Graphing an Exponential Function

Graph: [tex][tex]$f(x)=4^x$[/tex][/tex]

Step 1: Calculate the initial value of the function.
[tex]\[ f(0)= \][/tex]
[tex][tex]$\square$[/tex][/tex]

Step 2: Plot the initial value of the function at [tex][tex]$(0,1)$[/tex][/tex].

Step 3: Evaluate the function at two more points.
[tex]\[ f(1)=4 \quad \quad f(-1)= \][/tex]
[tex][tex]$\square$[/tex][/tex]

Step 4: Plot the points [tex][tex]$(1,4)$[/tex][/tex] and [tex][tex]$(-1,1/4)$[/tex][/tex].

[tex]\[
\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
0 & 1 \\
0.5 & 2 \\
& \\
& \\
\hline
\end{tabular}
\][/tex]



Answer :

Alright, let's go through the steps to graph the exponential function [tex]\( f(x) = 4^x \)[/tex].

### Step 1:
Calculate the initial value of the function at [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = 4^0 = 1 \][/tex]

So, the initial value is [tex]\( f(0) = 1 \)[/tex].

### Step 2:
Plot this initial value on the graph at the point [tex]\( (0, 1) \)[/tex].

### Step 3:
Evaluate the function at two more points:

First, at [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 4^1 = 4 \][/tex]

Second, at [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = 4^{-1} = \frac{1}{4} = 0.25 \][/tex]

So, we have [tex]\( f(1) = 4 \)[/tex] and [tex]\( f(-1) = 0.25 \)[/tex].

### Step 4:
Plot the points [tex]\( (1, 4) \)[/tex] and [tex]\( (-1, 0.25) \)[/tex] on the graph.

### Step 5:
Evaluate the function at [tex]\( x = 0.5 \)[/tex] for additional accuracy:
[tex]\[ f(0.5) = 4^{0.5} = 2.0 \][/tex]

### Step 6:
Update the table with the given points and the additional point for [tex]\( x = 0.5 \)[/tex].

[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 1 \\ 0.5 & 2 \\ 1 & 4 \\ -1 & 0.25 \\ \hline \end{array} \][/tex]

By plotting these points [tex]\((0, 1)\)[/tex], [tex]\((0.5, 2)\)[/tex], [tex]\((1, 4)\)[/tex], and [tex]\((-1, 0.25)\)[/tex], you can accurately sketch the graph of the exponential function [tex]\( f(x) = 4^x \)[/tex], which will show rapid growth as [tex]\( x \)[/tex] increases and rapid decay as [tex]\( x \)[/tex] decreases.