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the function $$g(x)=\left(\frac{1}{2}\right)^x$$ is an example of exponential decay. in the exponential decay of $g(x)$, the function shrinks in half every time you add one to its input $x$. if we call this parameter $k$, we can write our exponential function $f$ as $$f(x)=b^{kx}.$$ you can explore the influence of both parameters $b$ and $k$ in the following applet. 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 $f$.

in fact, for any change you make to $k$, you can make a compensating change in $b$ to keep the function the same. then, if you change either $b$ or $k$, the applet will automatically make a compensatory change in the other parameter to keep the function the same. we’ll often use two parameters for the exponential function: $c$ and one of $b$ or $k$. for example, we might set $k=1$ and use $$f(x)=cb^x$$ or set $b=e$ and use $$f(x)=ce^{kx}.$$ you can add the parameter $c$ to the applet by checking the “scale function” checkbox.

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