B3.1 Collect and graph data modelled by an exponential function

Ontario Grade 11 Mathematics

Exponential Functions

MCR3U – Unit B3 | Exponential Functions

You have already worked with linear functions (straight-line graphs) and quadratic functions (parabolas) in Grade 10. This lesson adds a third family: exponential functions. Exponential functions appear whenever a quantity is repeatedly multiplied by the same number — for example, a bacterial colony doubling every hour, or the value of a car dropping by a fixed percentage each year. The goal right now is practical: read or collect real-style data, build a table of values, plot those values carefully, and recognize the characteristic curved shape that marks exponential behaviour. Careful arithmetic, neat plotting, and pattern recognition are all you need.

What you will learn

Recognize real-life situations that produce data growing or decaying exponentially.
Record and organize collected data in a clearly labelled table of values.
Plot data points on a coordinate grid and draw a smooth exponential curve through them.
Identify key features of an exponential graph, including the y-intercept, direction of change, and horizontal asymptote.
Decide whether a data set is best described by a linear, quadratic, or exponential model by examining first differences, second differences, and ratios of consecutive output values.

Prerequisite Bridge: What You Already Know

In Grade 10 you graphed functions by making a table of values — choosing

x

-values, calculating the matching

y

-values, and plotting the ordered pairs

(x, y)

. That exact skill is your starting point here.

You also learned that a linear function has a constant rate of change: every time

x

increases by 1,

y

changes by the same fixed amount. A quick way to check linearity is to look at the differences between consecutive

y

-values. These are called first differences. If all first differences are equal, the relationship is linear.

For exponential data, first differences are not constant. Instead, you look at ratios: divide each

y

-value by the one before it. If every ratio is the same, the data is exponential. This single idea is the key to recognizing exponential behaviour in a table.

Linear data has constant first differences.
Exponential data has a constant ratio between consecutive $y$ -values.
Plot ordered pairs $(x, y)$ from a table, then connect them with a smooth curve — not straight-line segments.

What Is an Exponential Function?

An exponential function has the form

f(x) = a \cdot b^x

, where

a

is the starting value (the output when

x = 0

) and

b

is the constant multiplier called the base. The conditions

b > 0

and

b \neq 1

must both hold.

When

b > 1

, multiplying by

b

repeatedly makes the output grow larger and larger. This is exponential growth. When

0 < b < 1

, repeated multiplication shrinks the output toward zero. This is exponential decay.

The variable

x

sits in the exponent, which is what separates this family from linear or quadratic functions. In

f(x) = 3 \cdot 2^x

, the base is

2

and the starting value is

3

. Each time

x

increases by 1, the output is multiplied by

2

.

One important feature of every exponential graph is the horizontal asymptote. For

f(x) = a \cdot b^x

with a positive

a

, the graph gets closer and closer to the line

y = 0

as

x

moves in the decay direction, but it never actually touches that line. The line

y = 0

is therefore called the horizontal asymptote of the function.

f(x) = a \cdot b^x

General form: $f(x) = a \cdot b^x$ , with $a \neq 0$ , $b > 0$ , $b \neq 1$ .
$b > 1$ gives exponential growth; $0 < b < 1$ gives exponential decay.
The $y$ -intercept is always the point $(0,\, a)$ .
The $x$ -axis, $y = 0$ , is the horizontal asymptote.
Consecutive $y$ -values (with equally spaced $x$ -values) always share the constant ratio $b$ .

Collecting and Organizing Exponential Data

Data that grows or decays exponentially often comes from a hands-on experiment or a real-world scenario. Two classic examples are: (1) paper folding — each fold doubles the number of layers; and (2) bouncing ball heights — each bounce reaches a fixed fraction of the previous height.

Once you have a scenario, collect your data systematically. Set up a table with two columns: the independent variable (often a count such as number of folds or bounces) and the dependent variable (the measured quantity). Label every column clearly, including units where applicable.

Before graphing, check whether the data is truly exponential by computing ratios. Divide the second

y

-value by the first, the third by the second, and so on. If every ratio is the same — or very close, if the data comes from a real experiment with small measurement errors — the relationship is exponential.

Choose a scale for your axes that fits all your data comfortably. Because exponential values can grow quickly, you may need a larger

y

-axis range than you would use for a linear graph. Plot each point carefully, then draw a single smooth curve through all the points. Do not connect them with straight-line segments.

Organize data in a clear two-column table before graphing.
Check for a constant ratio between consecutive $y$ -values to confirm exponential behaviour.
Use an appropriate scale — exponential growth can produce very large values quickly.
Connect plotted points with one smooth curve, not a series of straight segments.

