The image shows the following instructions:

• A graph of the function $y = \frac{1}{x^2}$ is displayed.
• The task is to transform this graph to create the graph of the function $y = -\frac{1}{x^2} + 4$.

Steps to transform the graph:

1. Reflection: The graph of $y = \frac{1}{x^2}$ will be reflected over the x-axis due to the negative sign in front of $\frac{1}{x^2}$. This changes the graph to $y = -\frac{1}{x^2}$, making the curve open downwards instead of upwards.

2. Vertical Translation: The term "+4" translates the entire graph of $y = -\frac{1}{x^2}$ vertically by 4 units upwards. As a result, the vertex of the graph, previously located at $(0,0)$, will move to $(0,4)$.

Final Transformation Summary:

• The graph of $y = -\frac{1}{x^2} + 4$ is a downward-facing curve with its highest point at $y = 4$. The asymptotes remain the same: a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 4$.

Would you like a detailed step-by-step graphing process for this transformation or any other details?

Here are 5 related questions for further exploration:

1. How do horizontal shifts affect the graph of a rational function like $\frac{1}{x^2}$?
2. What happens if you add a constant inside the denominator, i.e., $y = \frac{1}{(x - c)^2}$?
3. How would the transformation change if the function was $y = \frac{1}{x^3}$?
4. How do vertical asymptotes behave under different transformations?
5. What role do reflections play in graph transformations for rational functions?

Tip: Always look at both vertical and horizontal asymptotes when analyzing transformations of rational functions.