Linear or Exponential?

Contributor: Michelle Haver. Lesson ID: 13646

Certain situations can be modeled by linear functions while others can be modeled by exponential functions. Discover how to determine if it's linear or exponential.

categories

Algebra I

subject
Math
learning style
Visual
personality style
Lion, Beaver
Grade Level
Middle School (6-8), High School (9-12)
Lesson Type
Quick Query

Lesson Plan - Get It!

Audio:

You decide you'd like to start saving up some money but aren't entirely sure the best way to go about doing so. Your parents, acting as the bank, provide you two options.

Option 1: Your parents will put aside $700 right away and will add an additional $100 once a year.

Option 2: Your parents will put aside $100 right away and will double the amount set aside each year.

  • Which option will give you the most money after two years?
  • Which option will give you the most money after five years?
  • Which option is the best for your savings in the long run?

Let's take a closer look at both options to decide which is the best choice!

Option 1 may seem like a great option since you get $700 right off the bat! Let's see how much money you would save over the course of five years.

In this option, you start with $700 and add $100 every year. We can lay this out with a table:

  Contribution Total Savings
  initial contribution $700
  after 1 year $700 + $100 = $800
  after 2 years $800 + $100 = $900
  after 3 years $900 + $100 = $1000
  after 4 years $1000 + $100 = $1100
  after 5 years $1100 + $100 = $1200

 

We see that, after five years, you will have saved $1,200 with your parents' help!

Suppose you want to know how much money you would have saved after 10 years. You could continue to use the table, but that would take a chunk of time to do!

  • Can you think of a way to represent the amount of money saved after x years using a function?

In this case, we have a starting value ($700) and will increase the amount saved by an additional $100 every year. This constant addition ($100) to the independent variable ($700) leads to a linear function.

The slope is the increase each year, and the y-intercept is the starting value. So we can write the function:

s(x) = 100x + 700.

This function gives the amount of money saved x years after the initial investment with option 1.

Now, let's consider option 2 and see how this compares.

This one may not seem as good right off the bat because you only start with $100, but this time we will double the amount saved each year.

Let's look at this using a table:

  Contribution Total Savings
  initial contribution $100
  after 1 year $100 x 2 = $200
  after 2 years $200 x 2 = $400
  after 3 years $400 x 2 = $800
  after 4 years $800 x 2 = $1600
  after 5 years $1600 x 2 = $3200

 

We see here that, after five years, you will have saved $3,200 with your parents' help! This is definitely more than the previous option.

However, in the beginning, we see that there are several times where option 1 will yield more money. Again, suppose you want to know how much money you would have saved after 10 years.

  • Can you think of a way to represent the amount of money saved after x years using a function?

In this case, we have a starting value ($100) and will multiply by 2 each year. However, the amount being multiplied by 2 will not be the same each year but will instead be dependent on how many years have passed.

Since we are doubling the amount each year, that is represented by 2 to the power of how many years the money has been saved. This makes the scenario an exponential function because we are multiplying the independent variable by an exponent.

We can write a function that gives the amount of money saved x years after the initial investment with option 2:

m(x) = 100(2x)

This is loads more money than you would have saved with option 1! In fact, it's over 60 times more!

Let's explore the main differences a bit more.

In option 1, we are adding the same number for each increase in x. This leads us to use a linear function!

These functions do not grow as quickly as exponential functions, which are used in option 2. In option 2, we are multiplying the same number for each increase in x.

To see some more examples showcasing the differences between situations modeled by linear functions and situations modeled by exponential functions, check out Understanding linear and exponential models | Functions and their graphs | Algebra II | Khan Academy:

You may have noticed some differences in phrasing, which can help you determine if a model is linear or exponential.

For exponential models, you want to look for terms such as "increases by a factor of" or "decreases by 5%".

For linear models, you want to look for situations where there is a constant increase or decrease.

Now that we've seen the differences between linear and exponential models, go ahead to the Got It? section to check your understanding!

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