To graph the function [tex]\( y = \left( \frac{1}{2} \right)^{x-1} - 7 \)[/tex] and analyze its behavior using the given table of values, follow these steps:
### Step-by-Step Solution:
1. Understanding the Function:
- The function is [tex]\( y = \left( \frac{1}{2} \right)^{x-1} - 7 \)[/tex].
2. Table of Values:
- The given table of values shows the outputs for a wide range of [tex]\( x \)[/tex] values from [tex]\( -10 \)[/tex] to [tex]\( 10 \)[/tex].
- Notice the significant drop in [tex]\( y \)[/tex] values as [tex]\( x \)[/tex] increases.
[tex]\[
\begin{array}{|c|c|}
\hline x & y \\
\hline -10 & 2041 \\
\hline -9 & 1017 \\
\hline -8 & 505 \\
\hline -7 & 249 \\
\hline -6 & 121 \\
\hline -5 & 57 \\
\hline -4 & 25 \\
\hline -3 & 9 \\
\hline -2 & 1 \\
\hline -1 & -3 \\
\hline 0 & -5 \\
\hline 1 & -6 \\
\hline 2 & -6.5 \\
\hline 3 & -6.75 \\
\hline 4 & -6.875 \\
\hline 5 & -6.9375 \\
\hline 6 & -6.96875 \\
\hline 7 & -6.984375 \\
\hline 8 & -6.9921875 \\
\hline 9 & -6.9960937 \\
\hline 10 & -6.9980469 \\
\hline
\end{array}
\][/tex]
3. Identifying the Asymptote:
- An asymptote is a line that the graph of a function approaches but never touches.
- For exponential functions of the form [tex]\( y = a \cdot b^{x} + c \)[/tex], where [tex]\( b \)[/tex] is a fraction between 0 and 1, the horizontal asymptote [tex]\( y \)[/tex] is [tex]\( c \)[/tex].
- In this function, as [tex]\( x \)[/tex] approaches infinity, [tex]\( \left( \frac{1}{2} \right)^{x-1} \)[/tex] approaches 0.
- Therefore, the asymptote is [tex]\( y = -7 \)[/tex].
4. Final Answer:
[tex]\[
y = -7
\][/tex]
So, the equation of the asymptote is:
[tex]\[
y = -7
\][/tex]
