Algebra I: Exponential Functions

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Grades 8–10AlgebraGuided journey: Discover, Learn, Practice, Apply

Tell exponential from linear by a constant factor, read the start a and the factor b in f(x) = abˣ, write growth and decay functions from stories, tables and two points, graph them with their y-intercept and asymptote, and state domain and range as inequalities. Aligned to HSF-LE.A.1, HSF-LE.A.2, HSF-LE.B.5 and HSF-IF.C.7.

What you’ll learn

Linear functions add the same amount each step; exponential functions multiply by the same factor.
In $f(x)=ab^x$ , $a$ is the value at $x=0$ (the $y$ -intercept) and $b$ is the factor each step.
$b>1$ is growth and $0<b<1$ is decay; growth uses $b=1+r$ and decay $b=1-r$ .
From a table, $b$ is the quotient of neighbors; walk back to $x=0$ to find $a$ .
The graph passes through $(0, a)$ and hugs the asymptote $y=0$ without touching it.
Domain: all real numbers. Range: $y>0$ when $a>0$ , $y<0$ when $a<0$ .

Aligned to HSF-LE.A.1, HSF-LE.A.2, HSF-LE.B.5, HSF-IF.C.7

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Algebra I: Exponential Functions

Free

Grades 8–10AlgebraGuided journey: Discover, Learn, Practice, Apply

What you’ll learn

Linear functions add the same amount each step; exponential functions multiply by the same factor.
In $f(x)=ab^x$ , $a$ is the value at $x=0$ (the $y$ -intercept) and $b$ is the factor each step.
$b>1$ is growth and $0<b<1$ is decay; growth uses $b=1+r$ and decay $b=1-r$ .
From a table, $b$ is the quotient of neighbors; walk back to $x=0$ to find $a$ .
The graph passes through $(0, a)$ and hugs the asymptote $y=0$ without touching it.
Domain: all real numbers. Range: $y>0$ when $a>0$ , $y<0$ when $a<0$ .

Aligned to HSF-LE.A.1, HSF-LE.A.2, HSF-LE.B.5, HSF-IF.C.7

For teachers

Share to Google Classroom

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Reference

Algebra I: Exponential Functions

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Where you are

Units

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Discover

Think about this

Why does growing by $3\%$ a year add more people each year, when $3\%$ never changes?

The y-intercept is a = 3. To the left the curve hugs y = 0 but never reaches it: y = 0 is the asymptote.

Learn

Each card starts with a real-life picture of the idea, works one example step by step, then shows a mistake to avoid and a tool to try it with. Try the steps yourself before moving to Practice.

Spot an exponential function

The rumor

3

Worked example: is this table linear, exponential or neither? $x$ is $0, 1, 2, 3, 4$ and $y$ is $2, 6, 18, 54, 162\text{.}$

x

Each step multiplies y by the same number, 3. A constant factor means exponential: f(x) = 2(3)ˣ.

x

x

Big mistake to avoid

y=1, 4, 9, 16

Cool math secret

Divide neighbors for exponential, subtract neighbors for linear. If neither the difference nor the quotient stays the same, the table is neither.

Worked example

Is this table linear, exponential or neither?

x

0, 1, 2, 3

and

y

5, 10, 20, 40\text{.}

Check that $x$ goes up by the same amount each step.
$0 \to 1 \to 2 \to 3$
Divide each $y$ by the one before it.
$\frac{10}{5} = \frac{20}{10} = \frac{40}{20} = 2$
A constant factor above 1 means exponential growth.
$\text{exponential, } b = 2$

Answer

\text{exponential, } b = 2

Now you try

Now classify:

x

0, 1, 2, 3

and

y

81, 27, 9, 3\text{.}

Linear, falling by

54

Exponential, multiplying by

\tfrac{1}{3}

Exponential, multiplying by

3

Neither

Optional practice. This does not change your mastery result.

