Collect and graph data modelled exponentially

Ontario Grade 11 Mathematics

Mathematical Models

MBF3C | Specific expectation A3.1

A quantity can change by adding the same amount each time, or by multiplying by the same factor each time. These patterns produce different graphs. This lesson focuses on collecting and graphing data that can be modelled exponentially. You will use equal time intervals, a table, a scatter plot, and a simple model. No advanced algebra is needed.

A scatter plot places each measured pair of values on a coordinate grid. The horizontal coordinate can be time, and the vertical coordinate can be the measured quantity. A model is a rule that describes a pattern in the data. Real measurements will not always sit exactly on the model, so a useful model should follow the overall pattern without claiming perfect precision.

What you will learn

Collect or organize one-variable data that may follow an exponential pattern.
Recognize exponential change by checking for a repeated multiplication factor over equal intervals.
Represent the data in a table and scatter plot, and describe an exponential model using its starting value and repeated factor.
Use a graphing tool to compare an exponential model with collected data and judge whether it is a useful fit.

1. Review the data and graph basics

A variable is a quantity that can change. In a one-variable data investigation, you measure one main quantity, often at several times. Time is recorded alongside each measurement so the pattern can be graphed. For example, you might record the height of a plant every week or the amount of a substance remaining at regular intervals.

Use the same units and the same time spacing throughout a data set. If one measurement is taken every two days, label the time values in days and keep that interval consistent. Record measurements carefully, including units. If you collect the data yourself, use the same measuring method each time. If you use supplied data, check its headings and units before graphing.

On a coordinate grid, the horizontal axis is the

x

-axis and the vertical axis is the

y

-axis. For a time-based investigation, time usually goes on the horizontal axis and the measured quantity goes on the vertical axis. A point

(x,y)

places one time and its matching measurement on the graph. Choose a scale that fits all the values and label each axis with its quantity and unit.

Use equal time intervals when checking a repeated pattern.
Keep measurement units consistent and label the graph axes.
Plot measured pairs as separate points before drawing or selecting a model.

2. Identify a possible exponential pattern

A quantity changes exponentially when it is multiplied by the same factor over equal intervals. That repeated factor is called the growth factor when it is greater than

1

. When the factor is between

0

and

1

, the quantity decreases; it is often called a decay factor. The starting value is the value at time zero.

To check a table, compare each value with the one before it by dividing the later value by the earlier value. If these ratios are the same, or close to the same because of rounding or measurement variation, the data may be exponential. The intervals must be equal for this comparison to be meaningful. A constant difference instead suggests a different kind of pattern, so do not confuse repeated multiplication with repeated addition.

For an exponential model,

y=a(b)^x

is a compact way to describe the pattern. Here,

x

counts equal time intervals,

y

is the modelled quantity,

a

is the starting value, and

b

is the factor applied each interval. If the recorded times are in weeks, for example,

x=3

means three weekly intervals after the starting measurement. The model gives a predicted value; it does not guarantee that a real measurement will match exactly.

y=a(b)^x

Compare successive ratios, not just successive differences.
In $y=a(b)^x$ , $a$ is the starting value and $b$ is the repeated factor.
A close, rather than exact, repeated factor can occur in measured data.

3. Graph and assess the model

First make a scatter plot of the measured data. The plotted points show the observations directly. Then use a graphing tool, if available, to display the model on the same axes. Enter time as

x

and the measured quantity as

y

. Use the table to choose the starting value and factor for a simple model, or use the tool’s exponential fit feature if it is available in your course setting.

An exponential graph is generally curved rather than a straight line. With a growth factor greater than

1

, the values rise and the curve becomes steeper as

x

increases. With a factor between

0

and

1

, the values fall and move closer to zero. The plotted data and model should be viewed together: the points are observations, while the curve represents the pattern described by the rule.

A model is useful when its curve follows the overall direction and lies reasonably close to the data points for the purpose of the investigation. Consider whether there are points far from the curve, whether measurements may have been rounded, and whether the model makes sense for the context. Do not assume that a model remains accurate indefinitely. Report the time range and units covered by the collected data.

Plot observations as points; show the model as a curve or graphing-tool output.
Check whether the model follows the data’s overall pattern and note any limitations.
Keep conclusions within the measured time range unless there is a clear reason to predict beyond it.

4. Apply the process to real data

A careful investigation begins with a question that can be answered by repeated measurements. Choose what to measure, decide on equal time intervals, and record the observations in a table. Graph the measured pairs, check for a repeated factor, and use that factor and the starting value to form a model. Finally, compare the model with the points and explain whether it represents the data reasonably.

Some measurements are affected by rounding, weather, equipment, or other conditions. Such variation does not automatically rule out an exponential model. Instead, describe the pattern as approximate and make the limits of the data clear. The graph is a way to see the pattern; the context and the quality of the measurements help decide how much confidence to place in it.

Plan the measurement schedule before collecting data.
State whether the pattern and model are approximate.
A graph supports a conclusion but does not replace careful measurement.

