Topics
5. Exponential Functions and Equations
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United States of America
Grades 9-12

5.04 Modeling exponential relationships (M)

Lesson
Practice

Introduction

Exponential growth and decay can be used to model a variety of real-world situations, including compound interest, carbon dating with radioactive decay, population growth, depreciation, and more. When we use exponential models, however, we need to give attention to the limits that exist in the real world and how they impact the reliability of the model.

A modeling cycle. Starting with the phrase Identify the problem inside a circle. An arrow pointing to the right where the phrase Create a model is inside a rectangle. Next is an arrow pointing downward where the phrase Apply and analyze is inside a rectangle. Then, an arrow pointing to the right where the phrase Interpret results is inside a rectangle. Next is an arrow pointing upward where the phrase Verify the model is inside a triangle. Then, an arrow pointing to the right where the phrase Report findings is inside a circle. There is an arrow pointing to the left from the phrase Verify the model to Create a model.

Each time we model a real-world situation we should:

  1. Identify the essential features of the problem

  2. Create a model using a diagram, graph, table, equation or expression, or statistical representation

  3. Analyze and use the model to find solutions

  4. Interpret the results in the context of the problem

  5. Verify that the model works as intended and improve the model as needed

  6. Report on our findings and the reasoning behind them

Interpreting and verifying a model

Once a model has been created, we need to apply it to the context and verify that it works the way we had intended. If the results from our model do not appear to be accurate, we can revisit our assumptions and repeat the modeling cycle.

Examples

Example 1

Consider the data from the U.S. Census for each decade in the 20th century:

YearPopulation (in millions)
190076.2
191092.2
1920106
1930122.8
1940132.2
1950150.7
1960179.3
1970203.3
1980226.5
1990248.7
2000308.7

The function P(t)=76.2(1.136)^\frac{t}{10} has been created from the table to find the population t years after 1900.

a

Find the 10-year growth factor for each decade and compare it to the growth factor of the model.

Worked Solution
Create a strategy

The growth factor can be found by dividing each decade's population by the previous population.

For example, from 1900-1910: \dfrac{92.2}{76.2}=1.210

Apply the idea
YearPopulation (in millions)Growth factor
190076.2
191092.21.210
19201061.150
1930122.81.158
1940132.21.077
1950150.71.190
1960179.31.140
1970203.31.134
1980226.51.114
1990248.71.098
2000308.71.241

The growth factor of the model is 1.136 which is larger than the growth rates from 1930-1940 and 1960-1990, but less than the growth rates from 1900-1930, 1940-1950, and 1990-2000.

Reflect and check

The growth factor in the model appears to be an average of the growth rates across the century. The decade from 1930-1940 had a very small growth rate which is likely due to the Great Depression. What other historical events may have influenced population growth?

b

Determine the decades in which the model is the least accurate.

Worked Solution
Create a strategy

We will use the function f(x)=76.2(1.136)^\frac{t}{10} to complete a table to estimate the population. Then, we can find the difference between the estimated population and the given population.

Apply the idea
YearPopulation (in millions)Estimated population (in milions)Difference
190076.276.20
191092.286.6-5.6
192010698.3-7.7
1930122.8111.1-11.7
1940132.2126.9-5.3
1950150.7144.2-6.5
1960179.3163.8-15.5
1970203.3186.0-17.3
1980226.5211.3-15.5
1990248.7240.1-8.6
2000308.7272.7-36

The model underestimates the population in every decade except 1900. The least accurate decade is 2000, when the population estimate is off by 36 million.

Reflect and check

A graph may be a better model to visualize both the data and the function.

U.S. population data
10
20
30
40
50
60
70
80
90
100
\text{Year since 1900}
50
100
150
200
250
300
350
400
\text{Population (in millions)}
c

Explain how the model could be modified to be more accurate in predicting the population after the year 2000.

Worked Solution
Create a strategy

We can see that the model loses accuracy for later decades as it underestimates those populations the most. The graph needs to grow faster, so we should try a higher growth rate.

Apply the idea

Let's try a slightly higher growth factor of 1.2. The new model would be: f(x)=76.2(1.2)^\frac{t}{10}

We can graph the original data and the modeling function using technology:

U.S. population data
10
20
30
40
50
60
70
80
90
100
110
\text{Years since 1900}
40
80
120
160
200
240
280
320
360
\text{Population (in millions)}

We can see this new model overestimates the data, so our growth factor is too high. Using technology, we can change the growth factor until we get a close fit for the data. A better function model would be f(x)=76.2(1.15)^\frac{t}{10}

We can see this in the graph:

U.S. population data
10
20
30
40
50
60
70
80
90
100
110
\text{Years since 1900}
40
80
120
160
200
240
280
320
360
\text{Population (in millions)}
Reflect and check

There are other possible models that may model the data. We could have chosen to use only the data after the baby boom in 1950, changing both the initial value and the growth factor. Or, we could have used statistical software that uses matheatical calculations to find the best fit curve.

