Let's tackle each part of the problem step by step.
### Part A: Explanation (4 points)
To understand why the [tex]\(x\)[/tex]-coordinates of the points where the graphs of the equations [tex]\(y = 2^{-x}\)[/tex] and [tex]\(y = 4^{x+3}\)[/tex] intersect are the solutions of the equation [tex]\(2^{-x} = 4^{x+3}\)[/tex], consider the following:
When two graphs intersect at a point, their [tex]\(y\)[/tex]-coordinates at that point are equal. This means that if the graphs of [tex]\(y = 2^{-x}\)[/tex] and [tex]\(y = 4^{x+3}\)[/tex] intersect at some point [tex]\((x, y)\)[/tex], then at that point,
[tex]\[2^{-x} = 4^{x+3}\][/tex]
Thus, solving the equation [tex]\(2^{-x} = 4^{x+3}\)[/tex] will give us the [tex]\(x\)[/tex]-coordinates where the two graphs intersect.
### Part B: Creating Tables (4 points)
To find the solution for the equation [tex]\(2^{-x} = 4^{x+3}\)[/tex] using integer values of [tex]\(x\)[/tex] between -3 and 3, we can create a table that shows the values of [tex]\(y\)[/tex] for both functions for each [tex]\(x\)[/tex]. The tables for [tex]\(y = 2^{-x}\)[/tex] and [tex]\(y = 4^{x+3}\)[/tex] are as follows:
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & y = 2^{-x} & y = 4^{x+3} \\
\hline
-3 & 8 & 1 \\
-2 & 4 & 4 \\
-1 & 2 & 16 \\
0 & 1 & 64 \\
1 & 0.5 & 256 \\
2 & 0.25 & 1024 \\
3 & 0.125 & 4096 \\
\hline
\end{array}
\][/tex]
From the table, we observe that the values of [tex]\(2^{-x}\)[/tex] and [tex]\(4^{x+3}\)[/tex] are equal when [tex]\(x = -2\)[/tex]. This point of equality suggests that [tex]\(x = -2\)[/tex] is a solution to the equation [tex]\(2^{-x} = 4^{x+3}\)[/tex].
### Part C: Graphical Solution (2 points)
To solve the equation [tex]\(2^{-x} = 4^{x+3}\)[/tex] graphically, you would follow these steps:
1. Plot the graph of the function [tex]\(y = 2^{-x}\)[/tex].
2. Plot the graph of the function [tex]\(y = 4^{x+3}\)[/tex] on the same axes.
3. Identify the points where the two graphs intersect.
The [tex]\(x\)[/tex]-coordinates of the intersection points are the solutions to the equation [tex]\(2^{-x} = 4^{x+3}\)[/tex]. From the table in Part B, we can see that the graphs intersect at [tex]\(x = -2\)[/tex], which confirms our solution.
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This completes our detailed step-by-step solution to the problem.
