The maximum and minimum values of a quadratic function �(�)=��2+��+�f(x)=ax2+bx+c depend on the coefficient $a$ and the direction of the parabola.

1. When $a>0$ (Concave Upward):
   • The quadratic function opens upward, and its graph forms a “U” shape.
   • In this case, the function has a minimum value.
   • The minimum value occurs at the vertex of the parabola.
   • The minimum value can be found using the formula: Minimum value=−�2�Minimum value=2a−b​
   • There is no maximum value in this case.
2. When $a<0$ (Concave Downward):
   • The quadratic function opens downward, and its graph forms an inverted “U” shape.
   • In this case, the function has a maximum value.
   • The maximum value occurs at the vertex of the parabola.
   • The maximum value can be found using the formula: Maximum value=−�2�Maximum value=2a−b​
   • There is no minimum value in this case.
3. Vertex Form:
   • Quadratic functions can also be expressed in vertex form: �(�)=�(�−ℎ)2+�f(x)=a(x−h)2+k where $(h,k)$ represents the coordinates of the vertex.
   • In vertex form, the minimum or maximum value of the function is the value of $k$ (the $y$-coordinate of the vertex) when $a>0$ (for a minimum) or when $a<0$ (for a maximum).

In summary, the maximum or minimum value of a quadratic function depends on the direction of the parabola (determined by the coefficient $a$). The maximum occurs when the parabola opens downward ( $a<0$), and the minimum occurs when the parabola opens upward ( $a>0$). The vertex of the parabola represents either the maximum or minimum value of the function, depending on the direction of the parabola.