Let's analyze the given table of values, which represents an exponential function.
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$x$ & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\
\hline
$y$ & \frac{1}{512} & \frac{1}{64} & \frac{1}{8} & 1 & 8 & 64 & 512 \\
\hline
\end{tabular}
\][/tex]
1. Identify the Base of the Exponential Function:
Recall that for an exponential function of the form [tex]\( y = a \cdot b^x \)[/tex], where [tex]\(a\)[/tex] is the initial value (which we assume is 1 here because [tex]\( y \)[/tex] equals 1 when [tex]\( x \)[/tex] is 0), and [tex]\( b \)[/tex] is the base. We need to determine the base [tex]\( b \)[/tex].
2. Inspect the Values for [tex]\( x = 1 \)[/tex]:
When [tex]\( x = 1 \)[/tex],
[tex]\[
y = 8
\][/tex]
Therefore, the base [tex]\( b \)[/tex] of the exponential function can be identified from this particular value pair.
3. Verify the Base with Other Values:
To ensure consistency, let's examine the other values provided:
[tex]\[
\begin{align*}
x = 2 & : y = 64 \Rightarrow 8^2 = 64 \\
x = -1 & : y = \frac{1}{8} \Rightarrow 8^{-1} = \frac{1}{8} \\
x = -2 & : y = \frac{1}{64} \Rightarrow 8^{-2} = \frac{1}{64} \\
x = 3 & : y = 512 \Rightarrow 8^3 = 512 \\
x = -3 & : y = \frac{1}{512} \Rightarrow 8^{-3} = \frac{1}{512}
\end{align*}
\][/tex]
4. Determine the Nature of the Exponential Function:
The base [tex]\( b = 8 \)[/tex] is greater than 1, indicating exponential growth.
Consequently, the correct statement is:
- The function represents exponential growth because the base equals 8.
