B1.4 Describe key properties of exponential functions

Ontario Grade 11 Mathematics

Exponential Functions

Understanding growth, decay, and the shape of exponential graphs

In Grade 10, you learned about linear functions, where a value changes by adding the same amount each time, and quadratic functions, where a value changes according to a squared term. There is a third major function family in this course: the exponential function. Instead of adding a fixed amount each step, an exponential function multiplies by a fixed amount each step. This single difference gives exponential functions a very distinct shape, a distinct table pattern, and a distinct set of properties. In this lesson, we will describe those properties in plain language, then in a table, then in graph form, and finally using symbols.

What you will learn

Identify the equation form of an exponential function and describe what each part means
Describe the domain, range, intercept, and asymptote of an exponential function
Distinguish exponential growth from exponential decay using the base value
Use a table of values and a graph to explain how an exponential function behaves
Apply these properties to a real situation involving repeated multiplication

Prerequisite Bridge: From Linear Change to Multiplicative Change

Before we describe exponential functions, let's remember how linear functions behave. In a linear function such as

y = 2x + 3

, the value of

y

increases by the same fixed amount every time

x

increases by 1. This fixed amount is called the rate of change, and it stays constant throughout the whole table of values.

An exponential function works differently. Instead of adding the same amount each time, the value of

y

is multiplied by the same amount each time. This constant multiplier is called the base. Because multiplying repeatedly can make numbers grow very fast (or shrink very fast), exponential functions look and behave very differently from linear or quadratic functions.

A quick way to test whether a table of values might represent an exponential function is to divide each output value by the one before it. If you always get the same ratio, the pattern is exponential, not linear.

Linear functions change by constant addition; exponential functions change by constant multiplication.
The constant multiplier in an exponential pattern is called the base.
A constant ratio between consecutive outputs is the signal that a table is exponential.

Plain-Language Description of an Exponential Function

An exponential function is a function where the input variable, usually

x

, appears as an exponent. The basic form used in this course is

y = a \cdot b^{x}

, where

a

is the starting value (the value of

y

when

x = 0

), and

b

is the base, the number being repeatedly multiplied.

The base

b

must be a positive number, and it cannot equal 1, because if

b = 1

, multiplying by 1 repeatedly never changes the value, and the function becomes a flat horizontal line instead of an exponential curve.

If the base

b

is greater than 1, the function models exponential growth: each output is bigger than the one before. If the base

b

is between 0 and 1 (a proper fraction, like

0.5

), the function models exponential decay: each output is smaller than the one before, getting closer and closer to zero without ever reaching it.

This last idea, getting closer and closer to a value without touching it, describes a horizontal asymptote. For the basic exponential function

y = a \cdot b^{x}

with

a > 0

, the horizontal asymptote is the line

y = 0

, meaning the graph gets extremely close to the x-axis but never actually crosses or touches it.

y = a · b^{x}

General form used in this course: $y = a \cdot b^{x}$ , with $a$ the starting value and $b$ the base.
The base must satisfy $b > 0$ and $b \neq 1$ .
Base greater than 1 gives growth; base between 0 and 1 gives decay.
The graph approaches but never reaches the horizontal asymptote $y = 0$ (for positive $a$ ).

Multiple Representations: Table and Graph

Let's describe the properties using a concrete example first. Consider

y = 3 \cdot 2^{x}

. Here

a = 3

is the starting value, and

b = 2

is the base, so this function models growth because

b > 1

.

If you build a table of values for

x = -2, -1, 0, 1, 2, 3

, you will notice that as

x

increases by 1, each

y

-value is exactly double the one before it. This constant doubling is the multiplicative pattern we described earlier. Also notice that even for negative

x

-values,

y

stays positive; it just becomes a small fraction. This confirms that

y

never becomes zero or negative.

On a graph, this table produces a curve that stays above the x-axis at all times, rises slowly on the left side, crosses the y-axis at the starting value

a = 3

, and then rises steeply as

x

increases. The curve gets flatter and flatter on the left, squeezing toward the x-axis without ever touching it. That flattening behavior is the visual picture of the horizontal asymptote

y = 0

.

