B2.5 Represent an exponential function from its graph or properties

Ontario Grade 11 Mathematics

Exponential Functions

MCR3U – B2.5 | Building and Writing Exponential Function Equations

You have already worked with linear and quadratic functions in Grade 10. Both of those families have graphs you can recognise at a glance — a straight line or a parabola. In this lesson you will learn to recognise and represent a third family: exponential functions. These functions appear whenever a quantity grows or shrinks by a constant multiplier during equal time steps — for example, a population that doubles every year, or a battery charge that drops to half every hour. By the end of this lesson you will be able to look at a graph or a set of properties and write the equation of the exponential function it describes.

What you will learn

Identify the key properties of an exponential function — base, initial value, domain, range, and asymptote — from a graph.
Write an equation of the form $f(x) = a \cdot b^x$ by reading or using given properties.
Explain how changing $a$ or $b$ affects the shape and position of the graph.
Determine whether a graph or table of values represents an exponential function, and justify your reasoning.

Prerequisite Bridge: What You Already Know

Before meeting exponential functions as a family, make sure two ideas from earlier mathematics are solid in your mind.

First, recall function notation. Writing

f(x)

means 'the output of function

f

when the input is

x

.' So

f(0)

means the output when

x = 0

, and that output is the y-intercept of the graph.

Second, recall the meaning of a base raised to an exponent:

b^x

means

b

multiplied by itself

x

times when

x

is a positive whole number. For example,

2^3 = 2 \times 2 \times 2 = 8

. In Grade 11 the exponent

x

can be any real number — including fractions and negatives — but the same core idea of repeated multiplication still guides your intuition.

$f(x)$ notation: input goes in, output comes out.
$f(0)$ gives the y-intercept of the graph.
$b^x$ grows larger as $x$ increases when $b > 1$ , and shrinks toward zero when $0 < b < 1$ .

What Makes a Function Exponential?

An exponential function has the form

f(x) = a \cdot b^x

, where

a

is a non-zero real number called the initial value and

b

is a positive real number called the base, with the restriction

b \neq 1

. The variable

x

sits in the exponent — that is what makes this family different from polynomial functions, where the variable sits in the base.

The parameter

a

tells you the y-intercept. When

x = 0

, the calculation is

f(0) = a \cdot b^0 = a \cdot 1 = a

, so the graph always crosses the y-axis at the point

(0,\ a)

.

The parameter

b

controls whether the function grows or decays. When

b > 1

the outputs increase as

x

increases — this is called exponential growth. When

0 < b < 1

the outputs decrease as

x

increases — this is called exponential decay. The base can never be negative or zero, and

b = 1

is excluded because

1^x = 1

for every value of

x

, which produces a constant horizontal line rather than an exponential curve.

Every exponential function of this form approaches the x-axis as a horizontal asymptote. The graph gets closer and closer to the line

y = 0

but never touches or crosses it. Spotting this asymptote is a key way to recognise an exponential graph.

f(x) = a \cdot b^x

Standard form: $f(x) = a \cdot b^x$ , with $a \neq 0$ , $b > 0$ , $b \neq 1$ .
The y-intercept is always the point $(0,\ a)$ .
$b > 1$ gives a growth curve; $0 < b < 1$ gives a decay curve.
The horizontal asymptote is $y = 0$ for the base form.
Domain is all real numbers; range is $y > 0$ when $a > 0$ .

Reading Key Properties from a Graph

When you are given a graph and asked to find the equation, treat it like a puzzle with two unknowns:

a

and

b

. You need two pieces of information to pin down two unknowns — exactly like solving a system of two equations.

The first step is always to find

a

. Look for where the curve crosses the y-axis. That y-coordinate is

a

, because substituting

x = 0

into

f(x) = a \cdot b^x

gives

f(0) = a \cdot 1 = a

.

The second step is to find

b

. Pick any other clearly readable point on the graph — call it

(x_1,\ y_1)

. Substitute into the equation

y_1 = a \cdot b^{x_1}

. Because you already know

a

, divide both sides by

a

to isolate

b^{x_1}

. Then ask: what base raised to the power

x_1

gives that result? When

x_1 = 1

this is especially direct, because

b^1 = b

, so dividing by

a

gives

b

immediately. When

x_1 = 2

you need to recognise that

b^2 = k

means

b = \sqrt{k}

, keeping only the positive root because a base must be positive. Choosing a second point where

x_1 = 1

avoids this extra step whenever the graph allows it.

