5.2 Modeling Exponential Relationships with Algebra

5.2: Modeling Exponential Relationships with Algebra
Learning Objectives
Upon completion of this section, you should be able to
- Identify elements in an exponential model (equation) of the form
- Evaluate an exponential model
- Create an exponential model with algebra between two quantitative variables
- Graph an exponential model
- Solve application problems using an exponential model created with algebra
Exponential Models
When populations grow rapidly, we often say that the growth is “exponential”. To a mathematician, however, the term exponential growth has a very specific meaning. In this section, we will take a look at exponential models, which model this kind of rapid growth and also decay.
When exploring linear growth, we observed a constant rate of change—a constant number by which the output increased by addition for each unit increase in input. For example, in the equation
Exponential Model
An exponential model (equation) has a form
- If
, we say it models exponential growth. As increases, the outputs of the model will increase as well and continue to increase at a much faster rate without bound. - If
, the model shows exponential decay. As increases, the outputs for the model decrease and continue to decrease, but levels off to the -axis (gets close to zero, but will not equal to zero). - When
the output is , so we often call a the initial amount in applications (or the starting amount). We also say is the -intercept on the graph for the model. - The value
is what we call the growth multiplier. If we increase by 1, then is what we multiply the previous output value by to get the new output.
|
|
For us to gain a better understanding of exponential growth, let us contrast exponential growth with linear growth. We will construct two models. The first model is exponential. We will start with an input of 0, and increase each input by 1. The second model is linear. For both models they will start with the same outputs on the first two steps. See the table below
x | ||
---|---|---|
0 | 1 | 0 |
1 | 2 | 2 |
2 | 4 | 4 |
3 | 8 | 6 |
4 | 16 | 8 |
5 | 32 | 10 |
6 | 64 | 12 |
Looking just down the middle column we see that for exponential growth we are multiplying by the same value of 2 to go from one output to the next (this is what is meant by saying
Example 1
Which of the following equations are examples of exponential models? If it is an exponential model identify the starting amount (
Solution
- Not Exponential. This is a linear model.
- Not Exponential. The base for
is a variable and not a constant. - Exponential model. Starting amount is 5 and growth multiplier is 4.
- Exponential model. Starting amount is 3 and growth multiplier is
Try it Now 1
Which of the following equations are examples of exponential models? If it is an exponential model identify the starting amount (
Answer (click to Show/Hide)
- This is not an exponential model as the base is a variable and not a constant.
- This is an exponential model. The starting amount is 200 and the growth multiplier is 5.
- This is an exponential model. The starting amount is 1 and the growth multiplier is
. - Recall the base for an exponential model is always positive constant, and
. Thus, does not represent an exponential model because the base, , is less than 0. - This is a linear model and not an exponential model.
Evaluate an exponential model
To evaluate an exponential model with the form
Let
To evaluate an exponential model with a form
Let
It is important to follow the order of operations as you will get an incorrect result by not doing so:
Example 2
Evaluate the exponential model
Solution
Evaluate at
Evaluate at
Try it Now 2
The population of India was about 1.39 billion in the year 2021, with an annual growth rate of about
Hint 1 (click to Show/Hide)
First find the value of
Answer (click to Show/Hide)
To estimate the population in 2030, we evaluate the models for
There will be about 1.5202 billion people in India in the year 2030.
One thing to note in the above work is how we carried out many more decimal places than required while still doing further calculations. The reason for doing so was to avoid rounding error in an answer that would get compounded by further calculations.
Create an exponential model with algebra between two quantitative variables
Before we explore applications and the graph of an exponential model let us first look at strategies for finding the exponential equation from two points. We will only consider applications where both points have positive
Properties of Exponents
These properties of exponents are leveraged in this section, especially the quotient rule, which you will use immediately below to solve for the base of an exponential model when given two points.
