What is the difference between constant and parameter?

Answer 1

A constant is a fixed numerical value anywhere on the real number line.

constants may be arbitrary numbers, or they can be mathematically significant numbers. For instance, in the function: f(x)=5ex , "5" is an arbitrary constant, "e" is a mathematically significant constant, and "x" is a variable. Keep in mind that "x" is NOT a parameter.

An aside: The number, "e", is mathematically significant because it is the only number that can be raised to the power of "x" and yield a graph with an instantaneous rate of change equal to "1" at the point, "x=0". This number is irrational; it cannot be described as the ratio of two integers. When cut off after fifteen decimal places "e" is equal to "2.718281818284590". Pi is another mathematically significant irrational constant that is equal to "3.141592653589793" when cut off after fifteen decimal places. Every circle has the property that its circumference divided by its radius is equal to pi.

A parameter defines a set of adjustable constants.

So what exactly is a parameter if it is not a fixed constant such as "e" or a variable such as "x"? A parameter defines a set of adjustable constants. One might argue that this is exactly what a variable is but this is not entirely correct. Variables, together with constants, define the exact nature of a function; on a graph the variables define a particular shape and constants fix the dimensions of that shape. For instance, in equation (1) below the variables, "x" and "y", define a circle while the constant, "5", fixes the radius of the circle. Parameters, on the other hand, are used to replace these specific constants with a generic letter representing a range of constants. For instance, in equation (2) below "r" is a parameter that has replaced the constant, "5". Instead of defining one circle with a radius of 5 units, our equation now defines any sized circle. Thus, parameters supplement variables and can be used to define the generic nature of a class of functions; a generic set of the shapes on a graph. In other words, a parameter indicates an adjustable geometric dimension of the graph.

(1) x2 + y2 = 52

(2) x2+y2 = r2

Whether or not we call something a parameter verses a variable depends on the relative context of the situation.

Take equation (2) for example: This equation tells us the nature of the circle, x2 + y2, with respect to the parameter, "r". Suppose we wanted to understand how the parameter "r" changes with respect to the variables "x" and "y". We could graph the system in polar coordinates where the focus is shifted to the parameter "r" with respect to the angle theta that is equal to the arctangent of "y/x". In this way "r" becomes a variable that depends on the independent variable, theta. Polar coordinates are different from rectangular coordinates in that there are no "x" and "y" axes. Instead the radius is swept out as the angle from a horizontal increases from 0 to however many revolutions around the coordinate system one cares to make.