Can anyone please tell me why it is called Parametric equation?

Question

Parametric representation of the parabola $y^2 = x$ is $(t^2 , t)$.

Can anyone please tell me why it is called Parametric representation ?

Can I say $(x , \pm \sqrt x)$ is a parametric representation of $y^2 = x$ ?

I know for a family of equations for every value of parameter we get a particular member of that family.

$x^2+y^2 = a^2$ is the family of circles with center at origin. If we put $a= 1$ , we get a particular circle. $a$ is also varying but for a particular value of $a$ we are getting a member. That is why , this is little different from variable.

Answer 1

As with any language, mathematics is allowed to use the same word for different notions.

• One sense of "parameter" (the second in the question) is that of "characteristic" or perhaps "ingredient"; a value that helps describe or define an object. Like how the parameters of a debate define the scope of topics that can be discussed. So, parameters can include the radius and/or center of a circle, the aspect ratio of a rectangle, the $x$- and $y$-intercepts of a line. The "Earliest Known Uses of Some of the Words of Mathematics" pages mention that "parameter" was first used in 1631 specifically for the latus rectum of a conic.

• The "parametric equation" sense of "parameter" is that of "other measurement", in particular with regard to graphing curves and surfaces. (See this ancient answer of mine to the question "What is a the intuition behind a parametric equation?".) Sticking with curves ... While we often graph by expressing $y$ in terms of $x$, or $x$ in terms of $y$, there are significant advantages (beyond the scope of discussion here) to expressing both $x$ and $y$ in terms of something else ---an "other measurement", or parameter--- that helps convey a graph as a path of travel. Whereas the cartesian equation $x^2+y^2=1$ for the unit circle is perfectly effective at defining a static set of points, the parametric forms $(\cos t,\sin t)$ and $\left(\frac{1-t^2}{1+t^2},\frac{2t}{1+t^2}\right)$ describe (very) different dynamic journeys along the curve.

So, the $a$ in $x^2+y^2=a^2$ and the $t$ in $(t,t^2)$ are both parameters, but in different senses of the word. (Attempts to fashion a single meaning to the term that covers both use cases are a bit strained.)

Consequently, we have the situation that $(a\cos t, b\sin t)$ uses the parameter $t$ to traverse an elliptical path defined by parameters (namely, semi-axes) $a$ and $b$. Language is funny like that sometimes. (See "Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo".)

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As to whether $(x,\pm \sqrt{x})$ can be considered a parametric representation of parabola $x=y^2$ ...

It's a not-unreasonable representation of the parabola, in the same way as one writes $y=\pm\sqrt{x}$. It tells you something about the points on the curve. So, fine.

However, $(x,\pm\sqrt{x})$ doesn't describe a path of travel along the curve; rather, at each $x$, an ostensible traveler either chooses between candidate $y$-values, or somehow visits both of them simultaneously. That defeats the purpose of describing a path. A proper "parametric representation" doesn't give a choice; the value of the parameter should tell you exactly where you are. Formally: the $x$ and $y$ coordinates should be functions of the parameter.

Answer 2

A parameter is one or more numbers that is varied to give a collection of "things". Each value of the parameter(s) should give a single thing. For instance,

• In $(t,t^2)$, the parameter $t$ can be varied to give a collection of points
• In $x^2+y^2=a^2$, the parameter $a$ can be varied to give a collection of circles
• In $\displaystyle\frac1{\sigma\sqrt{2\pi}}e^{-\frac12\left(\frac{x-\mu}{\sigma}\right)^2}$, the two parameters $\sigma$ and $\mu$ can be varied to give different distribution functions for different normal distributions

In the case where the collection of things is a collection of points in the plane, or in higher dimensional space, then we call that a parametric representation of that collection of points. Because we use a parameter to represent them.

Answer 3

The idea with a parametric equation (with a single real parameter) is that it "draws the curve" as you vary the parameter.

For the parabola $y^2=x$, your attempt $(x,\pm\sqrt x)$ makes sense, but multi-valued expressions like $\pm...$ aren't allowed. This would "draw two curves at once" as $x$ moves through positive values (and draw nothing when $x$ is negative). In contrast, the parametrization $(t^2,t)$ gives a clean one-to-one correspondence (in fact a homeomorphism) between the parameter space $\Bbb R$ and the parabola.

$x^2+y^2=a^2$ is an equation with a parameter that describes a family of circles, and positive values of $a$ are in one-to-one correspondence with the members of the family. But since it's not a family of points, I probably wouldn't call it a parametric equation.

Answer 4

It is called a parametric equation, because we can relate the values of x and y (or even z, if we consider a 3-D space) to a single parameter (let's call it t).

Consider the example you gave, of the parabola $y^{2}=x$. We can relate the values of x and y, by a single parameter t, which gives the value of x and y, as the parameter changes, both x and y change also, such that it satisfies the original equation.

And no, the example you gave isn't a parametric equation because it doesn't satisfy the original equation.