From this table and graph, we can now state four key properties clearly: the domain (all allowed input values) is all real numbers, because you can substitute any

x

, positive, negative, or zero. The range (all possible output values) is

y > 0

, because a positive base raised to any exponent, multiplied by a positive starting value, always stays positive. The y-intercept is always at

(0, a)

, since

b^0 = 1

. There is no x-intercept, because the graph never touches

y = 0

.

Domain of $y = a \cdot b^x$ (with $a>0$ ) is all real numbers.
Range is $y > 0$ ; the graph never reaches or crosses the x-axis.
The y-intercept is always $(0, a)$ , since any positive base raised to the exponent 0 equals 1.
There is no x-intercept for this basic exponential form.

Growth vs Decay: Comparing Two Exponential Functions

Property	Growth Example: $y = 3 \cdot 2^{x}$	Decay Example: $y = 5 \cdot (0.5)^{x}$
Base value $b$	2 (greater than 1)	0.5 (between 0 and 1)
Behaviour as $x$ increases	Output values get larger	Output values get smaller
Domain	all real numbers	all real numbers
Range	$y > 0$	$y > 0$
Y-intercept	$(0, 3)$	$(0, 5)$
Horizontal asymptote	$y = 0$	$y = 0$

Worked example

Describing the Properties of an Exponential Function

A function is given by

y = 5 \cdot (0.5)^{x}

. Describe its starting value, base, growth or decay behaviour, domain, range, intercept, and asymptote.

Identify the starting value and base
Compare the equation to the general form $y = a \cdot b^{x}$ . Here $a = 5$ and $b = 0.5$ . Since $b$ is a number between 0 and 1, this tells us right away what kind of behaviour to expect.
$a = 5, \ b = 0.5$
Decide growth or decay
Because the base $0.5$ is between 0 and 1, each time $x$ increases by 1, the output is multiplied by $0.5$ , which makes it smaller. This means the function represents exponential decay, not growth.
$0 < b < 1 \Rightarrow \text{decay}$
Find the y-intercept
To find where the graph crosses the y-axis, substitute $x = 0$ . Any positive base raised to the exponent 0 equals 1, so the y-intercept equals the starting value $a$ .
$y = 5 · (0.5)^{0} = 5$
State the domain
Since we can substitute any real number for $x$ , positive, negative, fraction, or zero, and always get a valid output, the domain includes every real number.
$x ∈ \mathbb{R}$
State the range and asymptote
Because $a = 5$ is positive and the base is positive, every output value stays positive no matter how large or small $x$ becomes. As $x$ increases, the outputs shrink toward 0 but never reach it, so $y = 0$ is the horizontal asymptote and the range is all positive real numbers.
$y > 0, \ \text{asymptote: } y = 0$
Summarize with a quick table check
Testing $x = 0, 1, 2$ gives $y = 5, 2.5, 1.25$ . Each value is exactly half of the one before, confirming the decay pattern matches the base of $0.5$ .
$5, \ 2.5, \ 1.25$

Answer: The function

y = 5 \cdot (0.5)^{x}

has starting value

a = 5

, base

b = 0.5

, and represents exponential decay. Its domain is all real numbers, its range is

y > 0

, its y-intercept is

(0, 5)

, there is no x-intercept, and its horizontal asymptote is

y = 0

.

Check: Substituting increasing values of

x

(0, 1, 2, 3) gives 5, 2.5, 1.25, 0.625. Each output is exactly half of the previous one, confirming a constant ratio of 0.5, which matches the identified base and confirms decay behaviour toward the asymptote

y=0

.

Common mistakes and how to avoid them

Thinking the horizontal asymptote means the graph eventually touches or crosses the x-axis at a very large x-value.

Correction: The graph gets closer and closer to

y = 0

forever but never actually reaches it. This is different from an x-intercept, which the basic exponential function does not have.

Believing that a base between 0 and 1 means the function is negative or that outputs become negative.

Correction: A base between 0 and 1 still produces positive outputs; it just makes the outputs shrink toward zero instead of growing. The range stays

y > 0

in both growth and decay cases.

Confusing the starting value

a

with the base

b

when reading an equation like

y = a \cdot b^{x}

.

