Exponential Function

L2MAEL

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An exponential equation is basically the inverted function of a Logarithmic Function; exponentials are related to logarithmics.

All exponential equation has an unknown exponent — for example: $y=2^x$ (graph below)

the generic equation is:

$y=\alpha {\beta }^{x-p}+q$

..where you can translate the graph by modifying the $p$ and $q$ coordinates.

By modifying the value of $p$, you can translate the graph along the $x$ axis,

..and by modifying the value of $q$, you can translate the graph along the $y$ axis.

Tip: If you don’t have an idea of what a graph might look like, draw a table of coordinates ( $x$ and $y$ pairs) because it will then tell you the points of the graph which then you can connect the points.

$\beta$ is the scale factor of the function, where if you increase it, the function turns into a more steep slope.

However, $\beta$ cannot be lower, or equal to 1, as it would make the graph linear, or undefined.

The asymptote is a boundary of the exponential function, described as “a line that goes near, but never touches”. All exponential functions have (a) asymtote(s). In the graph above ( $y=2^x$ ), the asymptote is $y=0$.

You might be able to see a connection between $q$ on the generic equation and the asymptote. Yes, the asymptote of a exponential function is the value of $q$.