Functions are the base of Calculus. There are so many types of functions that to understand a polynomial function, one must understand what polynomials are. Let us recall the definition of a polynomial.
Definition : A polynomial is defined as an expression formed by the sum of powers of one or more variables multiplied to coefficients. In its standard form, it is represented as:
\(a_n x^n~+~a_{n1} x^{n1}~+.……….…+~a_2 x^2~+~a_1 x~+~a_0\)
where all the powers are nonnegative integers.
\(a_0~,~a_1~,………,~a_n~∈~R.\)
A polynomial is called a univariate or multivariate if number of variables is one or more respectively. So, the variables of a polynomial can have only positive powers. A polynomial function is a function that can be expressed in form of a polynomial. The definition can be derived from the definition of a polynomial. A polynomial is generally represented as P(x). The highest power of the variable of P(x)is known as its degree. Degree of a polynomial function is very important as it tells us about the behavior of the function P(x) when x becomes very large. The domain of a polynomial function is entire real numbers (R).
If P(x) = \(a_n x^n~+~a_{n1} x^{n1}~+.……….…+~a_2 x^2~+~a_1 x~+~a_0\)
Polynomial Function Graph with Examples
The graph of P(x) depends upon its degree.
A polynomial having one variable which has the largest exponent is called as degree of the polynomial.
Let us look at P(x) with different degrees.
Zero Polynomial


Degree 0 (Constant Functions) – Standard form: P(x) = a = \( a.x^0\)
, where a is a constant Graph: A horizontal line indicates that the output of the function is constant. It doesn’t depend on the input.

E.g. y = 4, (see fig. 1)
Figure 1: y = 4
Linear Polynomial

Degree 1, Linear Functions – Standard form: P(x) = ax + b, where a and b are constants. It forms a straight line.
Graph: Linear functions have one dependent variable and one independent which are x and y respectively.
In the standard formula for degree 1, a represents the slope of a line, the constant b represent the yintercept of a line.
E.g., y = 2x+3(see fig. 2)
here a = 2 and b = 3
Figure 2: y = 2x + 3
Note: All constant functions are linear functions.

Degree 2, Quadratic Functions – Standard form: P(x) = \(ax^2~+~bx~+~c\)
, where a, b and c are constants Graph: A parabola is a curve with one extreme point called as vertex. A parabola is mirrorsymmetric curve where any point is at an equal distance from a fixed point known as Focus.
In the standard form, the constant ,’a’ represents the wideness of the parabola. As a decreases, the wideness of the parabola increases. This can be visualized by considering the boundary case when a=0, the parabola becomes a straight line. The constant c represents the yintercept of the parabola. The vertex of the parabola is given by
(h,k) = \( ~\left( – \frac {b}{2a}~,~ \frac{D}{4a} \right) \)
where D is the discriminant given by (\(b^2~~4ac\)
Note: Whether the parabola is facing upwards or downwards, depends on the nature of a.
 If a > 0, the parabola faces upward.
 If a < 0, the parabola faces downwards.
E.g. y = \( x^2~+~2x~~3\)
y = \( x^2~~2x~+~3\)
See fig. 3
Figure 3: y = \(x^2~+~2x~~3~ (black)~ and ~y\)
 Higher degree: Standard form – P(x) = \(a_n x^n~+~a_{n1} x^{n1}~+.……….+a_0\)
, where \(a_0~,~a_1,………,a_n\) are all constants.
Graph: Depends on the degree, If P(x) has degree n, then any straight line can intersect it at a maximum of n points. The constant term in the polynomial expression i.e. \(a_0\)
E.g. y = \(x^4~~2x^2~+~x~~2\)
Figure 4: y = \( x^4~~2x^2~+x~~2\)
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