Correction: The value in front (before the multiplication) is the starting value

a

, found at

x = 0

. The value being raised to the power of

x

is the base

b

, which controls growth or decay.

Assuming exponential functions change by adding a fixed amount, like linear functions.

Correction: Always check by dividing consecutive outputs, not subtracting them. A constant ratio (not a constant difference) confirms an exponential pattern.

Lesson summary

An exponential function has the form $y = a \cdot b^{x}$ , where $a$ is the starting value and $b$ is the base, with $b > 0$ and $b \neq 1$ .
If $b > 1$ , the function shows exponential growth; if $0 < b < 1$ , it shows exponential decay.
The domain is all real numbers, and (for positive $a$ ) the range is $y > 0$ .
The y-intercept is always $(0, a)$ , since $b^0 = 1$ , and there is no x-intercept.
The horizontal asymptote is $y = 0$ : the graph approaches this line but never touches or crosses it.
A constant ratio between consecutive table outputs is the key sign of exponential behaviour, unlike the constant difference seen in linear functions.

Check your understanding

Question 1

Which equation represents exponential growth?

$y = 4 \cdot (0.2)^{x}$
$y = 4 \cdot (1.5)^{x}$
$y = 4x + 2$
$y = 4x^{2}$

Show answer and explanation

y = 4 \cdot (1.5)^{x}

Exponential growth requires a base greater than 1. Here

b = 1.5 > 1

, so outputs increase as

x

increases. The other exponential option has a base less than 1 (decay), and the last two are not exponential at all.

Question 2

For the function

y = 7 \cdot (0.8)^{x}

, what is the y-intercept?

$(0, 0.8)$
$(0, 7)$
$(7, 0)$
$(0, 5.6)$

Show answer and explanation

(0, 7)

Substituting

x = 0

gives

y = 7 \cdot (0.8)^0 = 7 \cdot 1 = 7

, so the y-intercept is

(0, 7)

, matching the starting value

a

.

Question 3

What is the range of the function

y = 6 \cdot 3^{x}

?

All real numbers
$y > 0$
$y \geq 0$
$y < 6$

Show answer and explanation

y > 0

Since the starting value 6 is positive and the base 3 is positive, every output stays positive. The graph never reaches zero, so the range is

y > 0

, not

y \geq 0

.

Question 4

A table shows outputs 2, 6, 18, 54 for consecutive whole-number inputs. What does this pattern tell you?

It is linear, because the difference is always increasing
It is exponential, because each output is 3 times the one before
It is quadratic, because the outputs are increasing
It cannot be determined without a graph

Show answer and explanation

It is exponential, because each output is 3 times the one before

Dividing consecutive outputs gives a constant ratio of 3 each time (6 divided by 2, 18 divided by 6, 54 divided by 18), which is the defining sign of an exponential pattern with base 3.

Key terms

Exponential function: A function where the input variable appears as an exponent, written in this course as $y = a \cdot b^{x}$ .
Base: The fixed number $b$ that is repeatedly multiplied in an exponential function; it must be positive and not equal to 1.
Starting value: The value $a$ in $y = a \cdot b^{x}$ , equal to the output when $x = 0$ .
Exponential growth: Behaviour where output values increase as $x$ increases, occurring when the base $b$ is greater than 1.
Exponential decay: Behaviour where output values decrease toward zero as $x$ increases, occurring when the base $b$ is between 0 and 1.
Domain: The complete set of allowed input values ( $x$ -values) for a function.
Range: The complete set of possible output values ( $y$ -values) for a function.
Horizontal asymptote: A horizontal line that a graph gets closer and closer to but never touches or crosses.

Continue through MCF3M

View the complete Ontario Grade 11 Mathematics learning path

About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation B1.4. It is a study resource, not an official curriculum publication.

Official curriculum reference

Report a correction or ask a question

B1.4 · Describe key properties of exponential functions

Ontario Grade 11 Mathematics

Understanding growth, decay, and the shape of exponential graphs

What you will learn

Prerequisite Bridge: From Linear Change to Multiplicative Change

Plain-Language Description of an Exponential Function

Multiple Representations: Table and Graph

Growth vs Decay: Comparing Two Exponential Functions

Describing the Properties of an Exponential Function

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MCF3M

About this lesson