DoAssignment.ca

A3.2 · Describe characteristics of applied exponential relations

Learn to describe characteristics of applied exponential relations through clear examples and targeted practice.

Ontario Grade 11 Mathematics

Mathematical Models

Recognize the starting amount, repeated change, and practical meaning of an exponential model

Many real situations change by the same percentage over equal time periods. A savings balance may grow by a fixed percentage each year, or a quantity of medicine in the body may fall by a fixed percentage each hour. These changes are different from adding or subtracting the same amount each time. In this lesson, you will describe the characteristics of such applied exponential relations. An applied relation is a mathematical relationship used to represent a real situation. We will focus on what its values, pattern, and graph tell us, not on finding a complicated formula.

What you will learn

Identify the initial value and multiplicative change in an applied exponential relation.
Describe whether an exponential relation increases or decreases and explain why.
Connect a model’s values and graph features to the situation it represents.
State sensible restrictions on the input based on the context.

1. Review the change pattern

A quantity changes additively when the same amount is added or removed each time. For example, a tank that gains 4 litres each minute follows an additive pattern: its amount goes up by 4 litres per minute.

A quantity changes multiplicatively when it is multiplied by the same number each time. Suppose a culture has 100 cells and doubles every hour. The amounts are 100, 200, 400, and 800. Each new amount is twice the previous one. That repeated multiplication is the key pattern in an exponential relation.

A percentage change is also multiplicative. An increase of 5% means multiplying by 1.05, because the new amount is 100% of the old amount plus 5% more. A decrease of 20% means multiplying by 0.80, because 80% remains. These multipliers are called growth and decay factors, respectively. A factor is the number used to multiply an amount.

1 + \frac{r}{100}

Same amount added or removed each period: additive pattern.
Same factor used each period: multiplicative pattern.
A repeated percentage change creates a repeated multiplier.

2. Read the main characteristics

A common form for an exponential relation is

y=a(b)^x

. Here,

x

counts equal time periods,

y

is the amount after those periods,

a

is the initial value, and

b

is the multiplier for each period. The initial value is the amount when the count of periods is zero. The multiplier is also called the common ratio because each output is found by multiplying the previous output by the same number.

When

b>1

, the relation increases: each period makes the amount larger. When

0<b<1

, the relation decreases: each period leaves a fraction of the previous amount. In both cases, the amounts change by a constant factor, not by a constant difference. The value of

a

gives the starting point, while

b

describes how the amount changes from one period to the next.

A table or graph can help describe these features. The point at

x=0

has value

a

. For a growing relation, the graph rises as

x

increases. For a decreasing relation, it falls and gets closer to zero when the multiplier is between zero and one. In a real situation, however, the model may only make sense for a limited range. A model for a number of years after a purchase does not describe years before the purchase unless the situation says it should.

The input values also depend on the situation. If the amount is recorded once per day, the meaningful inputs may be whole numbers of days. If the relation describes a smoothly changing measurement, other time values may make sense too. Always use the units and timing given in the situation.

y=a(b)^x

The initial value is the amount at the start, when $x=0$ .
The multiplier describes the repeated change for one period.
A multiplier above 1 gives growth; a multiplier between 0 and 1 gives decay.
The context determines which input values and units are sensible.

3. Connect a model to its context

To describe an applied exponential relation, name what the quantities represent and include their units. Then state the initial value and the repeated multiplier. If the multiplier is given as a percentage change, explain the percentage in ordinary language as well. For example, a multiplier of 0.92 means that 92% remains each period, so the amount decreases by 8% per period.

Next, describe the direction of change and what happens over time. A growing model with a fixed multiplier rises faster in amount from one period to the next, even though its percentage rate stays the same. A decreasing model with a fixed multiplier loses a percentage of its current amount each period, so the amount removed is not the same each time.

Finally, check whether the model is reasonable for the situation. A number of people, objects, or visits is normally counted in whole numbers, even if a calculation gives a decimal estimate. Money can be represented with decimal amounts. A model may also stop being realistic after a certain time because real conditions can change. Describing a model includes recognizing what it tells us and what the context limits.

State the units for the input and output.
Explain the multiplier as a factor or a percentage change.
Describe the trend and check whether the model fits the real situation.

Sample values for the two situations

Periods after start	Nursery estimate (seedlings)	Battery charge (%)
0	80	64
1	100	48
2	125	36
3	156.25	27

Worked example

Growth in a plant nursery

A nursery begins with 80 seedlings in a tray. The number of seedlings in a monitored group is estimated to increase by 25% each week. Describe the exponential relation and estimate the group size after 3 weeks.

