B1.3 Graph and define exponential functions | Ontario Grade 11 Mathematics

Ontario Grade 11 Mathematics

Exponential Functions

Connect repeated multiplication to an equation, table, and graph.

Suppose a savings balance grows by the same percentage each year. The amount added is not the same each year because it depends on the current balance. Instead, the balance is multiplied by the same factor each year. This repeated multiplication creates an exponential pattern. You will connect that pattern to an equation, a table, and a graph.

What you will learn

Recognize an exponential function from its equation or repeated-multiplication pattern.
Define a basic exponential function using its initial value and base.
Use a table to graph an exponential function.
Describe the domain, range, intercept, and horizontal asymptote of a basic exponential graph.

1. Prerequisite bridge: repeated multiplication

An exponent tells how many times a base is used as a factor. For example,

3^4

means

3\times3\times3\times3

, which equals

81

. The base is

3

, and the exponent is

4

.

A function is a rule that assigns one output to each allowed input. In

y=3^x

, the input is

x

and the output is

y

. Since the variable is in the exponent, this is an exponential function.

Compare it with

y=3x

. Here, the variable is multiplied by a fixed number, so the rule is linear. In

y=3^x

, increasing the input by

1

multiplies the output by

3

.

3^4=81

An exponent counts repeated factors.
In an exponential function, the variable is in the exponent.
A constant multiplier between outputs one input unit apart is a clue to an exponential pattern.

2. Define an exponential function

A basic exponential function can be written as

f(x)=a b^x

. The number

a

is the initial value: it is the output when the input is

0

. The number

b

is the base. It is the factor that multiplies the output when the input increases by

1

.

For this form, the base must be positive and cannot equal

1

. If

b=1

, the output does not change. When

b>1

, the function shows exponential growth: its outputs increase as the input increases. When

0<b<1

, it shows exponential decay: its outputs decrease as the input increases.

For example,

f(x)=5(2)^x

has initial value

5

and base

2

. The function

g(x)=80(0.5)^x

has initial value

80

and base

0.5

. In the second function, each output is half the previous output when the input increases by

1

.

The domain is the set of inputs a function accepts. For this basic form, the domain is all real numbers. The range is the set of possible outputs. If

a>0

, the range is all positive real numbers. The graph stays above the horizontal line

y=0

and approaches it without reaching it. This line is called a horizontal asymptote.

f(x)=a b^x,\quad b>0,\quad b\ne1

In $f(x)=a b^x$ , $a$ is the initial value and $b$ is the base.
A base greater than $1$ gives growth; a base between $0$ and $1$ gives decay.
When $a>0$ , the graph has positive outputs and approaches $y=0$ .

3. Connect equation, table, and graph

A table shows input and output pairs. Choose input values, substitute them into the rule, and calculate the outputs. Starting with

x=0

is useful because any nonzero base raised to the power

0

equals

1

. So the output at

x=0

is the initial value,

a

.

For each one-unit increase in input, multiply the output by the base. For a decay function, the base is less than

1

, but it is still the multiplier. Moving one unit to the left reverses the pattern: divide by the base.

Each input-output pair is an ordered pair, written as

(x,y)

. Plot the pairs and draw a smooth curve through them. The curve should approach the horizontal asymptote without touching it. Do not connect the points with separate straight segments.

The vertical intercept is where the graph crosses the vertical axis. For the basic exponential function, this point is

(0,a)

. The equation, table, and graph show the same function in different ways.

f(0)=a b^0=a

The vertical intercept is $(0,a)$ .
Moving one unit right multiplies the output by the base.
A basic exponential graph is a smooth curve that approaches its horizontal asymptote.

4. Application: repeated percentage change

Exponential functions can model repeated percentage changes. Suppose an amount increases by 20% each time period. The new amount is the old amount plus 20% of the old amount. That equals 120% of the old amount, or

1.2

times the old amount.

If the starting amount is

a

, then after

x

periods the model is

a(1.2)^x

. For a decrease of 20%, 80% remains, so the multiplier is

0.8

. The model is then

a(0.8)^x

.

The exponent counts the number of equal time periods. Check whether the situation uses a repeated factor or a repeated fixed addition. Repeated percentage changes use a factor. Growth uses a factor above

1

; decay uses a positive factor below

1

.

A(x)=a b^x

A repeated increase of r% uses multiplier $1+r$ , with $r$ written as a decimal.
A repeated decrease of r% uses multiplier $1-r$ , with $r$ written as a decimal.
The initial amount is the output at $x=0$ .

Worked example

Build and sketch an exponential graph

For

f(x)=3(2)^x

, make a table for

x=-2,-1,0,1,2

. Describe how to sketch the graph and state its basic features.

