It looks like you are working with a graphing problem involving the function $y = \left( \frac{1}{2} \right)^{x - 4} - 7$, and you need to plot points, find the asymptote, and graph the function.

Steps to follow:

1. Identify the horizontal asymptote:
   The given function is of the form $y = a \cdot b^{x - h} + k$. In this case, the equation represents an exponential decay, and the horizontal asymptote is the value of $y$ as $x$ approaches infinity or negative infinity. From the equation, we can see that the horizontal asymptote is at: $y = -7$ This is because the function approaches $y = -7$ as $x \to \infty$.

2. Plot key points:
   To plot points on the graph, you can substitute values for $x$ and calculate the corresponding $y$-coordinates. Let's calculate some key points:

   • For $x = 0$:
     $y = \left( \frac{1}{2} \right)^{0 - 4} - 7 = \left( \frac{1}{2} \right)^{-4} - 7 = 16 - 7 = 9$
   • For $x = 1$:
     $y = \left( \frac{1}{2} \right)^{1 - 4} - 7 = \left( \frac{1}{2} \right)^{-3} - 7 = 8 - 7 = 1$
   • For $x = 2$:
     $y = \left( \frac{1}{2} \right)^{2 - 4} - 7 = \left( \frac{1}{2} \right)^{-2} - 7 = 4 - 7 = -3$
   • For $x = 3$:
     $y = \left( \frac{1}{2} \right)^{3 - 4} - 7 = \left( \frac{1}{2} \right)^{-1} - 7 = 2 - 7 = -5$
   • For $x = 4$:
     $y = \left( \frac{1}{2} \right)^{4 - 4} - 7 = 1 - 7 = -6$

3. Graph the function:
   You can now plot the points $(0, 9)$, $(1, 1)$, $(2, -3)$, $(3, -5)$, and $(4, -6)$ on the graph provided. The curve should approach the asymptote $y = -7$ as it moves left and right.

Would you like more details on how to plot the points or any specific calculations? Or do you need assistance with graphing the function further?

Here are some related questions that can help you dive deeper:

1. What happens to the graph as $x \to -\infty$?
2. How can you transform the graph if you change the exponent to $x - 2$ instead of $x - 4$?
3. What effect does changing the base from $\frac{1}{2}$ to $2$ have on the graph?
4. How can you find the domain and range of an exponential function like this one?
5. Can you explain the difference between horizontal and vertical asymptotes?

Tip: When plotting exponential functions, always check the asymptote first, as it gives you a boundary for the graph and can help guide where the function will approach as $x$ moves to large positive or negative values.