We have studied how to evaluate $a^n$ when $n$ is a rational number. But what about $a^n$ when $n$ is real but not necessarily rational? To determine this question, we can look at graphs of exponential functions, which are part of transcendental functions. Transcendental functions produce values that may not be expressed as rational numbers or roots of rational numbers.

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 $y=d$

$\textit{Properties of Exponential Graphs}$

- 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.

## Example 1

Sketch the graphs of $y=2^x$ and $\color{red}{y=-2^x}$.

Reflected the $x$-axis.

## Example 2

Sketch the graphs of $y=2^x$ and $\color{red}{y=2^{-x}}$.

Reflected the $y$-axis.

## Example 3

Sketch the graphs of $y=2^x$ and $\color{red}{y=-2^{-x}}$.

Origin rotated.

## Example 4

Sketch the graphs of $y=2^x$ and $\color{red}{y=3 \times 2^{x}}$.

Multiplied 3 times to the $y$ values.

## Example 5

Sketch the graphs of $y=2^x$ and $\color{red}{y=2^{x+1}}$.

Translated to left.

## Example 6

Sketch the graphs of $y=2^x$ and $\color{red}{y=2^{x-1}}$.

Translated to right.

## Example 7

Sketch the graphs of $y=2^x$ and $\color{red}{y=2^x+1}$.

Translated to up.

## Example 8

Sketch the graphs of $y=2^x$ and $\color{red}{y=2^x-1}$.

Translated to down.

## Example 9

Sketch the graphs of $y=2^x$ and $\color{red}{y=2^{x+1}-1}$.

Translated to left and down.

## Example 10

Sketch the graphs of $y=2^x$ and $\color{red}{y=2^{x-1}+1}$.

Translated to right and up.

## Example 11

Sketch the graphs of $y=2^x$ and $\color{red}{y=3^x}$.

The graph of $y=3^x$ is steeper than $y=2^x$.

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