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.
