Quadratic function definition & Types of Quadratic functions


This article gives the definition of a quadratic function and explains how to identify a quadratic function, given a polynomial function.

Following are the key features of this article. Please do attempt the quiz at the end.

  • Quadratic function definition
  • Identifying  a quadratic function from a given polynomial function
  • Classification of quadratic functions based on the number of terms
  • Standard and vertex form Classification
  • Worked out examples
  • Practice problems
  • Quiz

1)Quadratic function definition:-

In short, The Quadratic function definition is,”A polynomial function involving a term with a second degree and 3 terms at most “.

A quadratic function is a polynomial function, with the highest order as 2.

For example ,a polynomial function  x^{2}+3, can be called as a quadratic function ,since the highest order of   x^{2} is 2.

In this example, 3x^{3}+4 .We observe that the highest order is 3.

Therefore, referring to the Quadratic function definition, we can conclude that given polynomial function is not a quadratic.


2)Polynomial and a Quadratic Function:-

Quadratic equation belongs to the family of a Single variable polynomial function.

A single variable polynomial equation in “x” is an equation of the given form

 f\left( x\right) =a_{0}x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+.\ldots a_{n}

The equation has only one independent variable, which is “x” in this case. And there would be no negative exponents are allowed for the variable “x”.

a_{0},a_{1},a_{2}\ldots a_{n} are the coefficients  of the polynomial.

The order of the polynomial is the highest power of the variable in the given equation which is “n” in this case.

The single variable polynomial equation must have positive powers only and can have a constant.

As mentioned in Quadratic Definition, There will be a maximum of 3 terms in a Quadratic Expression.

3) Classification of Quadratic Functions

A Quadratic Expression can be classified into 3 categories based on the number of terms it contains.

As discussed, the maximum number of terms will not exceed 3 and the order must be equal to 2


i)Monomial Quadratic Expression:- If the quadratic Equation or expression has a single term, it is called as “Monomial quadratic”.

  • This expression contains only  x^{2} term.

            Examples:  x^{2}, 3x^{2}, 4x^{2},-2x^{2}.


      ii)Binomial Quadratic Expression:-If the quadratic expression has 2 terms, it is called as “Binomial Quadratic”.

  • This expression will have a   x^{2} term with a constant number or a “x” term.

            Examples:  x^{2}+3,2x^{2}+1,3x^{2}+x,x^{2}+4x.


      iii)Trinomial Quadratic Expression:-If the quadratic expression has 3 terms, it is called as “trinomial quadratic”.

            Examples:  x^{2}+2x+1, 2x^{2}-x+4, x^{2}-x+1.


Classification is also done based on the representation of the quadratic function in The standard form or The vertex form is explained in detail in further articles.

4)Practice Problems:-

Which of the  polynomials given below are quadratics? Justify your answer.

i) x+1

ii) x^{2}+4.

iii) x^{3}+2y^{2}+1.





  Strategy Step-by-Step:-

Before we dive into the solutions, let us construct a step-by-step approach to identify the quadratic expression from the given polynomial expression.

Here are the steps:-

     Step1) Observe the order of the given expression, the highest power present on the variable.

     Step2) Check whether the order is 2, ie the highest power on the variable is 2.

     Step3) In addition to step3, check whether there is any negative exponent given in the expression.

     Step4) Therefore, If the order is 2 and there is no negative exponent, the given polynomial is a quadratic


i) x+3.we observe that there is variable “x” which can be written as x^{1} and the default.

Since the order of the function given is 1 which is less than 2,  x+3 is not a quadratic function


ii)x^ {2}+4.Here we have the power of variable “x” as 2 and a constant number 4.

Therefore the conclusion is that x^{2}+4 is a quadratic expression.


iii) y^{3}+2y^{2}+1. Since the polynomial variable is given in “y” and the highest power of y is 3.

Therefore the conclusion is that the given polynomial is not a quadratic expression.


iv)[latex] x^{-2}+5.[/latex] Here we need to bit cautious. Since the degree of the variable “x” is a negative number “-2”.

Therefore we can conclude that the given polynomial is not a quadratic.


v)z^{2}+2z+5. The variable of the given polynomial is “z” and we observe that the order of the variable “z” is 2.

Hence we can conclude that the given polynomial is a quadratic.


vi) x^{2}+3x^{-2}+4x+5. Here we observe that there are two powers of “x” .-2 and +2.

Since there is a negative power, this expression itself is not a polynomial. In conclusion, this is not a quadratic expression


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