I believe the following 4 questions I have, are all related to eachother.
Question 1: Of course I've been using constants, variables and parameters for a long time, but I sometimes get confused with the definition. It seems to me that these terms are used very loosely.
Let's say we have a second degree polynomial e.g. $ax² + bx + c$. I've heard people say $a, b, c$ are both constants and parameters.
I understand they are parameters, since they allow us to have a 'family' of second degree polynomials. What I mean by that is that if e.g. $c$ is incremented by one, it becomes a different function, since the graph of the function would be shifted up by one.
Why do people call them constants as well though? AFAIK, constants are fixed, e.g. $\pi, e, \varphi$. They would be the same value in any given context, and never change. The parameters $a, b, c$ however, never change with respect to the function, but they represent multiple values, unlike $\pi$.
Question 2:
To make things even more confusing, there's also contrast between unknown and known variables. Are known variables the same thing as parameters? If they are known, why don't we just throw in the real value of the variable?
Question 3: If we have $ax + 3$, how does one know if a represents a variable like x, so it's a function that can take two inputs, or if it's a parameter? Should the context provide this information?
Question 4:
In the pythagoream theorem, are $a$, $b$ and $c$ constants, variables or parameters? If they are variables, then why are they represented by the letters a, b, c instead of x, y, z? I've read a, b and c are commonly used as known variables.
Best Answer
A constant is something like a "number". It doesn't change as variables change. For example $3$ is a constant as is $\pi$.
A parameter is a constant that defines a class of equations. $$\left(\frac xa\right)^2 + \left(\frac yb\right)^2 = 1$$ is the general equation for an ellipse. $a$ and $b$ are constants in this equation, but if we want to talk about the entire class of ellipses then they are also parameters -- because even though they are constant for any particular ellipse, they can take any positive real values.
A variable is an element of the domain or codomain of a relation. Remember that functions are just relations so the input and output of functions are variables. For example, if we talk about the function $x \mapsto ax +3$, then $x$ is a variable and $a$ is a parameter -- and thus a constant. $3$ is also a constant but it is not a parameter.
A "known" variable is typically a value that the conditions of the problem dictate the variable must take. For example if we are discussing an object an free fall, then acceleration is a variable. But physics puts a constraint on the value that that variable may take -- acceleration in free fall is $a=g\approx 9.8$. Thus, though $a$ may be defined as the input of a function, it must take a "known" value. Thus it is a known variable.
The Pythagorean theorem states that $a^2 + b^2 = c^2$ for sides $a,b$ and hypotenuse $c$ of a right triangle. These are parameters -- thus they are also constants.