Understanding Quadratic Functions: Introduction, Examples, and Formula

Study Notes

Understanding Quadratic Functions

Introduction

Quadratic functions are an essential part of algebraic mathematics, particularly in calculus and physics. They are polynomial functions of degree 2 and are represented graphically as parabolas. The term "quadratic" comes from the Latin word "quadrare," which means "to square," since these functions involve squared variables. Quadratic functions have many applications in various fields such as engineering and science, where they help obtain values of different parameters.

Standard Form of a Quadratic Function

The standard form of a quadratic function is given by the equation f(x) = ax^2 + bx + c, where a, b, and c are real numbers with a ≠ 0. This general form allows us to identify key characteristics like the vertex (point of minimum or maximum), intercepts, and symmetry properties based on the signs of the coefficients.

Example 1

Consider the quadratic function f(x) = x^2 - 4x + 4. Here, a = 1, b = -4, and c = 4. We can use this information to find important points about the function, such as the vertex (x, y) = (-2, 4), horizontal intercepts at x = 1 and x = 2, and vertical intercepts at y = 4.

Examples of Quadratic Functions

Here are some examples of quadratic functions:

f(x) = 2x^2 + 4x - 5
f(x) = 3x^2 - 9
f(x) = (x - 12)(x + 3)

Quadratic Formula

The quadratic formula is a common method used to solve quadratic equations of the form ax^2 + bx + c = 0. Given the coefficients a, b, and c, the formula provides the solutions for x in terms of the coefficients themselves. However, the application of the quadratic formula requires the equation to be in the correct form: ax^2 + bx + c = 0 or in standard form: ax^2 + bx + c = y, where a ≠ 0.

Description

Learn about quadratic functions, their standard form, key characteristics like the vertex and intercepts, and the quadratic formula used to solve equations. Explore various examples of quadratic functions and understand their applications in mathematics and other fields.