Which of the following is not the graph of a quadratic polynomial?

Options

A.
B.
C.
D.

Answer

To determine which of the given options is not the graph of a quadratic polynomial, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of a Quadratic Polynomial**:
A quadratic polynomial is a polynomial of degree 2, which can be expressed in the standard form as \( ax^2 + bx + c \), where \( a \neq 0 \). The graph of a quadratic polynomial is a parabola.

2. **Identify the Characteristics of the Graph**:
The key characteristic of the graph of a quadratic polynomial is that it can intersect the x-axis at most twice. This means:
- It can touch the x-axis at two points (two real roots).
- It can touch the x-axis at one point (one real root, also known as a double root).
- It can not touch the x-axis at all (no real roots).

3. **Analyze the Given Options**:
We need to analyze the graphs presented in the options:
- **Option 1**: If the graph touches the x-axis at two points, it is a quadratic polynomial.
- **Option 2**: If the graph touches the x-axis at two points, it is also a quadratic polynomial.
- **Option 3**: If the graph does not touch the x-axis at all (it lies completely above or below), it is not a quadratic polynomial.
- **Option 4**: If the graph touches the x-axis at three points, it is also not a quadratic polynomial.

4. **Conclusion**:
Based on the analysis:
- **Option 3** and **Option 4** do not represent graphs of quadratic polynomials because:
- Option 3 does not intersect the x-axis (no real roots).
- Option 4 intersects the x-axis at three points (which is impossible for a quadratic polynomial).

### Final Answer:
The options that are not graphs of a quadratic polynomial are **Option 3** and **Option 4**.

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