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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 \( x^2 \).

Which of the following are true about the graph of f(x) = -4x^2? Select all that apply.

Explore the properties of the quadratic function f(x) = -4x^2, including domain, range, intervals of increase and decrease, maximum points, and intercepts. Suitable for high school students learning about quadratic functions and their graphs.

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