Quadratic Functions and Their Graphs

Vertex of a Parabola

Quadratic functions are fundamental in algebra and precalculus, typically expressed in the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, and its most important feature is the vertex, which represents either the maximum or minimum point of the parabola depending on the sign of a.

• Standard Form:

• Vertex Formula: The vertex of the parabola can be found using:

• Direction of Opening:

  • If , the parabola opens upward (vertex is a minimum).

  • If , the parabola opens downward (vertex is a maximum).

Example

Given the quadratic function , identify the vertex:

• Here, , , .

• Calculate :

• Calculate :

• Vertex:

Additional info:

• For any quadratic function, the vertex can always be found using the formula above, regardless of the values of , , and .

• The vertex is a key point for graphing and analyzing the properties of quadratic functions.