The graph and the family of exponentials

It is convenient to keep in mind a “base” graph and to obtain the others by reflection.

Observation — A mental graph

A “base” graph to keep in mind is $y=2^x$: increasing, passing through $(0;1)$, through $(1;2)$, through $(-1;1/2)$, with the $x$-axis as asymptote on the left and divergence on the right. The graph of $y=(1/2)^x$ is that of $y=2^x$ reflected in the $y$-axis (because $(1/2)^x = 2^{-x}$): it still passes through $(0;1)$ but decreases, with an asymptote on the right.

Family of exponentials: all the curves pass through $(0;1)$ because $a^0=1$. The curves with $a>1$ increase towards the right, those with $0<a<1$ decrease; the $x$-axis is a horizontal asymptote on one side.

From this picture the practical rule emerges: the larger the base $a>1$, the steeper the growth; and passing from $a$ to $1/a$ amounts to reflecting the graph in the $y$-axis.

In the following simulation you can vary the base $a$ and watch how the graph of $y=a^x$ changes: growth for $a>1$, decay for $0<a<1$, and in every case passage through $(0;1)$.

Drag the slider to change the base $a$; the dashed curve is $y=2^x$.

Links

Topics: Exponential function
Concepts: Asymptote · Base · Exponential growth
Functions: Exponential function
Skills: Interpreting a graph · Sketching a graph