Is the Price Right?

Lesson Plan - Get It!

Audio:

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 = ab^x.

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!

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 = ab^x?

$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:

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:

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

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 = ab^x:

$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.