The third step is to verify your equation. Pick a third point from the graph and confirm the equation gives the correct output. If it does not match, recheck the coordinates you read.

y-intercept → find $a$ directly.
Second point with $x_1 = 1$ → divide to find $b$ in one step.
If $x_1 = 2$ , solve $b^2 = k$ by taking the positive square root.
Third point → verify the equation is correct.
The horizontal asymptote at $y = 0$ confirms the function is exponential.

How $a$ and $b$ Change the Graph

Changing

a

moves the y-intercept up or down and stretches or compresses the curve vertically. A larger positive

a

means the curve starts higher on the y-axis. A negative

a

reflects the entire curve below the x-axis; the range then becomes

y < 0

while the horizontal asymptote remains

y = 0

.

Changing

b

controls how fast the function grows or decays. Compare

f(x) = 2^x

with

g(x) = 3^x

: both pass through

(0, 1)

, but

g

climbs much faster for positive

x

because its base is larger. For decay, compare

h(x) = (0.9)^x

with

k(x) = (0.5)^x

: the second drops to half its value with every unit increase in

x

, so it decays much faster.

A useful mental check: increasing

x

by 1 always multiplies the output by

b

. This constant multiplier property is the defining characteristic of an exponential function — it is what separates it from linear functions (constant addition per step) and quadratic functions (outputs that grow in proportion to

x^2

).

\frac{f(x+1)}{f(x)} = b

$a$ sets the y-intercept and vertical scale; negative $a$ reflects the graph below the x-axis.
Larger $b$ (when $b > 1$ ) means faster growth.
Smaller $b$ (when $0 < b < 1$ ) means faster decay.
Increasing $x$ by 1 always multiplies the output by $b$ .

Recognising Exponential Behaviour in a Table of Values

Sometimes you receive a table of

(x, y)

pairs instead of a graph. You can test whether the data is exponential by checking whether the ratio of consecutive outputs is constant, provided the inputs are equally spaced.

Calculate

\frac{y_2}{y_1}

,

\frac{y_3}{y_2}

, and so on. If all of those ratios are equal, the data fits an exponential model. That constant ratio is the base

b

. The output when

x = 0

directly gives

a

.

Compare this test to the linear test from Grade 9: for a linear function the differences

y_2 - y_1

,

y_3 - y_2

, and so on are constant. For an exponential function the ratios are constant instead of the differences. Applying the wrong test is a common error, so be deliberate about which one you use.

Equal-ratio test: a constant ratio of consecutive outputs (with equally spaced inputs) signals exponential data.
That constant ratio equals $b$ .
The output when $x = 0$ equals $a$ .
Constant differences signal linear, not exponential, behaviour.

Comparing Key Properties: Growth vs. Decay

Property	Growth Example $f(x) = 5 \cdot 3^x$	Decay Example $f(x) = 80 \cdot \left(\frac{1}{4}\right)^x$
Base $b$	$3$ (greater than 1)	$\frac{1}{4}$ (between 0 and 1)
y-intercept $(0, a)$	$(0,\ 5)$	$(0,\ 80)$
Output change per unit increase in $x$	Multiplied by 3	Multiplied by $\frac{1}{4}$
Horizontal asymptote	$y = 0$	$y = 0$
Domain	All real numbers	All real numbers
Range (with $a > 0$ )	$y > 0$	$y > 0$

Worked example

Finding the Equation from Two Points on a Graph

A smooth curve passes through the points

(0, 5)

and

(2, 45)

. The curve has a horizontal asymptote at

y = 0

and no x-intercept. Find the equation of the exponential function in the form

f(x) = a \cdot b^x

.

Identify a from the y-intercept
The point $(0, 5)$ is on the y-axis, so it is the y-intercept. Because $f(0) = a \cdot b^0 = a \cdot 1 = a$ , we can read $a$ directly from this point.
$a = 5$
Write the equation with the known value of a
Replace $a$ with 5 in the standard form. The base $b$ is the only unknown left.
$f(x) = 5 \cdot b^x$
Substitute the second point into the equation
The point $(2, 45)$ tells us $f(2) = 45$ . Substitute $x = 2$ and $f(x) = 45$ into $f(x) = 5 \cdot b^x$ .
$45 = 5 \cdot b^2$
Isolate b squared
Divide both sides by 5 so that $b^2$ is alone on one side. This is valid because 5 is non-zero.
$b^2 = 9$
Solve for b by taking the positive square root
We need a number whose square is 9. Both $3$ and $-3$ satisfy $b^2 = 9$ , but the base of an exponential function must be positive, so we take $b = 3$ and discard $-3$ .
$b = 3$
Write the final equation
Substitute $a = 5$ and $b = 3$ back into the standard form to state the complete equation.
$f(x) = 5 \cdot 3^x$

