What is a Quadratic function?

Definition

A polynomial of degree two is called a quadratic function.

Introduction

A function is a mathematical expression and it contains terms. So, it is also called a polynomial. The term “Quadratic” was derived from a Latin term “Quadratus”. In Latin language, Quadratus means square. So, the term quadratic is used in English for expressing square. The combination of meanings of terms “Quadratic” and “Function” expresses a second-degree polynomial.

A quadratic function is also called by the following names alternatively in mathematics.

1. Quadratic polynomial
2. Polynomial of degree two

Similarly, a quadratic function is simply abbreviated as “Quadratic” in mathematics. Now, let’s know about the importance of a quadratic function in real life from an understandable example.

When you throw a ball into air, it reaches a maximum position, then it travels downwards and falls down on floor finally. It follows a path, known as parabola. Scientifically, an expression is required for us to denote this path. Mathematically, it can be expressed by a single variable second degree polynomial, called a quadratic function.

Likewise, there are also so many scientific examples of quadratic function in physics and engineering. Now, let’s learn how to express a quadratic function in mathematics.

Standard form

Let’s assume that $a, b$ and $c$ denote constants, and $x$ denotes a variable. Then, a quadratic function in standard form is written in mathematics as follows.

$ax^2+bx+c$

It is a mathematical expression having degree $2$. So, it is called the univariate quadratic function. Now, let’s know more about the general form of a quadratic function.

1. The constant $a$ is called the coefficient of $x$ square. Remember, the value of constant $a$ is not equal to zero ($a \ne 0$) because the quadratic function becomes a linear function.
2. The constant $b$ is called the coefficient of $x$.
3. The constant $c$ is called the constant term.

If a quadratic function is equated with zero, then the quadratic function becomes a quadratic equation.

Other form

More than one variable can also involve in forming a quadratic function. If $x$ and $y$ denote two variables, then a two variable quadratic function can be written as follows.

$ax^2+bxy+cy^2+dx+ey+f$

This type of quadratic equation is called by the following two names.

1. Bivariate Quadratic function
2. Multivariate Quadratic function

Remember, at least one of $a, b$ and $c$ are not equal to zero in this case.

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