If the graph intersects Y-axis below the X-axis then which of the following can be valid?

Options

A. Roots are real and parabola opens downward
B. Roots are non-real and parabola opens upward
C. Roots are real only
D. If the roots are real then parabola opens upward and if the roots are non-real then parabola opens downward

Answer

To solve the question, we need to analyze the conditions under which a graph intersects the Y-axis below the X-axis. This typically indicates that the value of the function at \( x = 0 \) (i.e., \( f(0) \)) is negative.

Let's break down the options provided:

1. **Option A: Roots are real and parabola opens downward.**
- If the parabola opens downward and has real roots, it means it will intersect the X-axis at two points and will be above the X-axis between these roots. Therefore, it cannot intersect the Y-axis below the X-axis. This option is **not valid**.

2. **Option B: Roots are non-real and parabola opens downward.**
- If the roots are non-real, the parabola does not intersect the X-axis at all. A downward-opening parabola that does not intersect the X-axis will be entirely below the X-axis. Therefore, it will intersect the Y-axis below the X-axis. This option is **valid**.

3. **Option C: Roots are real only.**
- This option is incomplete as it does not specify the direction of the parabola. However, if the roots are real, the parabola will intersect the X-axis, which means it cannot be entirely below the X-axis. Thus, this option is **not valid**.

4. **Option D: If roots are real, then the parabola opens upward; if roots are non-real, then the parabola opens downward.**
- This statement is true. If the roots are real, the parabola opens upward and intersects the X-axis. If the roots are non-real, the parabola opens downward and does not intersect the X-axis, which means it can intersect the Y-axis below the X-axis. This option is **valid**.

Now, summarizing the analysis:
- The valid options are **B** and **D**.

### Step-by-Step Solution:
1. **Identify the condition of the graph intersecting the Y-axis below the X-axis.**
- This means \( f(0) < 0 \).

2. **Analyze each option based on the behavior of the parabola.**
- **Option A:** Real roots and downward opening - **Not valid**.
- **Option B:** Non-real roots and downward opening - **Valid**.
- **Option C:** Real roots only - **Not valid**.
- **Option D:** Real roots (upward) or non-real roots (downward) - **Valid**.

3. **Conclude which options are valid.**
- The valid options are **B** and **D**.