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Exponential Decay

In this video, we are going to look at exponential decay.

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}.

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

Here, it decreases by a factor.

It is very different with a linear decay.

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

Or let’s decrease at 25g.

So we have 200g then decrase 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.