To determine the equation of the function $g(x)$ graphed above, we start by noting that it is a transformation of the base function $f(x) = x^2$.

Here are the observations based on the graph:

1. The parabola opens downward, which means there is a vertical reflection. This suggests a negative coefficient in front of the $x^2$ term.
2. The vertex of the parabola is located at $(0, -4)$. In the parent function $x^2$, the vertex is at $(0, 0)$, so this graph has been vertically shifted down by 4 units.

Considering these transformations:

• The reflection changes $x^2$ to $-x^2$.
• The downward shift moves the graph from $0$ to $-4$.

Therefore, the equation for the transformed function $g(x)$ is: $g(x) = -x^2 - 4$

Would you like more details or have any further questions?

Here are some related questions for further exploration:

1. How would the equation change if the parabola were shifted up by 3 units instead?
2. What transformations would be needed to shift the vertex to $(2, -4)$ instead of $(0, -4)$?
3. How does a horizontal shift affect the equation of a parabola?
4. If the parabola were stretched vertically, how would the equation reflect that change?
5. What happens to the graph of $g(x) = -x^2 - 4$ if you change it to $g(x) = -2x^2 - 4$?

Tip: When analyzing transformations, focus on changes in orientation (reflection), shifts (horizontal and vertical), and scaling (stretching or compressing).