Calculus How To

Parametric Function / Equation: Definition, Examples

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Types of Functions >

A parametric function is really just a different way of writing functions, just like explicit and implicit forms:

  • Explicit functions are in the form y = f(x), for a < x < b. These include:
  • Implicit functions, which describe shapes like circles, hyperbolas and parabolas. They take the form: F(x, y) = 0 for a < x < b and c < y < d.
  • The parametric function takes the form: p(t) = (f(t), g(t)) for a < t < b.

More specifically, a parametric function expresses certain quantities in terms of one or more independent variables called “parameters.” Multiple dependent variables x and y are treated as a single entity, which depend on an explicit independent variable (e.g. t). The range of a parameter function is a set of ordered pairs (x, y).


A parametric function is any function that follows this formula:
p(t) = (f(t), g(t))
for a < t < b.

Varying the time(t) gives differing values of coordinates (x,y).

In the above formula, f(t) and g(t) refer to x and y, respectively. Some authors choose to use x(t) and y(t), but this can cause confusion. That’s because if you use x(t) to describe the function value at t, x can also describe the input on the horizontal axis.


The implicit form for a circle is: x2 + y2 = r2. The parametric equation for a circle is:
parametric function

Parameterization and Implicitization

Suppose we want to rewrite the equation for a parabola, y = x2, as a parabolic function. The easiest way to do this is to introduce a new, free parameter—we can call it t. Then we can say:

We’ve just parameterized our function.

Implicitization is the opposite of parameterization. It means taking a parametric function and changing it back into a single formula with an implicit relationship between x and y.

For the parabola, it’s super simple: since x = t, replace t2 with x2 and you are back to your implicit formula.

Let’s look at something just a little more complicated. The parametric formula for a circle of radius a is

parametric equation example

We can divide both sides by a, and so rewrite this as

what is a parametric equation

The Pythagorean Triangle Identity gives us
what is a function

Substituting our parametric functions into that, we get
example of parametric functions

And so, finally

Which is the (standard) implicit equation for a circle a; so we’ve successfully implicitized it.

Use of Parametric Functions

In introductory calculus classes, parametric functions are usually taught as being representations of graphs of curves, but they can be used to model a much wider variety of situations. For example:

  • They are useful for Modeling the paths of moving objects,
  • They are necessary for optimizing multivariable functions.
  • In general, they enable complicated problems with multiple inputs to be reduced to a simpler function (Stalvey, (2014).


McQuarrie, B. Precalculus: Parametric Representations. Retrieved May 20, 2019 from:
Stalvey, H. (2014). The Teaching and Learning of Parametric Functions: A Baseline Study. Retrieved May 20, 2019 from:
Wilson, M. Assignment Ten: Investigating Parametric Functions. Retrieved May 20, 2019 from:

Stephanie Glen. "Parametric Function / Equation: Definition, Examples" From Calculus for the rest of us!

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