Identify the parts
The initial value is $3$ and the base is $2$ . Since the base is greater than $1$ , this function shows growth. Each one-unit increase in $x$ doubles the output.
$a=3,\quad b=2$
Calculate the values
Substitute each input into the equation. For negative inputs, use the reciprocal meaning of a negative exponent: $2^{-1}=\frac{1}{2}$ and $2^{-2}=\frac{1}{4}$ .
$\begin{array}{c|ccccc}x&-2&-1&0&1&2\\\hline f(x)&\frac{3}{4}&\frac{3}{2}&3&6&12\end{array}$
Plot and sketch
Plot $(-2,\frac{3}{4})$ , $(-1,\frac{3}{2})$ , $(0,3)$ , $(1,6)$ , and $(2,12)$ . Draw a smooth increasing curve through the points. Extend it to the left so it approaches $y=0$ without touching that line.
$(0,3)$
Describe the features
Every real number is an allowed input, so the domain is all real numbers. All outputs are positive, so the range is all positive real numbers. The vertical intercept is $(0,3)$ and the horizontal asymptote is $y=0$ .
$\text{Domain: }\mathbb{R},\quad \text{Range: }(0,\infty)$

Answer: The outputs for the listed inputs are

\frac{3}{4},\frac{3}{2},3,6,12

. The graph is an increasing curve through the table points. Its vertical intercept is

(0,3)

and its horizontal asymptote is

y=0

.

Check: Each output is twice the previous output as

x

increases by

1

. For example,

\frac{3}{2}

is twice

\frac{3}{4}

, which agrees with the base

2

.

Common mistakes and how to avoid them

Calling

y=4x

exponential because it contains a variable.

Correction: In

y=4x

, the variable is not an exponent, so the rule is linear. An exponential rule has the variable in the exponent, as in

y=4^x

.

Using the percentage increase itself as the base.

Correction: A 20% increase means the new amount is 120% of the old amount. Use the multiplier

1.2

, not

0.2

.

Treating the horizontal asymptote as a point the graph crosses.

Correction: For the basic function with positive initial value, the graph approaches

y=0

but does not reach it.

Adding the same amount at every step in an exponential table.

Correction: Multiply by the base to move one unit right. Exponential change uses a constant factor, while linear change uses a constant difference.

Lesson summary

An exponential function has its variable in the exponent and can be written as $f(x)=a b^x$ .
The initial value $a$ is the output at $x=0$ ; the base $b$ is the multiplier for each one-unit increase in input.
A base above $1$ gives growth. A positive base below $1$ gives decay.
Use a table to find points, plot them, and draw a smooth curve that approaches the horizontal asymptote.

Check your understanding

Question 1

Which equation is exponential?

$y=6x+2$
$y=6(1.5)^x$
$y=6+x^2$
$y=\frac{6}{x}$

Show answer and explanation

y=6(1.5)^x

The variable appears in the exponent in

y=6(1.5)^x

, so this equation is exponential.

Question 2

For

g(x)=7(0.4)^x

, what happens when

x

increases by

1

?

The output increases by $0.4$ .
The output is multiplied by $0.4$ .
The output is multiplied by $7$ .
The output decreases by exactly $7$ .

Show answer and explanation

The output is multiplied by

0.4

.

The base,

0.4

, is the factor between outputs for inputs one unit apart. Since it is between

0

and

1

, the function shows decay.

Question 3

What is the vertical intercept of

h(x)=4(3)^x

?

$(0,3)$
$(0,4)$
$(4,0)$
$(0,12)$

Show answer and explanation

(0,4)

At

x=0

,

3^0=1

, so

h(0)=4

. The vertical intercept is

(0,4)

.

Key terms

Exponential function: A function in which the variable appears in the exponent, commonly written as $f(x)=a b^x$ .
Initial value: The output when the input is $0$ ; in $f(x)=a b^x$ , it is $a$ .
Base: The fixed positive number raised to the variable exponent. It gives the multiplier for each one-unit increase in input.
Horizontal asymptote: A horizontal line that a graph approaches but does not reach in the basic exponential function.
Domain: The set of input values a function accepts.
Range: The set of output values a function can produce.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation B1.3. It is a study resource, not an official curriculum publication.

Official curriculum reference

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B1.3 · Graph and define exponential functions

Ontario Grade 11 Mathematics

Connect repeated multiplication to an equation, table, and graph.

What you will learn

1. Prerequisite bridge: repeated multiplication

2. Define an exponential function

3. Connect equation, table, and graph

4. Application: repeated percentage change

Build and sketch an exponential graph

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MCF3M

About this lesson