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B2.1 · Collect and graph data modelled exponentially

Learn to collect and graph data modelled exponentially through clear examples and targeted practice.

Ontario Grade 11 Mathematics

Exponential Functions

Finding the pattern of repeated multiplication in real data

Think about a rumour spreading at school. On day one, one person knows it. Each day, every person who knows it tells one new person, so the number of people who know doubles. By day five, hundreds of people know. This kind of growth does not add the same amount each time. It multiplies by the same amount each time. This lesson teaches you how to collect data that behaves this way, organize it in a table, graph it, and recognize the exponential pattern using numbers alone, before you write an equation for it.

What you will learn

Review what makes a relationship linear versus exponential using differences and ratios
Collect or read data that changes by repeated multiplication and organize it in a table
Graph exponential data by hand and recognize its curved shape
Use first differences and ratios of consecutive terms to confirm a data set is exponential
Describe real situations that produce exponential growth or decay

Prerequisite Bridge: Linear Patterns Add, Exponential Patterns Multiply

In Grade 10, you learned that a linear relationship has a constant first difference. This means if you subtract each output value from the next one, you always get the same number. For example, in the table where

x

goes CAD 1, 2, 3, 4 and

y

goes CAD 3, 5, 7, 9, the difference is always

2

. That constant difference told you the relationship was linear, matching an equation of the form

y = mx + b

An exponential relationship behaves differently. Instead of a constant difference, it has a constant ratio. A ratio compares two numbers by division. To find the ratio between consecutive terms, divide each output value by the one before it. If that division always gives the same number, the data is exponential, not linear.

This constant ratio is called the growth factor when it is greater than

1

, because the values are growing, or the decay factor when it is between

0

and

1

, because the values are shrinking toward zero. You will use this single idea, a constant ratio, as your main tool for recognizing exponential data all through this lesson.

Linear data has a constant first difference (values add the same amount)
Exponential data has a constant ratio (values multiply by the same amount)
A ratio greater than $1$ means growth; a ratio between $0$ and $1$ means decay

Plain Language: What Does Exponential Data Look Like?

Exponential data comes from situations where each new amount depends on multiplying the previous amount by a fixed number. Common real examples include a population of bacteria doubling every hour, a car's value dropping by the same percentage every year, or money in a savings account earning compound interest.

When you collect this kind of data, you usually record two columns: the input, often time, and the output, the amount being measured. As time increases by equal steps, like one hour at a time, the amount does not increase by the same amount each step. Instead, it increases by the same percentage or the same multiplying factor each step.

This is the key difference from linear growth. Linear growth adds a fixed amount, like gaining CAD 5 every week from an allowance. Exponential growth multiplies by a fixed factor, like a population growing by 10 percent every week. Even though both start small, exponential growth eventually becomes much larger, much faster, because the amount being multiplied keeps getting bigger.

Exponential data usually has time or trials as the input and an amount as the output
The output changes by a fixed percentage or fixed multiplying factor per equal step of input
Exponential growth eventually outpaces linear growth because the multiplier applies to a larger and larger amount

Multiple Representations: Table, Graph, and Ratio Test

Suppose a tablet's screen brightness setting doubles a display counter every second when a specific app runs a test. The table below shows collected data for seconds elapsed and the counter reading.

Look first at the differences between consecutive counter readings: CAD 2, 4, 8, 16. These differences are not constant, so this is not linear. Now look at the ratios: divide each reading by the one before it. Every ratio equals

2

. This constant ratio is strong evidence the data is exponential.

When you graph this data with seconds on the horizontal axis and counter reading on the vertical axis, you get points that curve sharply upward. Unlike a straight line from linear data, or a symmetric arch from quadratic data, exponential data forms a curve that stays close to the horizontal axis at first, then rises steeply. This shape is called exponential growth curve. If the ratio were between

0

and

1

instead, the curve would start high and flatten down toward the horizontal axis, showing exponential decay.

Always test both differences and ratios before deciding the data type
A constant ratio (not a constant difference) signals exponential data
The graph of exponential growth data curves upward slowly then steeply; decay curves downward toward zero

Tablet Counter Doubling Each Second

Seconds (input)	Counter reading (output)	Difference from previous	Ratio to previous
0	2	not applicable	not applicable
1	4	2	2
2	8	4	2
3	16	8	2
4	32	16	2

Worked example

Testing and Graphing a Bouncing Ball's Height

A student drops a ball and measures the height it reaches after each bounce, in centimetres. The results are: bounce 0 (the drop height) is 160 cm, bounce 1 is 120 cm, bounce 2 is 90 cm, bounce 3 is 67.5 cm. Determine whether this data is exponential, and describe the graph.