Property | Definition | Example |
---|---|---|
Property of 1 | ||
Property of 0 | ||
Product Property | ||
Property of Negatives | ||
Quotient Property | ||
Power of a Power Property |
There are many more exponent properties in algebra, but only the above is needed for our work with exponential growth.
The other facts from algebra we will need are given below about solving equations where the unknown variable is in the base and it is raised to an integer power along with a general strategy for solving for an exponential model.
Solving equation of the form
Let
General Strategy to find an Exponential Model given two points
To find an exponential model of the form
given two points
- Case 1: If one of the points is a
-intercept, i.e. , then the initial amount is already given to you. Using , substitute the other point into the exponential model and solve for . - Case 2: If neither points given has the form
you can solve for and by doing the following:- Step 1: Set up two equations by evaluating the model at both points.
- Step 2: Divide the second equation by the first (this will cancel out the unknown
term). - Step 3: Solve for
. - Step 4: Substitute
back into either original equation to find .
The above assumes both
Let us look now at a few examples of creating an exponential model. In the next example below we will be given two values where one is the
Example 3 (given a -intercept)
Given two points
Solution
Start with the exponential model
We now solve for
The exponential model is
Evaluating roots on the calculator
In the previous example, we had to calculate the 3rd root of a number. This is different than taking the basic square root, v. Many scientific calculators have a button for general roots. It is typically labeled like:
To evaluate the 3rd root of 8, for example, we’d either type 3
If your calculator does not have a general root button, all is not lost. You can instead use the property of exponents which states that
The parentheses tell the calculator to divide 1/3 before doing the exponent.
In the next example we will find the exponential model for two general points where neither is the
Example 4
Given two points
Solution
Because we don’t have the initial value, we substitute both points into an equation of the form
- Substituting
gives: - Substituting
gives
Next divide the second equation by the first (this step can also be swapped so you divide the first by the second – choose whichever appears to be easier to deal with):
Now cancel out
Rewrite the right hand side using the quotient property of exponents:
Next raise both sides of the equation to the power
At this point we can approximate
Next use the value of
The equation for the exponential model is:
Try it Now 3
Find an exponential equation that goes through the points
Hint 1 (click to Show/Hide)
Were either of the two points given a
Hint 2 (click to Show/Hide)
Neither point was a
Answer (click to Show/Hide)
Because we don’t have the initial value, we substitute both points into an equation of the form
- Substituting
gives: - Substituting
gives
Next divide the second equation by the first:
Now cancel out
Rewrite the right hand side using the quotient property of exponents:
Next raise both sides of the equation to the power
At this point we can approximate
Next use the value of
The equation for the exponential model is:
Graph an exponential model
When graphing linear models we said only two points are needed. The same can be said about an exponential model of the form
Let us examine the graph of
-2 | 0.25 |
-1 | 0.5 |
0 | 1 |
1 | 2 |
1 | 4 |
Some observations we can make is that if you look at the point
If we want to move to the left from
We also see that as
Graphing Exponential Models of the form
- Find the
-intercept and the point where the input is 1, - Use the base
to find additional points by moving left and right from one of the known points - Connect the points with a smooth curve
Example 5
Graph the exponential model defined by
Solution
Begin by choosing input values. As mentioned above it is enough to use the input values of 0 and 1. From there we can extend and add additional points by multiplying by the base 3 or dividing by the base 3.
To add an additional point at
To add an additional point at
Next plot the coordinate pairs and draw the exponential curve through the points. The graph below is for the model
The above graph has a base greater than 1, so we are seeing exponential growth.
Try it Now 4
Graph the exponential model given by
Hint 1 (click to Show/Hide)
Start with evaluating the model at
Answer (click to Show/Hide)
Evaluate at
Continue to add points by using the base to move forward and backward on the two points already found:
Find the coordinate at
To find the coordinate at
Plot the coordinates found and connect to form the exponential curve.
The above graph is showing exponential decay. For each unit increase in
Solve application problems using an exponential model
Many real-world phenomena exhibit growth that occurs not by a fixed amount in each time period, but rather by a fixed percentage increase. This type of growth, where a quantity increases by a consistent proportion over equal intervals, can be modeled using exponential equations.
