What is the difference between parameters and variables?

In mathmatical related sciences we often come accross the parameters and varibles.

A parameter for a particular value determines a particular function.

For different values of the parametrs we get a family of functions.

The variable for for values gives the paticular values of the paricular function.For the entire domain of the variable it defines a particular function.

Example:

y=5x+ c. Here c is a parameter. x is a variable.

y=5x + c reprepresents a family of parallel straight lines ,all having a slope of 5 ,for different values of c. for c=1, y=5x+1 is one member this family of straight lines.

x is a vaiable. For a particular  value of x=5, we get a particular functional value of the function y= 5x+1  and that value is 5*5+1=26.

For all  values of the variable x,    y = 5x+1  represents a particular function. In fact in y=5x+c is also a member of the family of functions y=mx+c for parmetric value of m=5.

x^2+y^2=a^2 represents a family  functions: y = +or- {Square root (x^2-a^2)}

For different values of a ,the parameter, we get a family of concentric circles but with different radii.

Hope this helps.