Identify the starting amount
At week zero, the group has 80 seedlings. This is the initial value, so it is the output when $x=0$ .
$a=80$
Find the weekly multiplier
A 25% increase means the new amount is 125% of the old amount. As a decimal multiplier, 125% is 1.25.
$b=1.25$
Describe the pattern
Because the multiplier is greater than 1, the model increases each week. The input is the number of weeks after the starting count, and the output is the estimated number of seedlings.
$N=80(1.25)^x$
Estimate after three weeks
Substitute 3 for the number of weeks. Multiplying by 1.25 three times gives an estimate of 156.25 seedlings. Since seedlings are counted as whole objects, this is about 156 seedlings.
$N=80(1.25)^3=156.25$

Answer: The initial estimate is 80 seedlings, and the model grows by 25% each week. After 3 weeks, it estimates about 156 seedlings.

Check: The estimate is greater than 80, as expected for repeated growth. The successive estimates are 80, 100, 125, and 156.25, so each step uses the same factor of 1.25.

Worked example

A battery losing charge

A device has 64% battery charge remaining. During a test, the remaining charge is 75% of the previous hour’s charge after each hour. Describe the relation and estimate the charge after 2 hours.

Identify the starting value
At the start of the test, which is hour zero, the device has 64% charge. This percentage is the initial value.
$a=64$
Interpret the multiplier
The multiplier is 0.75. It means 75% of the previous hour’s charge remains, so 25% is lost each hour.
$b=0.75$
Describe the direction
The multiplier is greater than zero and less than 1, so the model decreases. The output is the remaining charge percentage, and the input is the number of hours since the test began.
$C=64(0.75)^x$
Estimate after two hours
Apply the factor twice. After one hour the estimate is 48%; after two hours it is 36%.
$C=64(0.75)^2=36$

Answer: The model begins at 64% charge and retains 75% of the previous amount each hour, a decrease of 25% per hour. After 2 hours, the estimate is 36%.

Check: The amount falls from 64% to 48% to 36%. Each hour’s amount is 0.75 times the preceding amount, confirming the stated pattern.

Common mistakes and how to avoid them

Treating a 25% increase as a multiplier of 0.25.

Correction: A 25% increase leaves 125% of the original amount, so use 1.25 as the multiplier.

Saying a decreasing relation subtracts the same amount each period.

Correction: In an exponential decrease, the same fraction of the current amount remains each period. The amount lost can differ from period to period.

Calling the first recorded value the value at

x=0

without checking the context.

Correction: Confirm that the first value is measured at the stated start. If the count begins after a period has passed, identify that timing clearly.

Lesson summary

An applied exponential relation models repeated multiplication by a fixed factor.
The initial value is the amount at the start of the model.
A multiplier above 1 describes growth; a multiplier between 0 and 1 describes decrease.
Use the situation to identify units, meaningful input values, and limits of the model.

Check your understanding

Question 1

A town’s recorded number of visitors starts at 200 and rises by 10% each month. Which description is correct?

It starts at 200 and is multiplied by 1.10 each month.
It starts at 200 and is multiplied by 0.10 each month.
It starts at 10 and is multiplied by 200 each month.
It increases by exactly 10 visitors each month.

Show answer and explanation

It starts at 200 and is multiplied by 1.10 each month.

A 10% increase means the new amount is 110% of the previous amount, so the multiplier is 1.10. The starting amount is 200.

Question 2

A quantity is multiplied by 0.6 each day. What does this say about its trend?

It increases by 60% each day.
It decreases by 60% each day, leaving 40%.
It decreases by 40% each day, leaving 60%.
It decreases by the same fixed amount each day.

Show answer and explanation

It decreases by 40% each day, leaving 60%.

A factor of 0.6 means 60% of the previous amount remains. Therefore, 40% is lost each day and the relation decreases.

Question 3

A model has an initial value of 45 and a multiplier of 1.2. What is the model’s output after one period?

46.2
54
66
9

Show answer and explanation

After one period, multiply the initial value by the factor: 45 times 1.2 is 54. The factor is greater than 1, so this is growth.

Key terms

Applied relation: A mathematical relationship used to represent a situation.
Initial value: The amount at the start of a model, when the period count is zero.
Multiplier: The number used to multiply an amount to get the next amount.
Growth factor: A multiplier greater than 1 that represents repeated increase.
Decay factor: A multiplier between 0 and 1 that represents a repeated decrease.

Continue through MBF3C

View the complete Ontario Grade 11 Mathematics learning path

About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation A3.2. It is a study resource, not an official curriculum publication.

Official curriculum reference

Report a correction or ask a question