Measurements and successive factors in the decreasing-data example

Time (h)	Mass (g)	Ratio to previous mass
0	80	—
1	60	0.75
2	45	0.75
3	34	approximately 0.756

Worked example

A culture increasing over time

A school club records the number of visible colonies in a safe, prepared classroom data set. The counts at equal one-day intervals are

12

,

18

,

27

, and

40.5

. Organize the data, decide whether an exponential model is reasonable, and write a model with time

x

in days.

Set the time values
The first count is the starting observation, so assign it time zero. Each later count is one day after the previous one.
$(0,12),\ (1,18),\ (2,27),\ (3,40.5)$
Compare consecutive values
Divide each count by the count immediately before it. Each quotient is $1.5$ , so the data follow the same multiplication factor at every one-day interval.
$\frac{18}{12}=\frac{27}{18}=\frac{40.5}{27}=1.5$
Write and interpret the model
The starting value is $12$ and the daily factor is $1.5$ . Substitute these into the exponential model. On a graph, the points should rise along a curved pattern as time increases.
$y=12(1.5)^x$

Answer: An exponential model is reasonable for these exact data:

y=12(1.5)^x

, where

x

is days and

y

is the count predicted by the model.

Check: At day three the model gives

12(1.5)^3=40.5

, matching the recorded count.

Worked example

A quantity decreasing over time

A demonstration data set records the mass of a drying sample every hour. The measurements are

80

g,

60

g,

45

g, and

34

g at hours

0

,

1

,

2

, and

3

. Check for an exponential pattern and write a reasonable model using the first measurement and a repeated factor.

Compare successive ratios
Use each later mass divided by the previous mass. The first two ratios are $0.75$ . The last is about $0.756$ , which is close, so small measurement or rounding differences are plausible.
$\frac{60}{80}=0.75, \frac{45}{60}=0.75, \frac{34}{45}\approx0.756$
Choose a repeated factor
Use $0.75$ as an approximate hourly factor because it matches the first two ratios and is close to the third. A factor between zero and one represents a decrease each hour.
$b\approx0.75$
Form and check the model
The mass begins at $80$ g, so use $80$ as the starting value. The model predicts $33.75$ g at hour three, which is close to the measured $34$ g. The graph would show decreasing points and a falling curve.
$y=80(0.75)^x$

Answer: A reasonable approximate model is

y=80(0.75)^x

, where

x

is time in hours and

y

is mass in grams.

Check: At hour two the model predicts

80(0.75)^2=45

g, exactly matching that measurement; at hour three it predicts

33.75

g, close to

34

g.

Common mistakes and how to avoid them

Treating a repeated difference as proof that data are exponential.

Correction: Exponential data have a repeated multiplication factor over equal intervals. Compare successive ratios; a repeated difference is a different pattern.

Comparing values measured over unequal time intervals as if each ratio covered one equal interval.

Correction: Check the time spacing first. Use equal intervals when comparing successive factors.

Connecting observations with a straight line and calling that curve an exponential model.

Correction: Show the observations as points, then display an exponential model as a curve that follows their overall pattern.

Claiming that a model exactly predicts every real measurement or will remain accurate forever.

Correction: Describe the model as approximate when appropriate, and state the time range and measurement limits.

Lesson summary

Collect or organize measurements at equal intervals, using consistent units.
Graph each time-and-measurement pair as a point.
Check whether successive values have a common ratio; a close common ratio suggests an exponential pattern.
Use the starting value and repeated factor to describe the model, then assess the curve against the plotted data.

Check your understanding

Question 1

A quantity is measured at equal intervals:

20

,

30

, and

45

. Which statement best describes the pattern?

It is exponential with an approximate repeated factor of $1.5$ .
It is exponential with a repeated difference of $10$ .
It has a repeated factor of $2$ .
It cannot be graphed because the values increase.

Show answer and explanation

It is exponential with an approximate repeated factor of

1.5

.

The successive ratios are

30/20=1.5

and

45/30=1.5

. The repeated factor is

1.5

, so these data follow an exponential pattern.

Question 2

A quantity starts at

100

and is multiplied by

0.8

at each equal time interval. What does the factor tell you?

The quantity increases by $0.8$ units each interval.
The quantity is about 80% of its previous value each interval.
The starting value is $0.8$ .
The quantity decreases by exactly $20$ units each interval.

Show answer and explanation

The quantity is about 80% of its previous value each interval.

Multiplication by

0.8

means each new value is 80% of the previous one. The amount of decrease in units can change from interval to interval.

Question 3

Which graphing choice is most appropriate when checking a model against collected measurements?

Plot the measurements as points and compare them with the model curve.
Replace all measurements with the model curve.
Plot time on the vertical axis and leave both axes unlabeled.
Connect the points with a straight line and assume it is the model.

Show answer and explanation

Plot the measurements as points and compare them with the model curve.

The points preserve the observations, and the curve represents the proposed model. Comparing them shows whether the model follows the data.

Key terms

Data: Recorded observations or measurements.
Scatter plot: A graph that shows paired values as separate points.
Model: A rule or graph used to represent a pattern in data.
Growth factor: The repeated multiplication factor when a quantity increases.
Decay factor: A repeated multiplication factor between zero and one, which makes a quantity decrease.
Starting value: The modelled quantity at time zero.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation A3.1. 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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