To determine which function represents the graph, let's analyze the parabola shown:

1. Vertex Analysis: The vertex is located at $(0, -4)$. This means the equation of the parabola takes the form $f(x) = ax^2 + bx + c$, where $c = -4$ since the y-intercept is -4.

2. Direction of Opening: The parabola opens downward, so the coefficient of $x^2$, $a$, must be negative.

3. Options Elimination:

   • Option A: $f(x) = 2x^2 - 2$ opens upward (positive $a$), so this is incorrect.
   • Option B: $f(x) = x^2 - 4$ opens upward (positive $a$), so this is incorrect.
   • Option C: $f(x) = -x^2 - 4$ opens downward and matches the vertex.
   • Option D: $f(x) = -2x^2 + 2$ opens downward but has the wrong vertex $(0, 2)$.

4. Verification: Option C ($f(x) = -x^2 - 4$) is correct because it matches the vertex at $(0, -4)$ and opens downward.

Final Answer: C $f(x) = -x^2 - 4$

Would you like further clarification or detailed steps?

Here are 5 related questions to expand your understanding:

1. How can you find the equation of a parabola given its vertex and another point?
2. Why does the sign of the coefficient $a$ determine the direction of a parabola's opening?
3. How do you identify the vertex of a parabola from its equation in standard form?
4. What transformations are applied to the parent function $f(x) = x^2$ to get $f(x) = -x^2 - 4$?
5. How would the graph look if the equation were $f(x) = -2x^2 - 4$?

Tip: Always start by identifying key features of the graph, such as the vertex, direction of opening, and intercepts, to match it with the correct equation.