Idea summary

Exponential models can be used to predict growth and decay over a specified domain. Models can be verified by testing points whose values we know and then updated using different assumptions as needed.

Reporting the results

After we've used a model to solve a problem, it's time to write a report that summarizes the results and process. The stakeholders are the people who would be most interested in knowing the solution to the problem.

When writing a report, we generally want to include:

  1. The problem clearly defined and summarized.
  2. A brief explanation of the background of the problem (and why it needs to be solved).
  3. An explanation of the approach to solving the problem, the model used, and visuals to help the reader understand.
  4. The results, your conclusion, and the limitations of the assumptions and model.

Exploration

Consider the following problems:

  • How much does it cost to feed a family?
  • What is the most cost-effective way to recycle glass in your neighborhood?
  • How quickly will lice spread in an elementary school?

  1. For each problem, who would be the stakeholders? Give an explanation for why each problem might need to be solved.

Examples

Example 2

Aziz read a headline recently that said, "Local leaders fear affordable housing crisis could lead to unprecedented homelessness crisis of 8.1 \%." He decided to investigate the problem and model the shelter needs for the next ten years.

a

Summarize the problem Aziz is investigating.

Worked Solution
Create a strategy

A summary can include some background information, a clear problem statement, and any variable definitions that may be used to model the problem.

Apply the idea

According to endhomelessness.org, in 2020, 580\,466 people out of the 329.5 million living in the United States experienced homelessness. However, not all of these individuals and families were unsheltered. We want to investigate how many people may be experiencing homelessness in the next ten years and how many beds will be available to help them.

In order to determine if the number of beds available will match the number of beds needed and to track people experiencing homelessness in general, we will use the following variable definitions:

  • Let x represent the number of years since 2020.
  • Let H(x) represent the number of people experiencing homelessness as a function of x.
  • Let B(x) represent the number of available beds as a function of x.
Reflect and check

There are many ways to plan this summary; this is just one example.

b

Aziz developed the following models as a part of his solution:

  • H(x)=580466(1.027)^x
  • B(x)=941871(1.033)^x

Write an analysis and conclusion that Aziz could use in his report.

Worked Solution
Create a strategy

In our analysis, we should predict how the number of people experiencing homelessness and the number of beds will change over time. In our conclusion, we need to decide what this means for the future and what our next steps should be.

Apply the idea

Through research, we've determined that while homelessness over the last decade has been declining, in recent years, the rate of people experiencing homelessness has actually increased. We are focusing on data from 2018-2020 only for this analysis.

Predicting the future using only 3 years of data is risky, as the rates are likely to change. We will assume that the growth rates stay constant but also consider what would happen if they didn't in a worst-case scenario.

Number of homeless people and number of available beds
1
2
3
4
5
6
7
8
9
10
\text{Years since 2020}
600
700
800
900
1000
1100
1200
1300
1400
\text{Number, in thousands}

The number of beds in 2020 exceeds the homeless population and the data shows that the number of beds is increasing at a faster rate (3.3\%) than the number of people experiencing homelessness (2.7\%). If nothing changes, we will continue to have enough beds.

However, if the number of people experiencing homelessness started to grow at a faster rate as the headline implied, it would have to triple in rate to 8.1\% for the number of beds to run out, as seen in the following graph:

Number of homeless people and number of available beds
1
2
3
4
5
6
7
8
9
10
\text{Years since 2020}
600
700
800
900
1000
1100
1200
1300
1400
\text{Number, in thousands}

In conclusion, the number of beds across the U.S. should be enough to support the number of people experiencing homelessness.

Reflect and check

There are a few factors we ignored in the simplicity of our model and how we summarized the problem for Aziz. Although across the U.S. there are enough beds, that doesn't mean that those beds are available where the people need them. Looking at a map of the population of people experiencing homelessness, we see that most people are on the west coast, in Hawaii, or in New York:

A figure titled Number of Individuals Experiencing Homelessness in 2020 pero 10000 Individuals. It shows the U.S. map and each state corresponds to a number. A legend shows the colors and the corresponding number of individuals experiencing homelessness it represents. Speak to your teacher for more information.

If the beds are spread out across the United States, there is a good chance that some states don't have enough beds and others have too many. Aziz should reanalyze his model and consider rates by state instead of the country as a whole.

Idea summary

Reports should be written as professionally as possible. Present the results, supported by models both visual and algebraic. Summarize how those models were created and what those models tell us about the problem. Then, explain the limits of those models and how they might be improved for further investigation.

Outcomes

F.IF.B.4

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F.IF.B.5

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

F.IF.C.8.B

Use the properties of exponents to interpret expressions for exponential functions.

F.IF.C.9

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

F.LE.A.1.C

Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

F.LE.B.5

Interpret the parameters in a linear or exponential function in terms of a context.

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