Let's break down the given problem step-by-step:
1. Exponential Function: We're given the exponential function [tex]\( f(x) = 2(2)^x \)[/tex].
2. Calculate [tex]\( f(x) \)[/tex] for Given Values of [tex]\( x \)[/tex]:
- For [tex]\( x = -2 \)[/tex]: [tex]\( f(-2) = 2(2^{-2}) = 2 \cdot \frac{1}{4} = 0.5 \)[/tex]
- For [tex]\( x = -1 \)[/tex]: [tex]\( f(-1) = 2(2^{-1}) = 2 \cdot \frac{1}{2} = 1.0 \)[/tex]
- For [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = 2(2^0) = 2(1) = 2 \)[/tex]
- For [tex]\( x = 1 \)[/tex]: [tex]\( f(1) = 2(2^1) = 2(2) = 4 \)[/tex]
- For [tex]\( x = 2 \)[/tex]: [tex]\( f(2) = 2(2^2) = 2(4) = 8 \)[/tex]
Hence, the values are:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & -2 & -1 & 0 & 1 & 2 \\
\hline
$f(x)$ & 0.5 & 1.0 & 2 & 4 & 8 \\
\hline
\end{tabular}
\][/tex]
3. Identify Constants [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- For the function [tex]\( f(x) = a(b)^x \)[/tex], here we have [tex]\( a = 2 \)[/tex] and [tex]\( b = 2 \)[/tex].
4. Identify the [tex]\(y\)[/tex]-Intercept:
- The [tex]\( y \)[/tex]-intercept occurs where [tex]\( x = 0 \)[/tex].
- For [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 2 \)[/tex], so the [tex]\( y \)[/tex]-intercept is 2.
5. Determine the End Behavior:
- As [tex]\( x \rightarrow \infty \)[/tex]: Since we have an exponential growth function with base [tex]\( b > 1 \)[/tex], [tex]\( y \rightarrow \infty \)[/tex]. So, as [tex]\( x \rightarrow \infty, y \rightarrow \infty \)[/tex].
- As [tex]\( x \rightarrow -\infty \)[/tex]: For negative [tex]\( x \)[/tex] values, [tex]\( 2^x \)[/tex] approaches 0, and thus, [tex]\( y \rightarrow 0 \)[/tex]. So, as [tex]\( x \rightarrow -\infty, y \rightarrow 0 \)[/tex].
Here is the detailed solution summarized:
[tex]\[
\begin{array}{rl}
\text{Table of Values:} & \\
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & -2 & -1 & 0 & 1 & 2 \\
\hline
$f(x)$ & 0.5 & 1.0 & 2 & 4 & 8 \\
\hline
\end{tabular}
&
\\
\\
\text{Constants:} & a = 2, \, b = 2 \\
\text{y-intercept:} & y = 2 \\
\end{array}
\][/tex]
End Behavior:
[tex]\[
\text{As } x \rightarrow \infty, y \rightarrow \infty
\][/tex]
[tex]\[
\text{As } x \rightarrow -\infty, y \rightarrow 0
\][/tex]
So, filling in the blanks in the original question:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 2 \)[/tex]
- [tex]\( y \)[/tex]-Intercept = 2
- As [tex]\( x \rightarrow \infty, y \rightarrow \infty \)[/tex]
- As [tex]\( x \rightarrow -\infty, y \rightarrow 0 \)[/tex]
