What determines whether the quadratic function \( f(x) = ax^2 + bx + c \) has a maximum or minimum value?

The determination of whether the quadratic function \( f(x) = ax^2 + bx + c \) has a maximum or minimum value is based on the coefficient \( a \) in the quadratic expression. 1. **Coefficient \( a \)**: - If \( a > 0 \): The parabola opens upwards, and the function has a **minimum** value. The vertex of the parabola represents this minimum point. - If \( a < 0 \): The parabola opens downwards, and the function has a **maximum** value. Again, the vertex of the parabola represents this maximum point. 2. **Vertex**: - The vertex of the quadratic function can be found using the formula for the x-coordinate of the vertex, given by: \[ x = -\frac{b}{2a} \] - The corresponding y-coordinate (the maximum or minimum value) can be found by substituting this x-value back into the function \( f(x) \). In summary, the sign of the coefficient \( a \) determines whether the quadratic function has a maximum or minimum value.