Mathematics » Functions II » Exponential Functions

# Revision of Exponential Functions

## Revision of Exponential Functions

### Functions of the form $$y=a{b}^{x}+q$$

Functions of the general form $$y=a{b}^{x}+q$$, for $$b>0$$, are called exponential functions, where $$a$$, $$b$$ and $$q$$ are constants.

The effects of $$a$$, $$b$$ and $$q$$ on $$f(x) = ab^x + q$$:

• The effect of $$q$$ on vertical shift

• For $$q>0$$, $$f(x)$$ is shifted vertically upwards by $$q$$ units.

• For $$q<0$$, $$f(x)$$ is shifted vertically downwards by $$q$$ units.

• The horizontal asymptote is the line $$y = q$$.

• The effect of $$a$$ on shape

• For $$a>0$$, $$f(x)$$ is increasing.

• For $$a<0$$, $$f(x)$$ is decreasing. The graph is reflected about the horizontal asymptote.

• The effect of $$b$$ on direction

Assuming $$a > 0$$:

• If $$b > 1$$, $$f(x)$$ is an increasing function.
• If $$0 < b < 1$$, $$f(x)$$ is a decreasing function.
• If $$b \leq 0$$, $$f(x)$$ is not defined.
 $$b>1$$ $$a<0$$ $$a>0$$ $$q>0$$ $$q<0$$
 $$00$$ $$q>0$$ $$q<0$$