Reading and Interpreting the Graph

Once your curve is drawn, you can read off several key features. The

y

-intercept — where the curve crosses the

y

-axis — equals the starting value

a

in

f(x) = a \cdot b^x

. This is the output at

x = 0

.

For a growth curve where

b > 1

, the graph rises steeply to the right and flattens toward the horizontal asymptote on the left. For a decay curve where

0 < b < 1

, the opposite is true: the curve drops steeply from the left and levels off toward the asymptote as you move right.

You can also estimate values between plotted points directly from the graph. Because the function is continuous, every point on the smooth curve represents a valid output, not just the specific points you plotted.

Comparing the shape of your graph to linear and quadratic shapes is a useful identification skill. A linear graph is a straight line. A quadratic graph is a symmetric U-shape. An exponential growth graph curves upward and gets steeper toward the right, with no axis of symmetry. Its left and right sides look completely different from each other — the curve is much flatter on one side and much steeper on the other.

The $y$ -intercept of the graph equals $a$ , the starting value.
Growth curves ( $b > 1$ ) rise to the right; decay curves ( $0 < b < 1$ ) fall to the right.
The graph approaches but never crosses the horizontal asymptote $y = 0$ .
An exponential curve is not symmetric — its two sides have very different steepness.

Linear, Quadratic, or Exponential? Choosing the Right Model

When you are given a data set without being told the type of function, use a three-step test. First, check first differences: subtract each

y

-value from the next. If all first differences are equal, the model is linear. Second, if first differences are not equal, compute second differences — the differences between the first differences. If those are equal, the model is quadratic. Third, if neither set of differences is constant, compute ratios of consecutive

y

-values by dividing each one by the previous one. A constant ratio means the model is exponential.

This test works reliably only when the

x

-values are equally spaced, for example

x = 0, 1, 2, 3, \ldots

All data sets in this course will use equally spaced inputs, so you can apply the test directly.

In real collected data, ratios may not be perfectly identical because of small measurement errors. Look for a ratio that is approximately constant, and use the context of the problem — such as 'the population doubles each year' — to confirm your model choice.

Constant first differences → linear model.
Constant second differences → quadratic model.
Constant ratio of consecutive $y$ -values → exponential model.
This three-step test requires equally spaced $x$ -values.
In real data, approximately constant ratios still point to an exponential model.

Identifying the Function Model from a Data Table

Step	What to compute	If the result is constant, the model is…
1 — First differences	Subtract each $y$ -value from the next: $y_2 - y_1$ , $y_3 - y_2$ , …	Linear
2 — Second differences	Subtract consecutive first differences from each other	Quadratic
3 — Consecutive ratios	Divide each $y$ -value by the one before it: $y_2 \div y_1$ , $y_3 \div y_2$ , …	Exponential

Worked example

Example 1 (Growth): Paper-Folding Layers

A student folds a sheet of paper repeatedly. Before any folds there is 1 layer. Each fold doubles the number of layers. Record the number of layers for folds 0 through 5, confirm the data is exponential by checking ratios, identify the values of

a

and

b

, write the function rule, and describe the key features of the graph.

Build the table of values
Start with 1 layer at fold 0. Each fold multiplies the layer count by 2. Fill in the table by doubling each previous entry: fold 0 gives $L = 1$ , fold 1 gives $L = 2$ , fold 2 gives $L = 4$ , fold 3 gives $L = 8$ , fold 4 gives $L = 16$ , and fold 5 gives $L = 32$ .
Check ratios to confirm exponential behaviour
Divide each $L$ -value by the one before it. If the ratio is the same every time, the data is exponential.
$\frac{2}{1} = \frac{4}{2} = \frac{8}{4} = \frac{16}{8} = \frac{32}{16} = 2$
Identify the parameters and write the function rule
Every ratio equals 2, so the base is $b = 2$ . The value at fold 0 is $a = 1$ . Substituting into $f(x) = a \cdot b^x$ gives the rule $L(n) = 1 \cdot 2^n$ , which simplifies to $L(n) = 2^n$ .
$L(n) = 2^n$
Describe the key graph features
The $y$ -intercept is the point $(0, 1)$ , matching $a = 1$ . Because $b = 2 > 1$ , this is an exponential growth curve — it rises steeply to the right. The horizontal asymptote is the line $L = 0$ , visible on the left side of the graph where the curve flattens out. Plot the six points $(0,1)$ , $(1,2)$ , $(2,4)$ , $(3,8)$ , $(4,16)$ , $(5,32)$ and connect them with one smooth upward-curving line.

