Let's first calculate the values of the function [tex]\( f(x) = 3(0.5)^x \)[/tex] for the given [tex]\( x \)[/tex]-values.
Given [tex]\( x \)[/tex] values are: -2, -1, 0, 1, 2.
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = 3(0.5)^{-2} = 3 \cdot 4 = 12.0 \][/tex]
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = 3(0.5)^{-1} = 3 \cdot 2 = 6.0 \][/tex]
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 3(0.5)^0 = 3 \cdot 1 = 3.0 \][/tex]
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 3(0.5)^1 = 3 \cdot 0.5 = 1.5 \][/tex]
5. For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = 3(0.5)^2 = 3 \cdot 0.25 = 0.75 \][/tex]
Now, we can fill in the table:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & -2 & -1 & 0 & 1 & 2 \\
\hline
$f(x)$ & 12.0 & 6.0 & 3.0 & 1.5 & 0.75 \\
\hline
\end{tabular}
\][/tex]
Next, we'll identify the y-intercept. The y-intercept of a function [tex]\( f(x) \)[/tex] is the value of the function when [tex]\( x = 0 \)[/tex]:
[tex]\[ \text{y-intercept} = f(0) = 3.0 \][/tex]
For end behavior, we analyze [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] and [tex]\( -\infty \)[/tex]:
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( (0.5)^x \rightarrow 0 \)[/tex]. Therefore, [tex]\( f(x) \rightarrow 0 \)[/tex].
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( (0.5)^x \rightarrow \infty \)[/tex]. Therefore, [tex]\( f(x) \rightarrow \infty \)[/tex].
We can summarize the end behavior as:
- As [tex]\( x \rightarrow \infty, y \rightarrow 0 \)[/tex].
- And as [tex]\( x \rightarrow -\infty, y \rightarrow \infty \)[/tex].
So, our filled answer looks like:
[tex]\[
a = 12.0 \quad b = 6.0 \quad y\text{-intercept} = 3.0
\][/tex]
[tex]\[
\text{End Behavior: As } x \rightarrow \infty, y \rightarrow 0.
\][/tex]
[tex]\[
\text{And as } x \rightarrow -\infty, y \rightarrow \infty.
\][/tex]
