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1. Linear or exponential: equal differences or equal factors

Warm-up

Three quick questions from earlier lessons.

Discover

Offer A: 1,000 dollars a day for 30 days. Offer B: 1 cent the first day, then double every day. Which pays more on day 30?

Offer B
Offer A

Key idea

Linear adds, exponential multiplies.

Over equal steps in

x

, a linear function changes by equal differences (add

m

); an exponential function changes by equal factors (multiply by

b

). A growing exponential eventually overtakes any linear function.

Worked example

$y$ : 3, 6, 12, 24.

Not linear

Differences 3, 6, 12: not equal.

Exponential, factor 2

Ratios 2, 2, 2.

24 \cdot 2 = 48

Next value.

Your turn, guided

Your turn: an exponential function has $g(0) = 5$ , $g(1) = 15$ . Find $g(2)$ .

45

Factor 3.

Practice ★

A table gives $x = 0, 1, 2, 3$ and $y = 4, 12, 36, 108$ . What kind of relationship is it?
1. Exponential: $y$ is multiplied by 3 at each step
2. Linear: $y$ goes up by 8 at each step
3. Neither
A table gives $x = 0, 1, 2, 3$ and $y = 8, 24, 72, 216$ . What kind of relationship is it?
1. Exponential: $y$ is multiplied by 3 at each step
2. Linear: $y$ goes up by 16 at each step
3. Neither

An exponential function has $g(0) = 9$ and $g(1) = 18$ . Find $g(3)$ .

Watch out!

Practice ★★

$f(x) = 57x + 380$ and $g(x) = 1.04^x$ . For very large $x$ , which is greater?
1. $g$ : exponential growth eventually overtakes linear growth
2. $f$ : it is greater now and grows faster
3. They stay about equal
$f(x) = 73x + 397$ and $g(x) = 1.04^x$ . For very large $x$ , which is greater?
1. $g$ : exponential growth eventually overtakes linear growth
2. $f$ : it is greater now and grows faster
3. They stay about equal

Exit ticket

Three questions to check you have got it.

An exponential function has $g(0) = 9$ and $g(1) = 18$ . Find $g(3)$ .
An exponential function has $g(0) = 4$ and $g(1) = 20$ . Find $g(3)$ .
An exponential function has $g(0) = 5$ and $g(1) = 15$ . Find $g(3)$ .

Summary card

Linear or exponential?

Equal differences: linear. Equal factors: exponential.

Watch out: exponential growth wins in the end.

2. 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 $a_n$ .

3. 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 $n - 1$ .

4. Rational exponents in exponential functions

Warm-up

Three quick questions from earlier lessons.

Discover

A bacteria population doubles every day. By what factor does it grow in half a day?

Hint: whatever the factor $f$ is, growing for two half-days must double the population, so $f \cdot f = 2$ .

Key idea

Exponents that are fractions are roots.

b^{\frac{1}{n}} = \sqrt[n]{b}

and

b^{\frac{m}{n}} = \left(\sqrt[n]{b}\right)^m

So the half-day factor is $2^{\frac12} = \sqrt{2} \approx 1.41$ . A negative exponent means a reciprocal: $b^{-r} = \dfrac{1}{b^{r}}$ .

Worked example

Evaluate $8^{\frac{2}{3}}$ .

8^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2

The denominator 3 means a cube root; the numerator 2 means a square.

= 2^2

The cube root of 8 is 2.

= 4

Square it.

Your turn, guided

Your turn: evaluate $27^{\frac{2}{3}}$ . Fill in the missing line.

27^{\frac{2}{3}} = \left(\sqrt[3]{27}\right)^2 = 3^2

Cube root, then square.

= 9

3² = 9.

Practice ★

Evaluate $36^{ \frac{ 5 }{2} }$ .
Evaluate $81^{ \frac{ 3 }{2} }$ .
Evaluate $81^{ \frac{ 5 }{2} }$ .
Evaluate $4^{ \frac{ 3 }{2} }$ .
Evaluate $49^{ \frac{ 5 }{2} }$ .

