Graph exponential functions

Graph exponential functions of the form $y=ab^x$ and their basic transformations, interpreting $a$ as the initial value, $b$ as the multiplicative growth or decay factor, and identifying domain, range, intercepts, end behavior, and the horizontal asymptote. The graph connects constant ratios over equal input intervals with exponential growth when $b>1$ and decay when $0<b<1$ , distinguishing these from linear graphs with constant differences; advanced topics such as calculus-based analysis and complex or generalized exponential functions are not included.

Detailed Explanation: Graph exponential functions

To graph an exponential function, look for the form

y=ab^x.

$a$ is the initial value, because $y=a$ when $x=0$ .
$b$ $b$ is the multiplicative factor:
- If $b>1$ , the graph shows growth.
- If $0<b<1$ , the graph shows decay.
Exponential functions have a constant ratio between consecutive $y$ -values, not a constant difference.

Example: Graph $y=2\cdot 3^x$

1. Identify the initial value and growth factor

Compare the function with $y=ab^x$ :

a=2,\qquad b=3.

The initial value is $2$ , and the growth factor is $3$ . This means that each time $x$ increases by $1$ , the $y$ -value is multiplied by $3$ .

2. Make a table of values

Choose several values for $x$ .

\begin{array}{c|c} x & y=2\cdot 3^x\\ \hline -2 & 2\cdot 3^{-2}=\frac{2}{9}\\ -1 & 2\cdot 3^{-1}=\frac{2}{3}\\ 0 & 2\cdot 3^0=2\\ 1 & 2\cdot 3^1=6\\ 2 & 2\cdot 3^2=18 \end{array}

Plot the points

\left(-2,\frac{2}{9}\right),\quad \left(-1,\frac{2}{3}\right),\quad (0,2),\quad (1,6),\quad (2,18).

Then connect them with a smooth curve. The curve rises quickly to the right and gets closer and closer to the $x$ -axis on the left.

3. Identify the important features

Domain: Any real number can be used for $x$ .

\text{Domain: }(-\infty,\infty)

Range: Since $2\cdot 3^x$ is always positive,

\text{Range: }(0,\infty)

$y$ -intercept: Set $x=0$ .

y=2\cdot 3^0=2

So the $y$ -intercept is $(0,2)$ ).

$x$ -intercept: The function never equals $0$ , so there is no $x$ -intercept.
Horizontal asymptote: The graph approaches, but never reaches, the line

y=0.

End behavior:

As $x\to\infty$ , $y\to\infty$ because the function grows.

As $x\to-\infty$ , $y\to 0$ because the graph approaches the horizontal asymptote.

Basic transformations can be added to the function. In

y=a b^{x-h}+k,

$h$ shifts the graph left or right,
$k$ shifts the graph up or down,
the horizontal asymptote becomes $y=k$ .

Always use a table or the growth factor to plot points, then draw a smooth curve that approaches the horizontal asymptote without touching it.

Learn by doing: Graph exponential functions

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Graphing - Function Graphs - Example Formula to Graph

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