Determine whether the a-value is positive or negative. Provide an appropriate reason.

Question

The a-value is positive because the graph is of a parabola that opens up.
The a-value is negative because the x-intercepts are negative.
The a-value is negative because the y-intercept is negative.
The a-value is negative because the graph is of a parabola that opens down.

Answer 1

To determine whether the a-value in a quadratic function \( f(x) = ax^2 + bx + c \) is positive or negative, we need to consider the properties of the graph of the parabola:

1. Positive a-value: If the graph of the parabola opens upwards, then the a-value is positive. This means the vertex is the lowest point, and the graph will rise on both sides.

2. Negative a-value: If the graph of the parabola opens downwards, then the a-value is negative. This means the vertex is the highest point, and the graph will fall on both sides.

Given the provided statements:

1. "The a-value is positive because the graph is of a parabola that opens up." - This is a correct statement indicating a positive a-value.
2. "The a-value is negative because the x-intercepts are negative." - This statement is incorrect. The x-intercepts can be negative regardless of the sign of the a-value.
3. "The a-value is negative because the y-intercept is negative." - This statement is also incorrect. The y-intercept being negative does not determine the sign of the a-value.
4. "The a-value is negative because the graph is of a parabola that opens down." - This is a correct statement indicating a negative a-value.

To summarize: The correct conclusion about the sign of the a-value only depends on whether the parabola opens upwards (positive a-value) or downwards (negative a-value). Thus, the first and fourth statements correctly describe the scenario.