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

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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