Question

when is the vertex a minimum? when is it a maximum?

Explanation:

For a quadratic function in the form \( y = ax^{2}+bx + c \) (or vertex form \( y=a(x - h)^{2}+k \), where \((h,k)\) is the vertex), the coefficient \( a \) determines the direction the parabola opens. If \( a>0 \), the parabola opens upward, so the vertex \((h,k)\) is the lowest point (a minimum). If \( a < 0 \), the parabola opens downward, so the vertex \((h,k)\) is the highest point (a maximum).

Answer:

For a quadratic function (parabola) with equation \( y = ax^{2}+bx + c \) (or vertex form \( y=a(x - h)^{2}+k \)):

• The vertex is a minimum when the coefficient of \( x^{2} \) (i.e., \( a \)) is positive (\( a>0 \)) because the parabola opens upward.
• The vertex is a maximum when the coefficient of \( x^{2} \) (i.e., \( a \)) is negative (\( a < 0 \)) because the parabola opens downward.