Watch out!

Practice ★★

Evaluate $125^{ -\frac{ 2 }{3} }$ .
Evaluate $125^{ -\frac{ 1 }{3} }$ .
Evaluate $64^{ -\frac{ 1 }{3} }$ .
Evaluate $8^{ -\frac{ 1 }{3} }$ .
Evaluate $27^{ -\frac{ 1 }{3} }$ .

Exit ticket

Three questions to check you have got it.

Evaluate $125^{ -\frac{ 2 }{3} }$ .
Evaluate $125^{ -\frac{ 1 }{3} }$ .
Evaluate $64^{ -\frac{ 1 }{3} }$ .

Summary card

Rational exponents

$b^{\frac{1}{n}} = \sqrt[n]{b}$ ; $b^{\frac{m}{n}} = \left(\sqrt[n]{b}\right)^m$ .
$b^{-r} = \dfrac{1}{b^{r}}$ : the value stays positive.
A factor $b$ per unit means a factor $b^{\frac12}$ per half-unit.

5. Exponential growth and decay: multiplying by a constant factor

Discover

Two ponds each start with 5 m² of algae. In pond A, the area grows by 5 m² each week. In pond B, the area doubles each week.

After 1 week both ponds have 10 m². Which pond has more algae after 4 weeks? Guess first.

Pond A
Pond B
They have the same

Then check with a small table for weeks 0 to 4.

Key idea

When a quantity is multiplied by the same factor $b$ at each step, it changes exponentially.

Start

a

, factor

b

each step: after

n

steps the amount is

a \cdot b^n

If $b > 1$ , the quantity grows (exponential growth).
If $0 < b < 1$ , it shrinks (exponential decay).
Losing $p\%$ each step keeps $(100 - p)\%$ , so the factor is $1 - \frac{p}{100}$ . Gaining $p\%$ gives the factor $1 + \frac{p}{100}$ .

Worked example

A photo is 1000 pixels wide. Each time it is resized to 80% of its width. How wide is it after 3 resizes?

W = 1000 \cdot 0.8 \cdot 0.8 \cdot 0.8

Keeping 80% means multiplying by 0.8 each time.

W = 1000 \cdot 0.8^3

Three equal factors make a power.

W = 512

pixels

0.8³ = 0.512.

Your turn, guided

Your turn: a plant is 3 cm tall, and its height triples every week. How tall is it after 4 weeks? Fill in the missing line.

h = 3 \cdot 3^4

Start at 3 and multiply by 3 four times.

h = 243

3⁴ = 81, and 3 · 81 = 243.

Practice ★

The amount of algae starts at 5 m² and is multiplied by 4 each week. Complete the table.
The amount of algae starts at 2 m² and is multiplied by 2 each week. Complete the table.

A culture starts with 7 grams of algae. Each week, the amount is multiplied by 3. How many grams are there after 3 weeks?
A culture starts with 5 grams of algae. Each week, the amount is multiplied by 2. How many grams are there after 5 weeks?
A culture starts with 3 grams of algae. Each week, the amount is multiplied by 3. How many grams are there after 5 weeks?
A culture starts with 6 grams of algae. Each week, the amount is multiplied by 5. How many grams are there after 5 weeks?
A culture starts with 5 grams of algae. Each week, the amount is multiplied by 2. How many grams are there after 3 weeks?

Watch out!

Practice ★★

A photo is 8000 pixels wide. Each time it is resized, its width loses 40%. Write the width $W(t)$ after $t$ resizes.
A photo is 2000 pixels wide. Each time it is resized, its width loses 30%. Write the width $W(t)$ after $t$ resizes.
A photo is 2000 pixels wide. Each time it is resized, its width loses 25%. Write the width $W(t)$ after $t$ resizes.
A photo is 8000 pixels wide. Each time it is resized, its width loses 10%. Write the width $W(t)$ after $t$ resizes.
A photo is 4000 pixels wide. Each time it is resized, its width loses 30%. Write the width $W(t)$ after $t$ resizes.