Imagine a scenario where the value of an investment grows by 5% each year, or a bacterial colony doubles in size every hour. These situations, where the rate of increase depends on the current size or amount, are prime candidates for exponential modeling.
Find an exponential model in applications
- Identify the variables in the given information
- Assign the variables letter names
- If the initial amount is given in the information assign that value to
in the model. Using the value of substitute the other information into the exponential model and solve . - If the initial amount is not given you can use the approach for solving an exponential model when given two points where neither are the
-intercept.
Example 6
The population in Tucson in 2019 was approximately 548.1 thousand people and 551.3 thousand people in 2020. If the population continues to grow by the same percent each year (grows exponentially), then what would be the population in the year 2030?
Solution
The information provided can be used to build an exponential model as we have two data points to work with. The first is an initial amount of the population in the year 2019 and the second is the population one year later. To build the model we will put it in the form
In the model
To find
We can write the model as
Note: we rounded the value of
To find the population in 2025 we evaluate the model at
The population is estimated to be 567.6 thousand people in the year 2025 based on our exponential model.
Why did we choose in the example above for
If we had not done that we would have same value for
Example 7
The number of employees at Amazon has been growing at an exponential rate. Find a model for the number of employees since 2006 given that there were 88,400 employees in 2012 and 154,100 in 2014. Using this model how many employees would be at amazon in 2020?
Solution
Let
Now we are asked to use 2006 to represent the starting year for the model, so we are not given any initial amount as the first amount known is for the year 2012 (
Create two equations based on the given points
- Substituting
gives - Substituting
gives
Start by dividing the second equation by the first equation:
Next simplify the right hand side using property of exponents:
Now solve for b:
Use the value of
The equation for the exponential model is
Use the model to predict the number of employees in 2020. Use
The model predicts that in 2020 Amazon would have had approximately 816,332 employees.
Try it Now 5
Sales of Apple Ipod peaked in fiscal year 2008 with 54.83 million units sold. By 2012 the number of sales had decreased to 35.17 million units. Assuming the decrease from 2008 was exponential find a model for the number of units sold (in millions)
Hint 1 (click to Show/Hide)
Let N be the number of Ipods sold (in millions) and t be the number of years since 2008. The model would be
Determine if you can enter the value for
Answer (click to Show/Hide)
Let N be the number of Ipods sold (in millions) and t be the number of years since 2008. The model would be
We are given that in 2008 a total of 54.83 million units were sold. Since
To find the value of
The exponential model for Applie Ipod saled since 2008 is
To find the predicted number of ipod sales in 2014 we evaluate the model at
The model predicted the sales would be 28.17 million units.
Exercises
- Identify which are exponential models. If it is an exponential model state if it is exponential growth or decay.
Answer (click to Show/Hide)
- Not an exponential model.
- Exponential model with base 9, so exponential growth since base is greater than 1.
- Not an exponential model. This is a linear model.
- Exponential model with base 0.0763, so exponential decay since base is less than 1.
- Exponential model with base
, so exponential growth since base is greater than 1.
- Given an exponential model of the form
how do we determine if shows exponential growth or decay.
Answer (click to Show/Hide)
In the model we look at the base
and see if it is between 0 and 1 or greater than 1. If the base is between 0 and 1 we say it is exponential decay. If the base is greater than 1 we say it is exponential growth. - Determine if the scenario can be modeled with an exponential equation.
- The tuition increases by $2 each year per credit hour.
- The tuition increases by 1% each year per credit hour.
- The value of a car depreciates by 15% annually over the last 10 years.
- For each delivery stop the driver increases the cost by $5.00 for the delivery.
- Population in a culture of bacterial double every four hours.
Answer (click to Show/Hide)
- Not exponential. Example of linear growth by adding a fixed constant.
- Exponential. Would be multiplying by a percent increase each year.
