Exponential graph

Exponential functions are in the form of , where is a positive constant and (because if then the function would be a constant).

Asymptotes

An asymptote is a line that a graph approaches but never touches.

• They have a horizontal asymptote at (the x-axis) because the result of will never be or negative.

Asymptote at

Growth

As you increase the value of of an exponential graph, the value increases very quickly. That’s because an increase of just in means that the value is multiplied by .

For example, if and increases from to , the value increases from to - it multiplies by .

Domain

The domain of an exponential graph is (anything) because you can go as far left or right as you want on the x-axis (all values of will give a valid output to the function).

Domain:

Range

The range is (positive numbers) because the output of will always be positive.

Range:

Solving where

At no point on the graph is because the output of is always positive (never 0).

That means there are no solutions to the equation .

There can be solutions to the equation (where is a negative constant) because the graph of is just the graph of shifted down by units, so it can cross the x-axis.

But the normal graph of has no solutions to .

Solving for a given value

• We have our equation:
• If we want to find the value of for a given value, we can rearrange the equation into a logarithm:

the solution to is .

Intercept

The intercept is the value when .

(because any number to the power of is )

So the intercept is at .

Growth or decay?

• The value of will increase with the value of if (in the function ). That’s called growth.
• The value of will decrease with the value of if (in the function ). That’s called decay.

: growth
: decay