The function f is a parabolic function that intersects the x-axis. Which of the following statements must be true ?

Options

A. The function has at least one real root.
B. The function has no real roots.
C. The function intersects the positive y axis.
D. The function has two zeros.

Answer

To determine which statements must be true for a parabolic function \( f \) that intersects the x-axis, let's analyze the properties of parabolas and the implications of intersecting the x-axis.

### Step-by-Step Solution:

1. **Understanding Parabolic Functions**:
A parabolic function can be represented in the form \( f(x) = ax^2 + bx + c \), where \( a \neq 0 \). The graph of this function is a parabola, which can open upwards (if \( a > 0 \)) or downwards (if \( a < 0 \)).

**Hint**: Remember that the direction of the parabola is determined by the coefficient \( a \).

2. **Intersection with the x-axis**:
If a parabolic function intersects the x-axis, it means that there are values of \( x \) for which \( f(x) = 0 \). These values are known as the roots or zeros of the function.

**Hint**: The points where the parabola crosses the x-axis correspond to the real roots of the equation \( f(x) = 0 \).

3. **Analyzing the Options**:
- **Option A**: The function has at least one real root.
- Since the function intersects the x-axis, it must have at least one real root. This statement is true.

- **Option B**: The function has no real root.
- This statement contradicts the premise that the function intersects the x-axis. Therefore, this option is false.

- **Option C**: The function intersects the positive y-axis.
- A parabola can intersect the y-axis at any point, positive or negative. Therefore, this statement is not necessarily true for all parabolas.

- **Option D**: The function has two roots.
- A parabola can have two roots (intersecting the x-axis at two points), one root (touching the x-axis at one point), or no roots (not intersecting the x-axis). Since we know it intersects the x-axis, it can have either one or two roots. Thus, this statement is not necessarily true.

4. **Conclusion**:
The only statement that must be true for a parabolic function that intersects the x-axis is **Option A**: The function has at least one real root.

**Final Answer**: The correct statement is **A**: The function has at least one real root.