Is the Price Right?

Contributor: Jamie Hagler. Lesson ID: 13838

If your favorite candy bar gets 3% more expensive every year, how can you figure out what it will cost in 5 years? Find out with this lesson!

categories

Algebra

subject
Math
learning style
Visual
personality style
Golden Retriever
Grade Level
High School (9-12)
Lesson Type
Skill Sharpener

Lesson Plan - Get It!

Audio:
  • Did you know that the price for some of the things you buy can go up and down?
  • Why do things that we buy lose or gain value so quickly?

This is called exponential growth or decay.

When numbers increase or decrease rapidly, it is called exponential growth or decay.

Instead of the equation being linear like y = mx + b; it will be in the form y = abx.

Here's the setup of the equation:

  • Any number that is in the place of a is the initial amount, the beginning price, the original number of items, etc.
  • We place the amount of growth or decay in the place of b.

For growth, we enter the number b + 1.

For decay, we enter the number 1 - b.

  • Then, the time period, or number of times the event is repeated, goes in the place of x.

Notice that x is the exponent in this equation. That is why it is called an exponential function!

exponential function

Let's try one!

The initial price of a boat is $25,000. It depreciates (decays, loses value) by approximately 10% every year.

  • Where would you put the numbers into the equation y = abx?

$25,000 is the initial amount, so it will go in the place of a. It decays 10%, so we will enter 1 - 0.10 or 0.90.

The equation will look like this:

y = 25000(0.90)x

  • Why do we need to set it up as an equation?

There are several reasons; however, the main one is so we can predict the value of the boat each year.

Let's create a table to show the value each year. We will substitute the number of years into the place of the exponent x:

  x y = 25000(0.9)x y  
  0 25000*(0.9)0    
  1 25000*(0.9)1    
  2 25000*(0.9)2    
  3 25000*(0.9)3    
  4 25000*(0.9)4    
  5 25000*(0.9)5    
  6 25000*(0.9)6    

 

Then, we solve. These results will be our y values.

We can also write the coordinates of (x, y) so we can see these points on a graph:

  x y = 25000(0.9)x y (x, y)
  0 25000*(0.9)0 25000 (0, 25000)
  1 25000*(0.9)1 22500 (1, 22500)
  2 25000*(0.9)2 20250 (2, 20250)
  3 25000*(0.9)3 18225 (3, 18225)
  4 25000*(0.9)4 16402.5 (4, 16402.03)
  5 25000*(0.9)5 14762.25 (5, 14762.03)
  6 25000*(0.9)6 13286.03 (6, 13286.03)

 

If we plot these on a graph, we will see the exponential data is not linear, but makes a curve:

graph

Let's try writing another exponential equation:

Stuart made an initial deposit of $5,000 in a savings account earning a yearly compounded interest rate of 3%. Assuming:

  • Stuart makes no additional deposits or withdrawals, what function can model this growth?

Let's put the numbers into the equation y = abx:

$5,000 is the initial amount, so it will go in the place of a.

The money earns interest of 3%, so we will enter 0.03 + 1 or 1.03.

The equation will look like this:

y = 5000(1.03)x

Now that you have a handle on exponential growth and decay, let's move on to the Got It? section.

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