Exit ticket

Three questions to check you have got it.

A photo is 8000 pixels wide. Each time it is resized, its width loses 40%. Write the width $W(t)$ after $t$ resizes.
A photo is 2000 pixels wide. Each time it is resized, its width loses 30%. Write the width $W(t)$ after $t$ resizes.
A photo is 2000 pixels wide. Each time it is resized, its width loses 25%. Write the width $W(t)$ after $t$ resizes.

Summary card

Exponential change: the same factor at every step

Start $a$ , factor $b$ per step: after $n$ steps, $a \cdot b^n$ .
Gain $p\%$ : factor $1 + \frac{p}{100}$ . Lose $p\%$ : factor $1 - \frac{p}{100}$ .

Watch out: adding the same amount each step is linear, not exponential.

6. Interpret exponential functions and their graphs

Warm-up

Three quick questions from earlier lessons.

Discover

$f(x) = 40 \cdot \left(\tfrac{1}{2}\right)^x$ . What is $f(0)$ ?

$40$
$0$

Key idea

Read $a$ and $b$ from $f(x) = a \cdot b^x$ .

a = f(0)

is the starting value (the

y

-intercept).

b

is the growth factor:

b > 1

growth,

0 < b < 1

decay. Negative inputs go back in time:

f(-1) = a \div b

Worked example

Graph through $(0, 3)$ and $(1, 12)$ .

a = 3

Starting value.

b = 4

12 ÷ 3.

f(-1) = 3 \div 4 = 0.75

One step back.

Your turn, guided

Your turn: $f(x) = 50 \cdot 5^x$ . Find $f(-1)$ .

10

50 ÷ 5.

Practice ★

$f(x) = 24 \cdot 4^x$ . Find $f(-1)$ .
$f(x) = 18 \cdot 3^x$ . Find $f(-1)$ .

Is $f(x) = 4 \cdot \left(\tfrac{1}{ 4 }\right)^x$ exponential growth or decay?
1. Decay: each step multiplies by $\tfrac{1}{ 4 }$
2. Growth, because 4 is greater than 1
3. A linear decrease

Watch out!

Practice ★★

The graph of an exponential function passes through $(0, 8)$ and $(1, 16)$ . Write $f(x)$ .
The graph of an exponential function passes through $(0, 7)$ and $(1, 21)$ . Write $f(x)$ .

Exit ticket

Three questions to check you have got it.

$f(x) = 24 \cdot 4^x$ . Find $f(-1)$ .
$f(x) = 18 \cdot 3^x$ . Find $f(-1)$ .
$f(x) = 27 \cdot 3^x$ . Find $f(-1)$ .

Summary card

Reading exponential functions

$a$ : value at $x = 0$ ; $b$ : factor per step; $b^{-1} = 1/b$ .

Watch out: negative exponent, positive value.

7. Write an exponential function from a description or two points

Warm-up

Three quick questions from earlier lessons.

Discover

A table shows a function: $f(0) = 4$ , $f(1) = 12$ , $f(2) = 36$ , $f(3) = 108$ .

What do you multiply by to go from one value to the next? Can you write one formula that gives all four values?

Key idea

An exponential function can be written

f(x) = a \cdot b^x

, where

a = f(0)

and

b

is the factor for each increase of 1 in

x

If the quantity is multiplied by $b$ every $n$ units instead, divide the input by $n$ : $f(x) = a \cdot b^{\frac{x}{n}}$ .

Worked example

Write $f$ for the table above.

a = 4

The starting value is f(0).

b = 3

Divide consecutive values: 12 ÷ 4 = 3.

f(x) = 4 \cdot 3^x

Put them in the form a · b^x.

