Parametric Function / Equation: Definition, Examples

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:
  • Rational functions (e.g. (x² − 1) / (x + 19)),
  • Exponential functions (e.g. x²
  • Polynomial functions (e.g. 2^x 5 + 12).
• 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).

Formula

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.

Example

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

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

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

The Pythagorean Triangle Identity gives us

Substituting our parametric functions into that, we get


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).

References

McQuarrie, B. Precalculus: Parametric Representations. Retrieved May 20, 2019 from: http://cda.mrs.umn.edu/~mcquarrb/teachingarchive/Precalculus/Lectures/ParametricRepresentations.pdf
Stalvey, H. (2014). The Teaching and Learning of Parametric Functions: A Baseline Study. Retrieved May 20, 2019 from: https://scholarworks.gsu.edu/math_diss/18/
Wilson, M. Assignment Ten: Investigating Parametric Functions. Retrieved May 20, 2019 from: http://jwilson.coe.uga.edu/EMAT6680Fa11/Wilson/MGW_10/mgw_10.html

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