# Parameters of Polynomial Functions

Function parameters are constant terms or variable terms in a function. The specific form of the function is generally determined by these parameters. We will look at the definitions of the parameters of polynomial functions in more detail and use examples to better understand the definitions.

##### ALGEBRA

Relevant for

Learning about the parameters of polynomial functions.

See definitions

##### ALGEBRA

Relevant for

Learning about the parameters of polynomial functions.

See definitions

## Definition of parameters of polynomials

A parameter is a quantity that influences the behavior of mathematical functions, but that can be considered as constant. Parameters are related to variables and sometimes the difference is just a matter of perspective. Variables are seen as something that changes, while parameters either do not change or change slowly.

Parameters appear within functions. For example, the following is a generic quadratic function:

$latex f(x)=a{{x}^2}+bx+c$

In this function, the variable x is the input to the function. The symbols a, b and c are the parameters that determine the behavior of the function f.

## Effects of parameters on polynomial functions

Let’s take the function f again with the three parameters a, b and c:

$latex f(x)=a{{x}^2}+bx+c$

If we have a set of parameters such as $latex a=b=c=1$, the function f behaves like a normal function. Given these parameters, we have the function $latex f(x)={{x}^2}+x+1$ and each time we use the input $latex x=3$, we obtain the output $latex f(3)={{3}^2}+3+1=13$.

When we change the parameters, we are also changing the function f to a different function. For example, if we now have the parameter $latex a=3$, the function f becomes $latex f(x)=3{{x}^2}+x+1$ and we have $latex f(3)=3{{(3)}^2}+3+1=31$.

This means that a parameterized function represents an entire family of functions, where each parameter value produces a different function.

## Difference between parameters and input variables

We can think about what happens when we make the parameters a, b, c in the function $latex f(x)=a{{x}^2}+bx+c$ vary while keeping a constant value of x. For example, we can set $latex x=3$ and look at how $latex f(3)$ varies as a changes.

If we do these manipulations and look at how the output of a function changes as a parameter changes, then we are treating the function as if it were a function where the parameter acts as an input variable. This is normal because the difference between parameters and input values is simply a matter of perspective.

The distinction between input values and parameters is only a matter of convenience. However, by convention, a variable is something that changes and a parameter is commonly used to represent objects statically.

A parameter is normally a constant and is only changed when we need to adjust the behavior of the function.  