In the equation

$$y=ax^2+bx+c$$

there are five letters, and they are not on the same level. The $x$ and the $y$ vary: they run through the points of a parabola. The $a$, the $b$ and the $c$ stay still while this happens, but they can be chosen at will beforehand. Each choice gives a different parabola. Letters like these are called parameters, from the Greek pará, beside, and métron, measure: what stands beside the measure, regulating it.

Moving a parameter is not the same as moving a variable. Moving $x$ is walking along the curve. Moving $a$ is changing curves: the whole parabola opens or closes.

One equation, one family

This was Viète’s great invention, told in text 3: using letters also for the given numbers of the problem. The equation

$$ax^2+bx+c=0$$

is not one equation but a family of equations, one for each triple $(a,b,c)$ with $a\ne0$. In it the roles are clear: $x$ is the unknown, and $a$, $b$, $c$ are the parameters. The formula

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

solves the whole family at once, writing the unknown in terms of the parameters. This allows questions that no single equation allows: for which values of the parameters are there two solutions? The answer, $b^2-4ac>0$, is a statement about the family, not about one equation.

The letters of physics

Physics is the kingdom of parameters. In uniform motion,

$$s=s_0+vt,$$

the time $t$ and the position $s$ vary during the motion, while the initial position $s_0$ and the velocity $v$ stay fixed. But change the motion and they change. In the ideal gas law, $pV=nRT$, $R$ is a universal constant, the same for all gases; $n$ is fixed for a given sample; $p$, $V$ and $T$ vary. Three levels of “standing still” in a single line, and no sign to tell them apart.

The other parameter

To complicate things, the word parameter has a second meaning. In the equations

$$x=\cos t,\qquad y=\sin t,$$

the letter $t$ is called a parameter, and the equations are parametric. But here $t$ does not stand still: it runs, from $0$ to $2\pi$, and as it runs the point $(x,y)$ draws a circle. It is as if $t$ were time and the point a moving body. In this sense the parameter is the hidden variable that drives the other two.

Exercise: for which values of k is there a single solution?

In $x^2+kx+9=0$, $x$ is the unknown and $k$ is a parameter. There is a single solution when the discriminant is zero: $k^2-4\cdot9=0$, that is, $k=6$ or $k=-6$. For $k=6$ the solution is $x=-3$; for $k=-6$ it is $x=3$. The question is about the parameter, and the unknown appears only at the end.

Exercise: the family y = mx

For each value of $m$, $y=mx$ is a straight line through the origin. All the lines in the family pass through $(0,0)$, because there $y=m\cdot0=0$ for any $m$. For the line to pass also through $(2,6)$ we need $6=2m$, that is, $m=3$. Notice the swap of roles: to find the parameter, we treat $x=2$ and $y=6$ as given and $m$ as the unknown.

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