Answer:

f(x) = 5 \cdot 3^x

Check: Verify with both given points. At

x = 0

:

5 \cdot 3^0 = 5 \cdot 1 = 5

✓. At

x = 2

:

5 \cdot 3^2 = 5 \cdot 9 = 45

✓. Since

b = 3 > 1

, this is a growth curve, which is consistent with the point

(2, 45)

being above

(0, 5)

.

Worked example

Writing an Equation from a Table of Values

The table below shows values of a function. Determine whether the function is exponential. If it is, write its equation in the form

f(x) = a \cdot b^x

.

x

: 0, 1, 2, 3

y

: 80, 20, 5, 1.25

Test for a constant ratio between consecutive outputs
For equally spaced inputs (here the inputs increase by 1 each time), an exponential function multiplies its output by the same number at every step. Calculate each ratio of consecutive $y$ -values to check.
$\frac{20}{80} = 0.25, \quad \frac{5}{20} = 0.25, \quad \frac{1.25}{5} = 0.25$
Confirm the data is exponential and identify b
All three ratios equal 0.25, which is constant. This confirms the data fits an exponential model. The constant ratio is the base, so $b = 0.25$ . Writing 0.25 as a fraction gives a cleaner form.
$b = 0.25 = \frac{1}{4}$
Read a from the table
The y-intercept occurs at $x = 0$ . The table shows $y = 80$ when $x = 0$ , so $a = 80$ .
$a = 80$
Write the equation
Substitute $a = 80$ and $b = \frac{1}{4}$ into the standard form $f(x) = a \cdot b^x$ .
$f(x) = 80 \cdot \left(\frac{1}{4}\right)^x$

Answer:

f(x) = 80 \cdot \left(\frac{1}{4}\right)^x

Check: Check with the last table entry at

x = 3

:

80 \cdot \left(\frac{1}{4}\right)^3 = 80 \cdot \frac{1}{64} = \frac{80}{64} = 1.25

✓. Since

b = \frac{1}{4} < 1

, this is a decay function, which matches the steadily decreasing values in the table.

Common mistakes and how to avoid them

Using the y-intercept value as

b

instead of

a

, then trying to find

a

from a second point.

Correction: Always read

a

first from the y-intercept, since

f(0) = a

. Then use a second point to determine

b

.

Accepting a negative value for

b

when solving

b^2 = k

.

Correction: The base

b

of an exponential function must be positive. When solving

b^2 = k

, take only the positive square root. Choosing

b = 1

as the second point avoids this extra step altogether.

Checking only one point when verifying the equation and assuming it must be correct.

Correction: Always verify with at least one additional point beyond the two used to build the equation. A third-point check catches arithmetic errors that a single check would miss.

Applying the differences test (linear test) to a table instead of the ratios test when checking for exponential behaviour.

Correction: Subtract consecutive outputs to test for linear; divide consecutive outputs to test for exponential. Be deliberate about which test matches the function family you are investigating.

Believing

b

is the y-intercept because

b^x

appears first when the equation is written as

b^x \cdot a

.

Correction: Multiplication is commutative, so

a \cdot b^x = b^x \cdot a

. The y-intercept is always

a

, regardless of order. Substitute

x = 0

to confirm:

f(0) = a \cdot b^0 = a

.

Lesson summary

An exponential function has the form $f(x) = a \cdot b^x$ , where $a$ is the initial value (y-intercept) and $b$ is the base (constant multiplier per unit step).
The base must satisfy $b > 0$ and $b \neq 1$ ; the initial value $a$ must be non-zero.
To find the equation from a graph: read $a$ from the y-intercept, substitute a second known point to determine $b$ , then verify with a third point.
When finding $b$ from a second point, choose a point where $x = 1$ whenever possible — this gives $b$ directly by dividing $y_1$ by $a$ . If the exponent is 2, take the positive square root of the ratio.
In a table with equally spaced inputs, a constant ratio of consecutive outputs confirms exponential behaviour; that ratio is the base $b$ .
Growth occurs when $b > 1$ ; decay occurs when $0 < b < 1$ ; both cases share a horizontal asymptote at $y = 0$ .