Your turn, guided

Your turn: $f(0) = 5$ and $f(1) = 10$ . Fill in the missing line.

a = 5

b = 2

10 ÷ 5 = 2.

f(x) = 5 \cdot 2^x

Use the form a · b^x.

Practice ★

Plot the points of the graph of $f(x) = 4 \cdot 2^x$ at $x = 0$ and $x = 1$ .
Plot the points of the graph of $f(x) = 3 \cdot 4^x$ at $x = 0$ and $x = 1$ .

An exponential function $f$ has $f(0) = 9$ , and each time $x$ increases by 1, the value of $f$ is multiplied by 5. Write $f(x)$ .
An exponential function $f$ has $f(0) = 9$ , and each time $x$ increases by 1, the value of $f$ is multiplied by 4. Write $f(x)$ .
An exponential function $f$ has $f(0) = 6$ , and each time $x$ increases by 1, the value of $f$ is multiplied by 2. Write $f(x)$ .
An exponential function $f$ has $f(0) = 6$ , and each time $x$ increases by 1, the value of $f$ is multiplied by 4. Write $f(x)$ .
An exponential function $f$ has $f(0) = 4$ , and each time $x$ increases by 1, the value of $f$ is multiplied by 2. Write $f(x)$ .

Watch out!

Practice ★★

A population is 7 thousand at $t = 0$ and is multiplied by 3 every 4 years. Write the population $P(t)$ , in thousands, after $t$ years.
A population is 8 thousand at $t = 0$ and is multiplied by 3 every 3 years. Write the population $P(t)$ , in thousands, after $t$ years.
A population is 4 thousand at $t = 0$ and is multiplied by 2 every 3 years. Write the population $P(t)$ , in thousands, after $t$ years.
A population is 8 thousand at $t = 0$ and is multiplied by 2 every 2 years. Write the population $P(t)$ , in thousands, after $t$ years.
A population is 9 thousand at $t = 0$ and is multiplied by 3 every 5 years. Write the population $P(t)$ , in thousands, after $t$ years.

Exit ticket

Three questions to check you have got it.

A population is 7 thousand at $t = 0$ and is multiplied by 3 every 4 years. Write the population $P(t)$ , in thousands, after $t$ years.
A population is 8 thousand at $t = 0$ and is multiplied by 3 every 3 years. Write the population $P(t)$ , in thousands, after $t$ years.
A population is 4 thousand at $t = 0$ and is multiplied by 2 every 3 years. Write the population $P(t)$ , in thousands, after $t$ years.

Summary card

Writing an exponential function

$f(x) = a \cdot b^x$ : $a = f(0)$ , $b = \dfrac{f(1)}{f(0)}$ .
Multiplied by $b$ every $n$ units: $f(x) = a \cdot b^{\frac{x}{n}}$ .

Watch out: the starting value is $f(0)$ , and it goes in front, not in the base.

8. Percent change and compound interest

Warm-up

Three quick questions from earlier lessons.

Discover

A 50-dollar game goes up by 10%. Which calculation gives the new price in one step?

$50 \cdot 1.1$
$50 \cdot 0.1$

Key idea

Percent change is a growth factor.

An increase of

r

(as a decimal) multiplies by

1 + r

; a decrease multiplies by

1 - r

. Compound interest at yearly rate

r

, compounded

n

times a year:

A = P\left(1 + \tfrac{r}{n}\right)^{nt}

; yearly:

A = P(1 + r)^t

Worked example

800 dollars at 5% per year, compounded yearly, for 3 years.

1.05

Factor.

800 \cdot 1.05^3

Three years.

\approx 926.10

dollars

Calculator.

Your turn, guided

Your turn: the growth factor for a 7% increase.

1.07

1 + 0.07.

Practice ★

A quantity grows by 9% each year. By what number is it multiplied each year (the growth factor)?
A quantity grows by 16% each year. By what number is it multiplied each year (the growth factor)?

882 dollars are invested at 5% per year, compounded yearly. Write the value after 4 years (an exact expression is fine).

