Let's analyze each statement about the function $f(x) = -4x^2$:

1. Option A: The domain is $\{x | x \text{ is a real number}\}$.

   • True. Since $f(x) = -4x^2$ is a quadratic function, it is defined for all real values of $x$. So, the domain is indeed all real numbers.

2. Option B: The range of $f$ is $(-\infty, -4]$.

   • True. The function $f(x) = -4x^2$ is a downward-opening parabola, and the maximum value it reaches is 0 (when $x = 0$). Thus, the range is $(-\infty, 0]$, not $(-\infty, -4]$.

3. Option C: $f$ is decreasing over the interval $(-\infty, \infty)$.

   • False. The function decreases on the interval $(0, \infty)$ and increases on $(-\infty, 0)$, due to the symmetry of the parabola around the vertex at $x = 0$.

4. Option D: The point $(0, 0)$ is a maximum.

   • True. The vertex of $f(x) = -4x^2$ is at $(0, 0)$, which is the highest point on the graph since it is a downward-opening parabola.

5. Option E: There are two x-intercepts.

   • True. Setting $f(x) = 0$ gives $-4x^2 = 0$, leading to $x = 0$ as the only x-intercept, not two.

Correct Answers

The correct statements are:

• A and D.

Would you like a detailed explanation on any part of this solution, or do you have further questions?

Further Related Questions:

1. How do we determine the domain and range of a function?
2. Why does the symmetry of the parabola affect whether it’s increasing or decreasing?
3. How do we determine if a point is a maximum or minimum in a quadratic function?
4. What are the steps to find the x-intercepts of a quadratic function?
5. How does the leading coefficient of a quadratic function affect the graph’s direction?

Tip:

When identifying the range of a quadratic function, always check the direction of the parabola (upward or downward opening) based on the sign of the coefficient of $x^2$.