Pose and solve exponential problems using graphs

Ontario Grade 11 Mathematics

Mathematical Models

Ontario Grade 11 MBF3C — study topic A3.3

Exponential relationships describe situations where a quantity changes by the same factor over equal time steps. A growing social media audience, a cooling substance, or a machine’s resale value may follow this kind of pattern. In this lesson, you will pose questions about such situations and use graphs to estimate answers. A graph is especially useful when the answer is not easy to find by counting time steps.

What you will learn

Recognize an exponential pattern in a practical situation.
Represent an exponential relationship with a graph and a rule.
Use a graph to estimate when a quantity reaches a target.
Check whether a graph-based answer makes sense in context.

1. Review: patterns and coordinates

A graph shows how two quantities are related. The horizontal axis usually shows the input, such as time. The vertical axis shows the output, such as the number of bacteria or the value of an item. A point on the graph gives one paired input and output.

A coordinate is written as an ordered pair, such as

(2, 18)

. It means that when the input is

2

, the output is

18

. To read a graph, find the input on the horizontal axis, move to the curve, and then read the matching output on the vertical axis.

In a linear pattern, a quantity changes by the same amount each step. In an exponential pattern, it changes by the same factor each step. For example, multiplying by

2

each day is exponential growth. Multiplying by

0.8

each year is exponential decay. Decay means the quantity gets smaller by a repeated factor.

The input is usually on the horizontal axis; the output is on the vertical axis.
Equal repeated factors signal an exponential pattern.
A graph can show values between the entries in a table.

2. Build and read an exponential graph

An exponential rule can be written as

y=a(b)^x

. Here,

x

is the input,

y

is the output,

a

is the starting amount, and

b

is the factor applied for each one-unit increase in

x

. When

b

is greater than

1

, the graph rises. When

b

is between

0

and

1

, the graph falls.

For a practical model, first identify the starting amount and the repeated factor. Then choose meaningful input values, calculate the outputs, and plot the points. A smooth curve through those points helps show the pattern between the values. Use a sensible scale on each axis so the important part of the graph is easy to read.

To pose a problem, ask a question that connects the input and output, such as: “After how many years will the value fall to CAD 4,000?” To solve it using a graph, locate the target output, move across to the curve, and then down to the horizontal axis. The reading is an estimate, so report a suitable unit and level of precision.

y=a(b)^x

Label both axes with the quantity and its unit.
Plot points from the rule before drawing a smooth curve.
For a target-output question, read across from the output to the curve, then down to the input axis.

3. Choose a useful model and judge the answer

A graph-based model is useful when the situation changes by a repeated factor and the question asks about a value at a particular time or the time needed to reach a target. For example, a quantity that increases by 20% per month is multiplied by

1.20

each month. A quantity that loses 20% each month keeps 80%, so it is multiplied by

0.80

.

A graph gives an estimate, not a guarantee that the real situation follows the rule perfectly. Before accepting an answer, check that the input is in the appropriate range, that the output is close to the target, and that the units make sense. If the graph is coarse, make the estimate more carefully by adding points near the crossing or using a graphing tool.

The point where the curve reaches a chosen target is an intersection with a horizontal level on the graph. Its horizontal coordinate answers a “when?” question. Its vertical coordinate answers a “how much?” question. Keep the question in mind so you report the correct coordinate.

A percentage increase or decrease can be expressed as a repeated factor.
A graph reading is an estimate; precision depends on the scale and plotting.
Check the coordinate you report against the wording of the question.

Worked example

Growing a community garden’s seedling count

A greenhouse starts with 50 seedlings. The count is expected to multiply by 1.5 each week. Use a graph-based model to estimate when the count reaches 250 seedlings.

