The function $f(x)=a^x$ and its graph

The function $f(x)=a^x$ is defined for all $x$ whenever $a > 0$. Here are some features of its graph:

• If $a > 1$, this function grows very quickly to the right and shrinks very quickly to the left. Exponential growth is fast.
  
  

  Different values of $a >1$ give essentially the same graph, only stretched horizontally. For instance, since $4^x=(2^2)^x=2^{2x}$, the graph $y=4^x$ is just like $y=2^x$, only shrunk horizontally. Similarly, since $(\sqrt{2})^x = 2^{x/2}$, the graph $y=(\sqrt2)^x$ is like $y=2^x$, only stretched.

DO:  Plug in some values of $x$ into $f(x)=2^x$, such as $x=-2,-1,0,1,2,5,10$ to see why the graph looks like the sketch above.

• If $a < 1$, then the function shrinks quickly to the right and grows quickly to the left. After all, $a^x=(a^{-1})^{-x}$.

DO:  Plug in some values of $x$ into $f(x)=(\frac{1}{2})^x$, such as $x=-2,-1,0,1,2,5,10$ to see why the graph looks like the sketch above.

• If $a=1$, then the function $f(x)=1^x=1$ is constant. 
  


• The graph $y=a^x$ always goes through $(0,1)$, no matter what $a$ is.  DO:  Why is this?
  


• The domain of the function $a^x$ is the whole real line.


• The range of the function $a^x$ is $(0,\infty)$ (unless $a=1$). This means that $a^x>0$ for all $x$.  DO:  Why is this?