University Mathematics Handbook (2015)

II. Functions

Chapter 3. Parametric Form of a Function

In the explicit form of function , variables and relate directly.

But in its parametric representation, and relate indirectly. In this case, variables and are dependent of another variable, , as presented by the two equations , , in a given domain for variable . Variable is called the parameter of the form. If function is invertible above the given domain, then it can be denoted as , and be denoted in the explicit form of the function .

Geometrical Interpretation

If parameter is regarded as time variable, then equations , describe point in a plane, where particle is situated at time . Therefore, the plain curve consisting of all points represents the trajectory of particle in a plane.

Examples:

a.  , , is a parametric representation of an upper semicircle, the explicit form of which is. If also the domain of is extended to interval , the result will be full circle , which is not a graph of a function.

b.  . , is a parametric form of a function. The function is invertible in interval . Then to directly relate between and , we extract from the former equation. The result is . Positioning it in the latter equation, we get

Another way of finding it here is using the trigonometric identity

.

The result is , since for all in its domain , it is the upper half of the aforementioned ellipse .

c.  , is a parametric form of function , since , and therefore .