Answer: The data is exponential with

a = 1

,

b = 2

, and rule

L(n) = 2^n

. The graph passes through

(0,1)

,

(1,2)

,

(2,4)

,

(3,8)

,

(4,16)

,

(5,32)

, curves steeply upward to the right, and has a horizontal asymptote at

L = 0

.

Check: Spot-check two points:

L(3) = 2^3 = 8

✓ and

L(5) = 2^5 = 32

✓. Ratio check:

32 \div 16 = 2

✓.

Worked example

Example 2 (Decay): Bouncing Ball Heights

A rubber ball is dropped from a height of 80 cm. After each bounce it reaches 75% of its previous height. Record the bounce heights for bounces 0 through 4, confirm the data is exponential by checking ratios, identify

a

and

b

, write the function rule, and state the equation of the horizontal asymptote.

Build the table of values
Bounce 0 is the starting drop height of 80 cm. Each bounce multiplies the height by 0.75. Calculate each entry in order: bounce 0 is 80 cm, bounce 1 is $80 \times 0.75 = 60$ cm, bounce 2 is $60 \times 0.75 = 45$ cm, bounce 3 is $45 \times 0.75 = 33.75$ cm, bounce 4 is $33.75 \times 0.75 = 25.3125$ cm.
Confirm the constant ratio
Divide each height by the one before it to check whether the ratio is always the same.
$\frac{60}{80} = \frac{45}{60} = \frac{33.75}{45} = \frac{25.3125}{33.75} = 0.75$
Identify the parameters and write the function rule
The constant ratio is $b = 0.75$ . Because $0 < 0.75 < 1$ , this is exponential decay. The starting value is $a = 80$ . Substituting gives $h(n) = 80 \cdot (0.75)^n$ .
$h(n) = 80 \cdot (0.75)^n$
State the asymptote and describe the graph
As the bounce count grows, the height gets closer and closer to zero but never actually reaches it. The horizontal asymptote is therefore the line $h = 0$ . The graph starts at the point $(0, 80)$ , drops quickly at first, then more and more slowly, always curving downward toward the asymptote. Plot the five points and draw one smooth decay curve through them.
$h = 0$

Answer: The data is exponential decay with

a = 80

,

b = 0.75

, and rule

h(n) = 80 \cdot (0.75)^n

. The graph passes through

(0,80)

,

(1,60)

,

(2,45)

,

(3,33.75)

,

(4,25.3125)

, curves downward, and has a horizontal asymptote at

h = 0

.

Check: Verify

h(2) = 80 \cdot (0.75)^2 = 80 \cdot 0.5625 = 45

✓. Verify

h(4) = 80 \cdot (0.75)^4 = 80 \cdot 0.31640625 = 25.3125

✓.

Common mistakes and how to avoid them

Connecting plotted points with straight-line segments instead of a smooth curve.

Correction: An exponential function is continuous and curved everywhere. Always draw one smooth curve through all the points, not a dot-to-dot zigzag.

Checking first differences instead of ratios when testing for exponential behaviour.

Correction: Exponential data has a constant ratio between consecutive

y

-values, not constant differences. Divide consecutive values — do not subtract — when testing for an exponential pattern.

Mixing up the base

b

and the starting value

a

in

f(x) = a \cdot b^x

.

Correction: The starting value

a

is the

y

-intercept — the output when

x = 0

. The base

b

is the constant ratio you found from the data table.

Thinking the curve eventually touches or crosses the horizontal asymptote for very large

x

-values.

Correction: The curve approaches

y = 0

but never reaches it. No matter how large

x

gets, the function value stays above zero (for positive

a

). The asymptote is a boundary the graph gets infinitely close to but never crosses.

Applying the ratio test when the

x

-values in the table are not equally spaced.

Correction: The constant-ratio test is only valid when

x

-values increase by the same step each time, such as CAD 0, 1, 2, 3. Always verify that inputs are equally spaced before applying the test.

Lesson summary

An exponential function has the form $f(x) = a \cdot b^x$ , where $a$ is the starting value (the $y$ -intercept) and $b$ is the constant multiplier called the base.
Exponential growth occurs when $b > 1$ ; exponential decay occurs when $0 < b < 1$ .
To check whether a data set is exponential, compute the ratio of each consecutive pair of $y$ -values with equally spaced $x$ -values — a constant ratio confirms exponential behaviour.
Key graph features include the $y$ -intercept at $(0, a)$ , a smooth curve rising steeply (growth) or falling and levelling off (decay), and a horizontal asymptote at $y = 0$ .
To build and graph exponential data: collect values, record them in a labelled table, confirm the constant ratio, plot the points, and draw one smooth curve.
To choose the right model: constant first differences → linear; constant second differences → quadratic; constant ratio of consecutive $y$ -values → exponential.