Check your understanding

Question 1

A graph of an exponential function crosses the y-axis at

(0, 6)

and also passes through

(1, 18)

. What is the equation of the function?

$f(x) = 18 \cdot 6^x$
$f(x) = 6 \cdot 3^x$
$f(x) = 6 \cdot 18^x$
$f(x) = 3 \cdot 6^x$

Show answer and explanation

f(x) = 6 \cdot 3^x

The y-intercept gives

a = 6

. Substituting the point

(1, 18)

:

18 = 6 \cdot b^1

, so

b = 18 \div 6 = 3

. The equation is

f(x) = 6 \cdot 3^x

.

Question 2

A table shows outputs 200, 50, 12.5, 3.125 for inputs

x = 0, 1, 2, 3

. What is the base

b

of the exponential function?

$b = 4$
$b = 150$
$b = 0.25$
$b = 0.5$

Show answer and explanation

b = 0.25

Calculate the ratio of consecutive outputs:

50 \div 200 = 0.25

,

12.5 \div 50 = 0.25

, and

3.125 \div 12.5 = 0.25

. The constant ratio is 0.25, so

b = 0.25

.

Question 3

Which of the following is NOT a valid base for an exponential function in the form

f(x) = a \cdot b^x

?

$b = 0.3$
$b = 1$
$b = 5$
$b = \frac{2}{3}$

Show answer and explanation

b = 1

b = 1

is excluded because

1^x = 1

for every value of

x

, making

f(x) = a

, which is a constant (horizontal line), not an exponential function.

Question 4

An exponential function satisfies

f(0) = 4

and

f(2) = 100

. A student writes the equation

f(x) = 4 \cdot 5^x

. Which check best confirms this equation is correct?

Substituting $x = 1$ gives $f(1) = 20$ , and 20 lies between 4 and 100, so the equation must be right.
Substituting $x = 2$ gives $4 \cdot 5^2 = 4 \cdot 25 = 100$ , which matches the given point.
The base 5 is greater than 1, confirming growth, which is sufficient evidence.
The y-intercept is 4, which matches $f(0)$ , so no further check is needed.

Show answer and explanation

Substituting

x = 2

gives

4 \cdot 5^2 = 4 \cdot 25 = 100

, which matches the given point.

The strongest confirmation is substituting

x = 2

and checking that the output equals the given value of 100.

4 \cdot 5^2 = 4 \cdot 25 = 100

✓. The other options are incomplete checks that do not directly verify the second given point.

Key terms

Exponential function: A function of the form $f(x) = a \cdot b^x$ , where the variable $x$ is in the exponent, $a \neq 0$ , $b > 0$ , and $b \neq 1$ .
Initial value ( $a$ ): The output of the function when $x = 0$ ; it equals the y-intercept of the graph.
Base ( $b$ ): The constant multiplier in an exponential function. Each time $x$ increases by 1, the output is multiplied by $b$ .
Exponential growth: The behaviour of $f(x) = a \cdot b^x$ when $b > 1$ and $a > 0$ : outputs increase as $x$ increases.
Exponential decay: The behaviour of $f(x) = a \cdot b^x$ when $0 < b < 1$ and $a > 0$ : outputs decrease as $x$ increases.
Horizontal asymptote: A horizontal line that the graph of a function approaches but never reaches or crosses. For $f(x) = a \cdot b^x$ , the asymptote is the line $y = 0$ .
Constant ratio: In a table of values with equally spaced inputs, the constant ratio of consecutive outputs is the base $b$ of the exponential function.
Domain and range: The domain of $f(x) = a \cdot b^x$ is all real numbers. The range is all positive real numbers when $a > 0$ , or all negative real numbers when $a < 0$ .

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation B2.5. It is a study resource, not an official curriculum publication.

Official curriculum reference

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B2.5 · Represent an exponential function from its graph or properties

Ontario Grade 11 Mathematics

MCR3U – B2.5 | Building and Writing Exponential Function Equations

What you will learn

Prerequisite Bridge: What You Already Know

What Makes a Function Exponential?

Reading Key Properties from a Graph

How $a$ and $b$ Change the Graph

Recognising Exponential Behaviour in a Table of Values

Comparing Key Properties: Growth vs. Decay

Finding the Equation from Two Points on a Graph

Writing an Equation from a Table of Values

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MCR3U

About this lesson