# Math Insight

### Applet: The exponential function

The exponential function $f(x)=b^{kx}$ for base $b >0$ and constant $k$ is plotted in green. You can change the parameters $b$ and $k$ by typing new values in the corresponding boxes. It turns out the parameters $b$ and $k$ can change the function $f$ in the same way, so you really only need to change one of them to see all the different functions $f$. To see how they do the same thing, you can click the “fix function” checkbox, which will fix the function $f(x)$. When that box is checked, if you change the parameters $b$ or $k$, the other parameter will change in a way to leave the function $f(x)$ unchanged.

For the function $f(x)=b^{kx}$, the value $f(0)=1$ for all parameters. To change the value of $f(0)$, you can allow scaling of the function by clicking the corresponding checkbox. Then, the function changes to $f(x)=c b^{kx}$ with an additional parameter $c$ that scales (multiplies) the whole function so that $f(0)=c$. You can change the value of $c$ by dragging the red point.

You can change range of the $x$ and $y$-axes buttons labeled $x+$, $x-$, $y+$, and $y-$. Since $f(x)$ is always non-negative, only the positive $y$-axis is shown.

Applet file: exponential_function.ggb