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# Exponential Functions

### Exponential Functions

Exponential functions describe situations where the rate of growth is constantly increasing or decreasing. This is a powerful concept that appears in areas like investments and cell growth, as exponential growth and decay.

Sometimes, quantities increase or decrease at a rate which is not constant. For example, you started out as a single cell. The number of cells has grown **exponentially** over your life to include the trillions of cells that make your body up today!

Algebraically, an exponential function is an equation of the following form, where $a$ can be any number. When $x$ is positive, this is called exponential growth

$y = a^x$

Conversely, when the value of $x$ is **negative**, the function describes exponential decay. The general equation for exponential decay is:

$y = a^{-x}$

Let's calculate a few more points for $y=2^{-x}$ .

What is $y$ when $x=2$?

When $x=2$, $y=\dfrac{1}{4}$

Remember your negative indices! For $x=2$, $y=2^{-2}=\dfrac{1}{2^2}=\dfrac{1}{4}$.

What will $y$ equal to when $x=-1$ ?

And what will the y-coordinate be for when the x-coordinate is $1$ ?

Nice!

We can see these points represented on the graph.

Both exponential growth and decay can be used to model situations in real life.