Which of the following condition is correct for the graph of quadratic polynomial p(x) = `ax^(2) + bx + c` to be an upward parabola?

Options

A. `a lt 0`
B. `a = 0`
C. `a gt 0`
D. `b = 0`

Answer

To determine the condition for the graph of the quadratic polynomial \( p(x) = ax^2 + bx + c \) to be an upward-opening parabola, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the standard form of a quadratic polynomial**:
The standard form of a quadratic polynomial is given by:
\[ p(x) = ax^2 + bx + c \]
where \( a \), \( b \), and \( c \) are real numbers and \( a \neq 0 \).

**Hint**: Recall that the term \( ax^2 \) determines the shape of the parabola.

2. **Understand the role of the coefficient \( a \)**:
The coefficient \( a \) in the quadratic polynomial plays a crucial role in determining the direction in which the parabola opens.

**Hint**: Think about how positive and negative values of \( a \) affect the graph.

3. **Determine the condition for an upward-opening parabola**:
- If \( a > 0 \) (i.e., the coefficient of \( x^2 \) is positive), the parabola opens upwards.
- If \( a < 0 \) (i.e., the coefficient of \( x^2 \) is negative), the parabola opens downwards.

**Hint**: Visualize the graph of a parabola with different values of \( a \).

4. **Conclusion**:
Therefore, for the graph of the quadratic polynomial \( p(x) = ax^2 + bx + c \) to be an upward-opening parabola, the condition is:
\[ a > 0 \]

**Final Answer**: The correct condition is that \( a \) should be positive.