The image contains a math problem where the first question asks to identify which function corresponds to the given parabola on a graph. The parabola is upside down (opens downward) and passes through the origin.

The function options are:

• $f(x) = x^2 - 2$
• $f(x) = -x^2 + 2$
• $f(x) = -x^2$
• $f(x) = -x^2 - 2$
• $f(x) = x^2 + 2$

Analysis:

The parabola opens downward, so the leading coefficient of the quadratic term must be negative. This immediately eliminates the options with positive leading coefficients, such as $f(x) = x^2 - 2$ and $f(x) = x^2 + 2$.

Now, let's examine the vertex. The vertex of the parabola is at the origin, which suggests that there is no constant term added or subtracted from the quadratic function. Therefore, the correct function must be $f(x) = -x^2$, as it has the correct orientation and vertex placement.

Conclusion:

The correct answer is $f(x) = -x^2$.

Would you like more details, or do you have any further questions?

Here are 5 related questions to expand your understanding:

1. What is the general form of a quadratic function?
2. How does the sign of the leading coefficient affect the direction a parabola opens?
3. How can you find the vertex of a parabola from its equation?
4. What effect does adding or subtracting a constant have on the graph of a quadratic function?
5. How can you determine the x-intercepts of a parabola?

Tip: The vertex form of a quadratic function, $f(x) = a(x-h)^2 + k$, is useful for identifying the vertex $(h,k)$ and determining the orientation of the parabola.