Watch out!

Practice ★★

660 dollars are invested at 4% per year, compounded monthly. Write the value after 2 years (an exact expression is fine).
1818 dollars are invested at 7% per year, compounded monthly. Write the value after 3 years (an exact expression is fine).

Boss problem. Ana puts 1800 dollars in a savings account that pays 8% per year, compounded yearly.

Exit ticket

Three questions to check you have got it.

A quantity grows by 9% each year. By what number is it multiplied each year (the growth factor)?
A quantity grows by 16% each year. By what number is it multiplied each year (the growth factor)?
A quantity grows by 34% each year. By what number is it multiplied each year (the growth factor)?

Summary card

Percent change and interest

Increase $r$ : × $(1 + r)$ ; decrease: × $(1 - r)$ ; compounding: rate and exponent per period.

Watch out: 1.1, not 0.1, for a 10% increase.

Answers: Introduction to exponential functions

1. Linear or exponential: equal differences or equal factors

Exponential: $y$ is multiplied by 3 at each step
Exponential: $y$ is multiplied by 3 at each step

$72$

$g$ : exponential growth eventually overtakes linear growth
$g$ : exponential growth eventually overtakes linear growth

$72$
$500$
$135$

2. Arithmetic sequences

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

Geometric, with ratio 2

$98$
$86$
$133$

3. Geometric sequences

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

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

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

4. Rational exponents in exponential functions

$7776$
$729$
$59049$
$8$
$16807$

$\frac{1}{25}$
$\frac{1}{5}$
$\frac{1}{4}$
$\frac{1}{2}$
$\frac{1}{3}$

$\frac{1}{25}$
$\frac{1}{5}$
$\frac{1}{4}$

5. Exponential growth and decay: multiplying by a constant factor

$20, 80, 320$
$4, 8, 16$

$189$
$160$
$729$
$18750$
$40$

$8000 \left(\frac{3}{5}\right)^{t}$
$2000 \left(\frac{7}{10}\right)^{t}$
$2000 \left(\frac{3}{4}\right)^{t}$
$8000 \left(\frac{9}{10}\right)^{t}$
$4000 \left(\frac{7}{10}\right)^{t}$

$8000 \left(\frac{3}{5}\right)^{t}$
$2000 \left(\frac{7}{10}\right)^{t}$
$2000 \left(\frac{3}{4}\right)^{t}$

6. Interpret exponential functions and their graphs

Decay: each step multiplies by $\tfrac{1}{ 4 }$

$8 \cdot 2^{x}$
$7 \cdot 3^{x}$

7. Write an exponential function from a description or two points

$(0, 4),\ (1, 8)$
$(0, 3),\ (1, 12)$

$9 \cdot 5^{x}$
$9 \cdot 4^{x}$
$6 \cdot 2^{x}$
$6 \cdot 4^{x}$
$4 \cdot 2^{x}$

$7 \cdot 3^{\frac{t}{4}}$
$8 \cdot 3^{\frac{t}{3}}$
$4 \cdot 2^{\frac{t}{3}}$
$8 \cdot 2^{\frac{t}{2}}$
$9 \cdot 3^{\frac{t}{5}}$

$7 \cdot 3^{\frac{t}{4}}$
$8 \cdot 3^{\frac{t}{3}}$
$4 \cdot 2^{\frac{t}{3}}$

8. Percent change and compound interest

$\frac{109}{100}$
$\frac{29}{25}$

$\frac{85766121}{80000}$

$\frac{3365027202877485792030412063535621133822430265203298001319211}{4707158941350000000000000000000000000000000000000000000000}$
$\frac{88263896780194910285616660379297501244924918384278626761032336923688742382456650426114169842822935756855808839301}{39377881943616213632296905944976064512000000000000000000000000000000000000000000000000000000000000000000000000}$

$\frac{52488}{25}$

$\frac{109}{100}$
$\frac{29}{25}$
$\frac{67}{50}$