$128$
$4374$
$64$

$2 \cdot 4^{n - 1}$
$9 \cdot 4^{n - 1}$
$8 \cdot 5^{n - 1}$

$2 \cdot 4^{n - 1}$
$9 \cdot 4^{n - 1}$
$8 \cdot 5^{n - 1}$