In this video, we are going to look at exponential decay. After you finish this lesson, view all of our Algebra 1 lessons and practice problems.

$y=a \times (b)^x$
For a graph to demonstrate exponential decay, the base of the exponent, b, must be between 0 and 1. These graphs decrease by a factor, and will not fall into the negatives.
For example:
The graph of
$y=200 \times (\frac{1}{2})^x$
curves slowly downward after plotting points as shown in the video, and demonstrates exponential decay. This graph decreases by a factor of $\frac{1}{2}$.

## Examples of Exponential Decay

### Example 1

Does the function $f=\dfrac{1}{3}\cdot \left(\dfrac{2}{9}\right)^x$ model exponential decay?

Yes, it shows an exponential decay since the value of $b$ is less than $1$ and the curve is slowly downward.

### Example 2

What is the shape of the graph of the function $y=0.32(0.5^x)$?
Now, let’s solve for y and plot the coordinates

The graph shows a decreasing function. This is an exponential decay.

## Video-Lesson Transcript

Let’s go over exponential decay.

Exponential decay is when something decreases by a certain factor over and over again.

You may see this in a chemistry problem or something like that nature.

For example:

We have $200$g of something. It decreases half of the amount so now we have $100$g. Then decrease by half again, we have $50$g. Then decrease by half again, we have $25$g. Then decrease by half again, so we have $12.5$g. And so on.

Here, it decreases by a factor.

It is very different with a linear decay.

Where we have $200$g and when we decrease it by $100$g, we’ll have $100$g. And if we decrease it by $100$g again, we’ll have $0$.

Or let’s decrease at $25$g.

So we have $200$g then decrease to $175$. Then decrease again by $25$, we’ll have $150$.
And so on.

In exponential decay, we keep decreasing and the amount we decrease by also keeps on decreasing.

Whereas in a linear decay, it keeps on decreasing by a certain amount over and over again.

If we think about the linear exponential function we have

$y = ab^x$

In exponential decay, $0 \textless b \textless 1$. So its somewhere between $0$ and $1$.

For example:

$y = 200 (\dfrac{1}{2})^x$

This one models the one we have before. Whereas $200$ decreases by $\dfrac{1}{2}$ over and over again.

So initially, at time $0$, we have $200$.

When $x = 1$ then $y = 100$,
$x = 2$ then $y = 50$,
$x = 3$ then $y = 25$,
$x = 4$ then $y = 12.5$,
and so on.

Let’s graph this.

But if we have a linear decrease, we’ll have a straight line decrease.

The exponential decay line will not pass through the $x$ axis.

Whereas, the linear decay line will go down into the negatives.