 Exponential functions > Exponential functions
12345Exponential functions

## Theory

Take a look at the applet: Exponential Function

The graph of the exponential function $f\left(x\right)=b\cdot {g}^{x}$ has the following characteristics:

• The graph intersects the $y$ -axis at the point $\left(0,b\right)$.

• If $b>0$ and $g>1$ the graph is increasing. To the left (for decreasing $x$) the graph approaches the $x$ -axis. You can take the function value as close to $0$ as you like by taking $x$ sufficiently small. The $x$ -axis is the horizontal asymptote.

• If $b>0$ and $0 the graph is decreasing. To the right (for increasing $x$) the graph approaches the $x$ -axis, the horizontal asymptote.

• If $b<0$ and $0 the graph is increasing. To the right (for increasing $x$) the graph approaches the $x$ -axis, the horizontal asymptote.

• If $b<0$ and $g>1$ the graph is decreasing. To the left (for decreasing $x$) the graph approaches the $x$ -axis, the horizontal asymptote.

• If $g=1$ the graph is the horizontal line $y=b$.

Exponential equations like $b\cdot {g}^{x}=a$ can be solved using the calculator.

For exponential inequalities one uses the properties mentioned above .