Quadratic Functions and Their Extrema

A parabola is the graphical representation of a quadratic function, a polynomial function of degree two. Its shape is determined by the coefficients of the quadratic equation. A key feature of a parabola is its vertex, representing either the maximum or minimum value of the function.

Standard Form of a Quadratic Function

The standard form of a quadratic function is expressed as: f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants, and 'a' ≠ 0. The value of 'a' determines the parabola's concavity (opening upwards if a > 0, downwards if a < 0).

Determining the x-coordinate of the Vertex

The x-coordinate of the vertex, often denoted as h, can be calculated using the formula: h = -b / 2a. This formula is derived from the process of completing the square to convert the quadratic equation into vertex form.

Determining the y-coordinate of the Vertex

Once the x-coordinate (h) is determined, the y-coordinate of the vertex, often denoted as k, is found by substituting 'h' back into the original quadratic function: k = f(h) = a(h)² + b(h) + c. Therefore, the vertex is located at the point (h, k).

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by: f(x) = a(x - h)² + k, where (h, k) represents the coordinates of the vertex. This form clearly displays the vertex and the parabola's concavity.

Finding the Vertex through Completing the Square

Alternatively, the vertex can be found by completing the square for the quadratic expression. This method involves manipulating the equation to achieve the vertex form, making the vertex coordinates directly apparent.

Applications

Determining the vertex is crucial in various applications, including optimization problems (finding maximum profit, minimum cost), projectile motion analysis (finding the maximum height), and curve fitting in data analysis.