10.5 Parametric Equations

Parametric equations are used to represent a curve in the plane, using a third variable.

 is a rectangular equation.

Although, this equation doesn't help us determine time. To determine the "when" at a given point (x,y) on the path, you introduce the third variable, t:

 

 

Definition of a Plane Curve

If f and g are continuous functions of t on an interval I, the set of ordered pairs (f(t), g(t)) is a plane curve C. The equations:

x = f(t)        and       y = g(t)

are parametric equations for C, and t is the parameter.

 

Sketching a Plane Curve

Each set of coordinates (x,y) is determined from a value chosen for the parameter t.

The orientation of the curve is plotting the resulting points in the order of increasing values of t, tracing the line in a specific equation.

 

Sketch the curve given by the parametric equations:
x = 2t - 4 and y = t + 3

 

T	-2	-1	0	1	2
X	-8	-6	-4	-2	0
Y	1	2	3	4	5

 

Eliminating the Parameter

1. start with the parametric equations:

 

2. solve for t in one equation:

t = 2y

3. Substitute in second equation

4. Rectangular equation

 
***note
Converting equations from parametric to rectangular form can change the ranges of x and y. In such cases, you should restrict x and y in the rectangular equation so that its graph matches the graph of the parametric equations.