- Exponential. Would be multiplying by a percent decrease each year.
- Not exponential. Example of linear growth by adding a fixed constant.
- Exponential. Would be multiplying by the population by 2 for every four hours.
- Find the formula for an exponential equation that passes through the two points
.
Answer (click to Show/Hide)
- Find the formula for an exponential equation that passes through the two points
.
Answer (click to Show/Hide)
- Find the formula for an exponential equation that passes through the two points
.
Answer (click to Show/Hide)
- Find the formula for an exponential equation that passes through the two points
. Round values to six places.
Answer (click to Show/Hide)
- Find the formula for an exponential equation that passes through the two points
.
Answer (click to Show/Hide)
- Find an equation for an exponential model that goes through the points
and graph this model showing the -intercept. Round all values to two decimal places.
Answer (click to Show/Hide)
- Find an equation for an exponential model that goes through the points
and graph this model showing the -intercept. Round all values to six decimal places.
Answer (click to Show/Hide)
- The Mexican gray wolf was reintroduced to Arizona and New Mexico regions in 1998. Authorities initially released 11 wolves at that time. In 2012 there were 58 wolves. Assuming the growth was exponential find a model for the number of wolves after the year 1998. Use this model to predict the number of wolves in the population in 2020. Round the values in the model to six places and population in 2020 to nearest integer.
Answer (click to Show/Hide)
The population in 2020 is projected to be 150.
- In 1990, the residential energy use in the US was responsible for 962 million metric tons of carbon dioxide emissions. By the year 2000, that number had risen to 1182 million metric tons. If the emissions grow exponentially and continue at the same rate, what will the emissions grow to by 2050?
Answer (click to Show/Hide)
Let
represent the number of years since 1990 and be the metric tons (in emissions) of carbon dioxide emissions.The exponential model is
.Evaluate the model at
to find that there is approximately 3308.4 million metric tons of carbon dioxide predicted by the model. - The population in a city in 2020 is approximately 200 thousand and has been growing by 2.7% each year. If we assume the growth continues at the same rate estimate the population in 2030.
Answer (click to Show/Hide)
Exponential model is
Population in 2030 is estimate to be 261 thousand people.
- According to the New York Times the 7-day average for number of new cases of covid 19 on March 2nd was 9. One week later the 7-day average was 92. If we assume the growth rate of new cases follow an exponential model and continues to follow the same rate, then approximate the number of new cases five weeks from the start of the data collection on March 2nd.
Answer (click to Show/Hide)
Let
be the number of days after March 2nd and be the 7-day average of new cases.The exponential model would be
The 7 day average for new cases in five weeks (35 days) after March 2nd would be approximately 1,004,540.
On April 7th the reported 7 day average was 30,133. Source: New York Times
- A bacterial culture starts with 300 bacteria. After 4 hours the population has grown to 500 bacterial. If the population is growing exponentially find how many bacteria there are after 24 hours from the start time.
Answer (click to Show/Hide)
Let
be the number of hours from start. The amount of bacterial present would be modeled byThe number of bacteria after 24 hours is projected to be 6,430.
- The population of a small town is modeled by
, where is the number of years after 2010. What was the population in 2010? Is the population increasing or decreasing?
Answer (click to Show/Hide)
The population in 2010 is 45,000. The population is growing as the base is greater than 1.
- Suppose that you have a bowl of 500 Skittles, and each day you eat
of the Skittles you have. Is the number of candies left changing linearly or exponentially? Write an equation to model the number of candies left after days.
Answer (click to Show/Hide)
The number of candies left would be changing exponentially. The model would be
Attributions
This page contains modified content from David Lippman, “Math In Society, 2nd Edition.” Licensed under CC BY-SA 4.0.
This page contains modified content from "College Algebra," Abramson, Jay et al., OpenStax. Licensed under CC BY 4.0.
This page contains content by Robert Foth, Math Faculty, Pima Community College, 2021. Licensed under CC BY 4.0.