Try it with a tool

y

Read a and b in f(x) = abˣ

The savings account

\$500

Worked example: what do $a$ and $b$ mean in $f(x)=300(0.8)^x\text{?}$

a=300

Check: $f(1)=300(0.8)=240\text{,}$ which is $60$ less, and $60$ is $20\%$ of $300\text{.}$

a is the value at x = 0. b is what each step multiplies by: above 1 grows, between 0 and 1 decays.

f(x)=50(1.25)^x\text{.}

f(x)=90(0.6)^x\text{.}

Big mistake to avoid

b

Cool math secret

1\text{.}

Worked example

A lab culture has

B(h) = 50(2)^h

bacteria after

h

hours. What do

50

and

2

mean?

At $h = 0$ the power is $1\text{,}$ so $a$ is the starting amount.
$B(0) = 50(2)^0 = 50$
Each hour multiplies by $b = 2\text{:}$ the culture doubles.
$b = 2 > 1 \quad \text{(growth)}$

Answer

a = 50 \text{ at the start}, \quad b = 2 \text{: doubles each hour}

Now you try

Now: a car is worth

V(t) = 24{,}000(0.85)^t

dollars after

t

years. What does

0.85

mean?

It loses

15\%

of its value each yearIt loses

85\%

of its value each yearIt gains

85\%

each yearIt is worth

\$0.85

after one year

Optional practice. This does not change your mastery result.

Try it with a tool

y=a(1+r)^t

Write an exponential function from a story

The growing town

12{,}000

Worked example: a doctor gives $200$ mg of a medicine, and $30\%$ of what is left leaves the body each hour. Write the amount $M$ after $t$ hours.

a=200\text{.}

Check: after $1$ hour, $200(0.7)=140$ mg, which is $60$ mg, $30\%$ of $200\text{,}$ gone.

Growth adds the rate to 1; decay subtracts it from 1. The factor is what is left after one step.

\$40

64

Big mistake to avoid

b

Cool math secret

a\text{),}

Worked example

A phone bought for

\$800

loses

20\%

of its value each year. Write its value

V

after

t

years.

The starting value is $a\text{.}$
$a = 800$
Losing $20\%$ keeps $80\%\text{.}$
$b = 1 - 0.20 = 0.8$
Write $V = ab^t\text{.}$
$V = 800(0.8)^t$

Answer

V = 800(0.8)^t

Now you try

Now: an account has

1{,}500

followers and grows

10\%

a month. Write the followers

F

after

m

months.

F = 1500(0.1)^m

F = 1500(1.1)^m

F = 1500 + 150m

F = 1500(0.9)^m

Optional practice. This does not change your mastery result.

Try it with a tool

• Exponential growth and decay : turn a percent rate into a factor. • Half-life : stories where the amount halves each period.

Write an exponential function from a table or two points

Retrace your steps

3\text{,}

Worked example: write $f(x)$ for the table $x$ is $1, 2, 3$ and $y$ is $6, 18, 54\text{.}$

\tfrac{18}{6}=3

Check: $f(3)=2(27)=54\text{,}$ which matches the table.

The factor is 18 ÷ 6 = 3. Undo one step to reach x = 0: 6 ÷ 3 = 2, so a = 2 and f(x) = 2(3)ˣ.

x

(1, 40)

Big mistake to avoid

a

Cool math secret

x

Worked example

Write

f(x)

for the table:

x

0, 1, 2

and

y

4, 12, 36\text{.}

Divide neighbouring $y$ -values for the factor.
$b = \frac{12}{4} = 3$
The $y$ under $x = 0$ is the start.
$a = f(0) = 4$
Write $f(x) = ab^x\text{.}$
$f(x) = 4(3)^x$

Answer

f(x) = 4(3)^x

Now you try

Now:

x

1, 2, 3

and

y

10, 5, 2.5\text{.}

f(x) = 10\left(\tfrac{1}{2}\right)^x

f(x) = 20\left(\tfrac{1}{2}\right)^x

f(x) = 20(2)^x

f(x) = 5\left(\tfrac{1}{2}\right)^x

Optional practice. This does not change your mastery result.

Try it with a tool

x=1, 2, 3

Graph growth and decay

The never-empty glass

8

Worked example: graph $f(x)=3(2)^x\text{.}$

x=-2, -1, 0, 1, 2\text{:}

$b=2>1\text{,}$ so the graph rises from left to right.

