To solve this problem, we need to reflect the function $f(x)$ about the line $y = 2$. From the graph, the original function $f(x)$ appears to be a parabola with its vertex at $(0, -4)$. A standard form for such a parabola is $f(x) = \frac{1}{2}x^2 - 4$.

Steps for Reflection:

1. Reflection Formula: To reflect the graph $f(x)$ about the line $y = 2$, we can use the formula for reflection: $g(x) = 2 - (f(x) - 2)$ This can be rewritten as: $g(x) = 4 - f(x)$
2. Substitute $f(x)$: Substituting $f(x) = \frac{1}{2}x^2 - 4$ into the reflection equation: $g(x) = 4 - \left(\frac{1}{2}x^2 - 4\right)$ Simplifying: $g(x) = 4 - \frac{1}{2}x^2 + 4$ $g(x) = 8 - \frac{1}{2}x^2$

Conclusion:

The equation of the reflected function is: $g(x) = -\frac{1}{2}x^2 + 8$ This matches option E.

Final Answer:

The correct answer is E.

Do you want more details, or do you have any questions?

Here are five follow-up questions to expand your understanding:

1. How does reflecting a function over a line change its equation?
2. What is the general form of a quadratic equation, and how do transformations affect it?
3. Why did we choose to reflect over the line $y = 2$ instead of any other line?
4. Can you derive the reflection formula used in this solution from basic geometric principles?
5. How would the graph change if we reflected it over the $x$-axis instead?

Tip: Always double-check the vertex of the original parabola when reflecting to ensure accuracy in transformations.