Identify the pattern
The starting count is 50. Each week, the count is multiplied by 1.5, so the rule uses that repeated factor.
$y=50(1.5)^x$
Make graph points
Calculate values for whole weeks. These points let you sketch the curve and see where it reaches 250.
$(0,50),\ (1,75),\ (2,112.5),\ (3,168.75),\ (4,253.125)$
Read the target
On the graph, draw or imagine a horizontal level at 250. It meets the curve just before week 4. The estimate is about 4 weeks; the curve is close to 250 at that time.
$y=250 \Rightarrow x\approx4$

Answer: The model reaches about 250 seedlings after approximately 4 weeks.

Check: At 4 weeks the model gives about 253 seedlings, which is close to the target. The estimate and unit are reasonable.

Worked example

A used bicycle losing value

A bicycle is worth CAD 900 when purchased. Its value is expected to be multiplied by 0.75 each year. Use a graph-based model to estimate when it will be worth CAD 400.

Write the model
The starting value is CAD 900. Keeping 75% of the value each year means multiplying by 0.75 for every year.
$y=900(0.75)^x$
Plot helpful values
Calculate several values to shape the decreasing curve. Plot these points with years on the horizontal axis and value in CAD on the vertical axis.
$(0,900),\ (1,675),\ (2,506.25),\ (3,379.69)$
Estimate the crossing
The target of CAD 400 lies between the values at years 2 and 3. On the graph, the curve reaches 400 a little before year 3, so an estimate of about 2.9 years is suitable.
$y=400 \Rightarrow x\approx2.9$

Answer: The bicycle is worth about CAD 400 after approximately 2.9 years, or close to 3 years.

Check: The model gives about CAD 506 after 2 years and CAD 380 after 3 years. CAD 400 is between them, supporting the graph estimate.

Common mistakes and how to avoid them

Reading the output coordinate when the question asks when something happens.

Correction: For a “when?” question, report the horizontal coordinate, which represents the input or time.

Using the percentage as the factor, such as using 0.25 for a 25% decrease.

Correction: A 25% decrease leaves 75%, so the repeated factor is 0.75.

Treating a graph estimate as exact.

Correction: Use words such as “about” or “approximately,” and match the precision to the graph scale.

Plotting time on the vertical axis without labeling the graph clearly.

Correction: Put the independent input, usually time, on the horizontal axis and label both axes with units.

Lesson summary

An exponential pattern changes by a repeated factor over equal input steps.
Use the starting amount and repeated factor to build a rule and graph.
To estimate when a target is reached, read from the target output across to the curve and down to the input axis.
Check the estimate using nearby graph points, units, and the situation.

Check your understanding

Question 1

A quantity follows the model

y=80(1.25)^x

, where

x

is weeks. Which points should be plotted for

x=0

and

x=1

?

$(0,80)$ and $(1,100)$
$(0,100)$ and $(1,80)$
$(0,80)$ and $(1,105)$
$(0,1.25)$ and $(1,80)$

Show answer and explanation

(0,80)

and

(1,100)

At

x=0

, the output is 80. At

x=1

, it is

80(1.25)=100

.

Question 2

A graph shows a target output of 300 meeting a curve at about

x=5.5

. If

x

represents months, what does the estimate mean?

The output is about 5.5 units.
The target is reached after about 5.5 months.
The starting amount is 5.5 months.
The output decreases by 5.5 each month.

Show answer and explanation

The target is reached after about 5.5 months.

The horizontal coordinate represents months. The graph estimates that the target output is reached after about 5.5 months.

Question 3

A quantity loses 10% of its value each year. Which repeated factor belongs in an exponential model?

$0.10$
$0.90$
$1.10$
$10$

Show answer and explanation

0.90

Losing 10% means 90% remains. As a decimal, 90% is 0.90.

Key terms

Exponential pattern: A pattern in which the quantity is multiplied by the same factor for equal steps of the input.
Repeated factor: The number multiplied by the current amount at each equal step.
Coordinate: A pair of numbers that identifies a point on a graph; the first is horizontal and the second is vertical.
Estimate: A value that is close to an answer but may not be exact.

Continue through MBF3C

View the complete MBF3C Ontario Grade 11 Mathematics curriculum and lessons

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation A3.3. 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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