The y-intercept is a = 3. To the left the curve hugs y = 0 but never reaches it: y = 0 is the asymptote.

f(x)=3(2)^x

f(x)=16\left(\tfrac{1}{2}\right)^x

Big mistake to avoid

y=0\text{.}

Cool math secret

f(x)=ab^x

Worked example

Name the

y

-intercept and asymptote of

f(x) = 5(0.5)^x\text{,}

and say whether it grows or decays.

The $y$ -intercept is $f(0) = a\text{.}$
$f(0) = 5(0.5)^0 = 5 \quad\Rightarrow\quad (0,\ 5)$
Halving again and again gets close to $0$ but never reaches it.
$\text{asymptote: } y = 0$
$b$ is between $0$ and $1\text{.}$
$0 < 0.5 < 1 \quad\Rightarrow\quad \text{decay}$

Answer

(0,\ 5), \quad y = 0, \quad \text{decay}

Now you try

Now do the same for

f(x) = 2(3)^x\text{.}

(0, 2)\text{,}

asymptote

y = 0\text{,}

growth

(0, 3)\text{,}

asymptote

y = 0\text{,}

growth

(0, 2)\text{,}

asymptote

y = 2\text{,}

growth

(2, 0)\text{,}

asymptote

x = 0\text{,}

decay

Optional practice. This does not change your mastery result.

Try it with a tool

x

Domain and range

Any input, only positive outputs

2

Worked example: state the domain and range of $f(x)=4(1.5)^x$ as inequalities.

-\infty<x<\infty\text{.}

Any x can go in: the domain is all real numbers. Every output is above 0: the range is y > 0.

f(x)=-2(3)^x\text{.}

B(h)=50(2)^h\text{,}

Big mistake to avoid

y>0\text{,}

Cool math secret

f(x)=ab^x\text{:}

Worked example

State the domain and range of

f(x) = 7(0.4)^x

as inequalities.

Any number can be an exponent.
$\text{domain: } -\infty < x < \infty$
A positive $a$ times positive powers stays above $0\text{.}$
$\text{range: } y > 0$

Answer

\text{domain: all real } x, \quad \text{range: } y > 0

Now you try

Now find the range of

f(x) = -3(2)^x\text{.}

y > 0

y < 0

y \ge -3

All real numbers

Optional practice. This does not change your mastery result.

Try it with a tool

x\text{;}

Key points to remember

Linear functions add the same amount each step; exponential functions multiply by the same factor.
In $f(x)=ab^x\text{,}$ $a$ is the value at $x=0$ (the $y$ -intercept) and $b$ is the factor each step.
$b>1$ is growth and $0<b<1$ is decay; growth uses $b=1+r$ and decay $b=1-r\text{.}$
From a table, $b$ is the quotient of neighbors; walk back to $x=0$ to find $a\text{.}$
The graph passes through $(0, a)$ and hugs the asymptote $y=0$ without touching it.
Domain: all real numbers. Range: $y>0$ when $a>0\text{,}$ $y<0$ when $a<0\text{.}$

Practice

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The best check is explaining Algebra I: Exponential Functions in your own words. The tutor listens like a curious friend and asks clarifying questions. No score, no grade.

Explain it in your own words

Explain Algebra I: Exponential Functions as if to someone who has never heard of it. The tutor responds like a curious learner with clarifying questions, not a score.

Reflection prompts (optional)

Why does growing by $3\%$ a year add more people each year, when $3\%$ never changes?
Why can an exponential graph get close to $y=0$ but never reach it?
When would a story limit the domain of an exponential model?

Optional: scored check

Want a scored check? Open it here. The teach-back above does not require finishing it.

The scored check opens up once you have spent a little time in Learn and Practice. Recall sticks best after you have engaged with the material first.

Algebra I: Exponential Functions

What you’ll learn

Algebra I: Exponential Functions

What you’ll learn

Where you are

Units

Get the free 7-Day Study Reset

Where you are

Units

Get the free 7-Day Study Reset

Learning acts

Spot an exponential function

Read a and b in f(x) = abˣ

Write an exponential function from a story

Write an exponential function from a table or two points

Graph growth and decay

Domain and range

Key points to remember

Practice what you learned

Practice for you

Flashcards

Apply what you learned

Explain it in your own words