MM - Linear, Exponential and Logistic Models Lesson

There are many types of graphical models that represent data. In this lesson we are going to talk about Linear, Exponential, Logistic and Piecewise.

Linear Model

A linear model has a constant rate of change or in other words, by the same amount in each time step. Here is an example of a linear model.

Starting at the age of 25, imagine if you could save $20 per week, every week, until you retire, how much money would you have stuffed under your mattress at age 65? To solve this problem, we could use a linear growth model. Linear growth has the characteristic of growing by the same amount in each unit of time. In this example, there is an increase of $20 per week; a constant amount is placed under the mattress in the same unit of time.

If we start with $0 under the mattress, then at the end of the first year we would have $20*52=$1040. So, this means you could add $1040 under your mattress every year. At the end of 40 years, you would have $$ 1040 \cdot 40 = $ 41, 600$ for retirement. This is not the best way to save money, but we can see that it is calculated in a systematic way.

A linear growth model can be represented by

P(t)=P₀+td

where P₀ is the initial amount, t = period of time, d=common difference or how much it changes per period of time.

In Derrick's case:

P₀=0, t=40 years, d=$1040 per month

P(40)=0+40(1040)=$41,600

or we would could months instead of years....

P₀=0, t=2080 months, d=$20 per month

P(2080)=0+2080(20)=$41,600

Graphically, we can see there is a constant rate of increase of $20 each month. Since most calculators use x and y, I will input y=0+x(20)=20x

Derrick Linear of input y=0+x(20)=20x

Exponential Model

An exponential model demonstrates constant multiplication. It has a higher rate of increase than the linear model. Here is an example of an exponential model.

Derrick is trying to save money for the down payment on a used car. His parents have said that, in an effort to help him put aside money, they will pay him 10% interest on the money Derrick accumulates each month. At the moment, he has saved $200. How long will it take Derrick to save at least $2,000 for the down payment if the only additions to his savings account are his parents' interest payments?

This is an exponential model because instead of adding a certain value as the months progress, we are multiplying by a number as the months progress.

Month 0: $220 (initial amount)
Month 1: $200*1.10*=$220
Month 2: $220*1.10=$242
Month 3: $242*1.10*266.20

* 10% = 0.10, so the percentage rate increase = 1.10

and so on...

This would take a long time to solve if we were to continue in this manner to get all the way up to $2000. It would be better to use a function model.

Let n=the number of months and d=the dollars saved

$220*(1.10)^n=d

If d = $2,000, the amount of money that Derrick wants we can solve the equation

Using the Functional Rule Model

*Note: "ln" is the natural log function. You can find the ln function on your calculator. The natural log function is often used when solving for an exponent in an equation. If you would like to make sure you are inputting this function into your calculator correctly, refer to your calculator manual. Make sure you get the answer as shown in the previous example.

With the addition of only the interest paid by his parents, it will take Derrick 24.16, or 25, months (just over two years) to earn enough for the $2,000 down payment.

Let's also take a look at the growth of Derrick's money on a graphical model. The function for the model is 200*(1.10)^n=d but most graphing calculators will require x's and y's so we can use 200*(1.10)^x=y

Graphing representation of 200*(1.10)^x=y

You will solve other exponential equations in this module and it will be helpful for you to have the formula for compound interest:

Compound interest, or 'interest on interest', is calculated using the compound interest formula A = P*(1+r/n)^(nt), where P is the principal balance, r is the interest rate (as a decimal), n represents the number of times interest is compounded per year and t is the number of years.

Here is another example:

In an effort to speed up the time needed to save $2,000, Derrick decides to take on some jobs in his community. Suppose he commits to adding $50 per month to his savings, starting with the initial deposit from his parents. Fill in the table, showing the amount of money Derrick will have over several months.

Recursive Rule Model image

The following is a scatterplot of the data you generated in the table and the graph to the function rule you found for Question 1. How does adding $50 per month to Derrick's savings change the way in which his money grows?

Scatterplot image

The scatterplot shows the savings with the added $50 per month, while the graph of the function rule shows the savings with only the interest paid by the parents. The graph shows that the scatterplot increases at a faster rate than the function rule. Adding $50 per month greatly affects the overall time needed to save $2,000.

How long will it take Derrick to save $2,000 for the down payment if he continues to add $50 every month?

Method 1:

Method 2:

According to this routine, it will take Derrick only 14 months to save up the money for the down payment.

You try:

Suppose Derrick changes the amount of money he adds to his savings each month to $100. How does this affect the time it takes to save $2,000? How much does he have to add to the savings each month to have enough money for the down payment on his car in six months?

By adding $100 to the savings each month, it will take Derrick 10 months to save $2,000. Using a table of values supports this conclusion. After testing several amounts greater than $100, it seems that Derrick needs to save $200 per month to have $2,000 in six months.

Logistic Model

There are some sets of data that have a pattern that does not fit a linear or exponential model, but it may fit a logistic model. The logistic model is frequently used in biology, ecology, chemistry, demography, economics probability, sociology, statistics and political science.

View the video below to learn more about logistic models.

¹Another interesting application of the logistic model can be used in identifying patterns of hospitalizations during the Covid pandemic. The blue data represents the number of cumulative hospitalizations. We can see that it will fit the logistic model well. The green data represents new daily hospitalizations. It's interesting how the data is transformed from daily to cumulative hospitalizations, so it fits a logistic model. There is value in seeing and comparing both representations.

File:20200413 Logistic growth COVID-19 Belgium.png - Wikimedia Commons. (2020, April 14). https://commons.wikimedia.org/wiki/File:20200413_Logistic_growth_COVID-19_Belgium.png

[CC BY-NC-SA 4.0] UNLESS OTHERWISE NOTED | IMAGES: LICENSED AND USED ACCORDING TO TERMS OF SUBSCRIPTION - INTENDED ONLY FOR USE WITHIN LESSON.