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

**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=ab $
- There is no maximum value in this case.

**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=ab $
- There is no minimum value in this case.

**Vertex Form:**- Quadratic functions can also be expressed in vertex form: $f(x)=a(x−h_{2}+k$ 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.