Organize the data in a table
Before testing anything, list the bounce number as the input and the height as the output, in order. This makes it easier to compare consecutive values without missing one.
Check first differences
Subtract each height from the height before it: $120 - 160 = -40$ , then $90 - 120 = -30$ , then $67.5 - 90 = -22.5$ . These differences are not equal, so the data is not linear.
$-40,\ -30,\ -22.5$
Check the ratio between consecutive terms
Divide each height by the height right before it in the table. This tests whether the data multiplies by a fixed factor each time.
$\frac{120}{160} = 0.75,\ \frac{90}{120} = 0.75,\ \frac{67.5}{90} = 0.75$
Interpret the constant ratio
Every ratio equals $0.75$ . Since this ratio is the same every time and it is between $0$ and $1$ , the data is exponential decay. This makes sense physically: each bounce reaches only 75 percent of the previous height, because the ball loses energy.
$0.75$
Describe the graph shape
If you plot bounce number on the horizontal axis and height on the vertical axis, the points start high at bounce 0 and drop quickly at first, then level off closer to the horizontal axis as bounce number increases. This curve never quite touches zero in the pattern, matching typical exponential decay behaviour.

Answer: The data is exponential, with a constant ratio of

0.75

between consecutive heights, representing exponential decay. The graph is a curve that drops quickly then flattens toward the horizontal axis.

Check: Multiply the first height by

0.75

repeatedly to see if you regenerate the table:

160 \times 0.75 = 120

120 \times 0.75 = 90

90 \times 0.75 = 67.5

. All values match the original data, confirming the constant ratio is correct.

Common mistakes and how to avoid them

Testing only the differences and concluding the data is not a nice pattern at all, without also testing ratios.

Correction: Always test ratios as a second step. Many data sets that fail the constant-difference test for linear data will pass the constant-ratio test for exponential data.

Dividing the terms in the wrong order, such as dividing an earlier term by a later term instead of later by earlier.

Correction: Always divide each term by the term that comes immediately before it in the table, meaning later value divided by earlier value, to get a consistent ratio.

Assuming any curved graph must be exponential.

Correction: Confirm with the ratio test using the actual table values. Quadratic data also produces curved graphs, but quadratic data has constant second differences, not a constant ratio.

Believing exponential decay data eventually reaches exactly zero.

Correction: Exponential decay values get smaller and smaller but keep multiplying by the same fraction, so they approach zero closely without ever landing exactly on zero in the pattern.

Lesson summary

Linear data has a constant difference; exponential data has a constant ratio between consecutive terms
A ratio greater than $1$ signals exponential growth; a ratio between $0$ and $1$ signals exponential decay
Collecting exponential data means recording equally spaced inputs, like time, alongside outputs that multiply by a fixed factor
Graphing exponential data produces a curve, either rising steeply (growth) or dropping and flattening (decay)
Always test both differences and ratios on real data before deciding it is exponential

Check your understanding

Question 1

A data set has outputs CAD 5, 15, 45, 135 for equally spaced inputs. What kind of relationship is this?

Linear, because the differences are constant
Exponential, because the ratio between consecutive terms is always 3
Exponential, because the difference between consecutive terms is always 3
Neither, because the values keep increasing

Show answer and explanation

Exponential, because the ratio between consecutive terms is always 3

Dividing each term by the one before it gives

15/5 = 3

45/15 = 3

135/45 = 3

. This constant ratio of 3 means the data is exponential growth, not linear.

Question 2

Which feature identifies data as exponential rather than linear?

A constant first difference between consecutive outputs
A constant ratio between consecutive outputs
A graph that is a perfectly straight line
Outputs that always increase in value

Show answer and explanation

A constant ratio between consecutive outputs

Linear data has a constant difference. Exponential data instead has a constant ratio, meaning each term is the previous term multiplied by the same fixed number.

Question 3

A machine's value drops from CAD 8000 to CAD 6400 to CAD 5120 to CAD 4096 over three equal years. What is the constant ratio, and what does it tell you?

The ratio is 0.8, showing exponential decay
The ratio is 1600, showing linear decrease
The ratio is 0.8, showing linear decrease
The ratio is 1.25, showing exponential growth

Show answer and explanation

The ratio is 0.8, showing exponential decay

Dividing consecutive values gives

6400/8000 = 0.8

5120/6400 = 0.8

4096/5120 = 0.8

. A constant ratio between 0 and 1 means exponential decay.

Question 4

Why is it not enough to look only at a curved graph shape to decide data is exponential?

Because exponential data always produces straight line graphs instead
Because quadratic data can also produce a curved graph, so the ratio test is needed to be sure
Because curved graphs never represent real situations
Because only exponential decay ever produces curves, never exponential growth

Show answer and explanation

Because quadratic data can also produce a curved graph, so the ratio test is needed to be sure

Both quadratic and exponential data can look curved on a graph. The reliable way to tell them apart is to test the actual table values: quadratic data has constant second differences, while exponential data has a constant ratio.

Key terms

First difference: The result of subtracting one output value from the next output value in an ordered table of data.
Ratio: The result of dividing one number by another, used here to compare consecutive output values in a table.
Constant ratio: A situation where dividing each output by the previous output always gives the same number, which identifies exponential data.
Growth factor: A constant ratio greater than $1$ that causes data values to increase, producing exponential growth.
Decay factor: A constant ratio between $0$ and $1$ that causes data values to decrease toward zero, producing exponential decay.
Exponential relationship: A relationship between input and output where the output is multiplied by the same fixed factor for every equal step in the input.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation B2.1. It is a study resource, not an official curriculum publication.

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