The simplest general exponential function has the form $y = a^x$ where $a > 0,a \ne 1$.

For example, $y=2^x$ is an exponential function.

We can construct a table of values from which we graph the function $y=2^x$. \begin{array}{|c|c|c|c|c|c|c|c|} \hline x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline y & \frac{1}{8} & \frac{1}{4} & \frac{1}{2} & 1 & 2 & 4 & 8 \\ \hline \end{array} When $x=-20$, $y = 2^{-20} \approx 0.00000095367 \cdots$

When $x=20$, $y = 2^{20} = 1048576$

As $x$ becomes large and negative, the graph of $y=2^x$ approaches the $x$-axis from above but never touches it, since $2^x$ becomes very small but $\textit{never}$ zero.

So, as $x \rightarrow -\infty,y \rightarrow 0^{+}$.

We say that $y=2^x$ in $\textit{asymptotic}$ to the $y$-axis or $y=0$ is a $\textit{horizontal asymptote}$.

For the general exponential fruntion $y=a \times b^{x-c}+d$ where $b>0,b\ne1,a\ne0$.

- $b$ controls how steeply the graph increases or decreases
- $c$ controls horizontal translation
- $d$ controls vertical translation
- the equation of the horizontal asymptote is $y=d$

- The graph of $y=a^x$ passes through a fixed point $(0,1)$.
- The domain of $y=a^x$ is all real numbers.
- The range of $y=a^x$ is $y>0$.
- The graph of $y=a^x$ is increasing.
- The graph of $y=a^x$ is asymptotic to the $x$-axis as $x$ approaches $-\infty$.
- The graph of $y=a^x$ increases without bound as $x$ approaches $+\infty$.
- The graph of $y=a^x$ is continuous.