Check your understanding

Question 1

A data set has

y

-values 5, 15, 45, 135 for equally spaced

x

-values. Which feature confirms this is exponential data?

The first differences are all equal to 10.
The ratio of each $y$ -value to the previous one is always 3.
The second differences are all equal to 20.
The $y$ -intercept is positive.

Show answer and explanation

The ratio of each

y

-value to the previous one is always 3.

Dividing consecutive values gives

15 \div 5 = 3

,

45 \div 15 = 3

, and

135 \div 45 = 3

. A constant ratio of 3 is the defining feature of exponential data. The first differences are 10, 30, 90 — not constant — so a linear model does not fit.

Question 2

The function

f(x) = 50 \cdot (0.6)^x

models a data set. What are the

y

-intercept and the equation of the horizontal asymptote?

$y$ -intercept: $(0,\, 0.6)$ ; asymptote: $y = 50$
$y$ -intercept: $(0,\, 50)$ ; asymptote: $y = 0$
$y$ -intercept: $(0,\, 50)$ ; asymptote: $y = 0.6$
$y$ -intercept: $(0,\, 30)$ ; asymptote: $y = 0$

Show answer and explanation

y

-intercept:

(0,\, 50)

; asymptote:

y = 0

Substituting

x = 0

gives

f(0) = 50 \cdot (0.6)^0 = 50 \cdot 1 = 50

, so the

y

-intercept is

(0, 50)

. Since

0 < 0.6 < 1

, the output decays toward zero but never reaches it, making the horizontal asymptote

y = 0

.

Question 3

A colony of bacteria starts with 200 cells and triples every hour. How many cells are present after 3 hours?

1 800 cells
600 cells
5 400 cells
2 400 cells

Show answer and explanation

5 400 cells

The rule is

f(x) = 200 \cdot 3^x

. At

x = 3

:

f(3) = 200 \cdot 3^3 = 200 \cdot 27 = 5400

cells. Tracing step by step: 200 \to 600 \to 1800 \to 5400 ✓.

Question 4

You plot data that curves sharply upward to the right and flattens toward the

x

-axis on the left. Which description matches this graph?

A straight line with a positive slope.
A U-shaped parabola opening upward.
An exponential growth curve.
An exponential decay curve.

Show answer and explanation

An exponential growth curve.

A curve that rises steeply to the right and approaches the

x

-axis (the horizontal asymptote

y = 0

) on the left is the shape of an exponential growth graph with

b > 1

. A decay graph would instead fall steeply from the left and approach the asymptote toward the right.

Key terms

Exponential function: A function of the form $f(x) = a \cdot b^x$ in which the variable $x$ appears as the exponent, $a$ is the starting value, and $b$ is the constant base with $b > 0$ and $b \neq 1$ .
Base ( $b$ ): The constant number that is repeatedly multiplied in an exponential function. It equals the ratio between any two consecutive output values when the inputs are equally spaced by 1.
Starting value ( $a$ ): The output of the exponential function when $x = 0$ . It equals the $y$ -intercept of the graph.
Exponential growth: The behaviour of $f(x) = a \cdot b^x$ when $b > 1$ : outputs increase rapidly as $x$ increases.
Exponential decay: The behaviour of $f(x) = a \cdot b^x$ when $0 < b < 1$ : outputs decrease and approach zero as $x$ increases.
Horizontal asymptote: A horizontal line that a graph approaches but never reaches or crosses. For $f(x) = a \cdot b^x$ with positive $a$ , the horizontal asymptote is the line $y = 0$ .
Constant ratio: The fixed number obtained by dividing any $y$ -value by the previous $y$ -value in a table with equally spaced $x$ -values. A constant ratio identifies exponential data.
First differences: Values found by subtracting each $y$ -value from the next in a table. Constant first differences identify a linear relationship.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation B3.1. It is a study resource, not an official curriculum publication.

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B3.1 · Collect and graph data modelled by an exponential function

Ontario Grade 11 Mathematics

MCR3U – Unit B3 | Exponential Functions

What you will learn

Prerequisite Bridge: What You Already Know

What Is an Exponential Function?

Collecting and Organizing Exponential Data

Reading and Interpreting the Graph

Linear, Quadratic, or Exponential? Choosing the Right Model

Identifying the Function Model from a Data Table

Example 1 (Growth): Paper-Folding Layers

Example 2 (Decay): Bouncing Ball Heights

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