Parametric Equations

by

Mike Rosonet

A parametric curve in the plane is a pair of functions

**x = f(t)**

**y = g(t)**

where the two continuous functions define ordered pairs **(x, y)**. The two equations are called the parametric equations of a curve. The extent of the curve depends on the range of **t**. In many applications, **x** and **y** are thought to be "varying with time **t**" or the angle of rotation that some line makes from an initial location.

Various graphing technology, such as graphing calculators or graphing computer software, can be easily used with parametric equations. This investigation of parametric equations will use the program Graphing Calculator 3.5, which is considered to be one of the most friendly graphing software programs available.

Consider the graph of the parametric equations:

**x = cos (t)**

**y = sin (t)**

As seen from the graph, these are the parametric equations of a circle. In order to investigate these equations deeper, it's necessary to apply constant values at different positions in the equations.

What if the constants **a** and **b** were placed as coefficients of **t** ?

**x = cos (at)**

**y = sin (bt)**

Let

a = b =2.This graph is the same as the graph of the parametric equations without the constants

aandb. Is this due to the fact thata = b?To investigate, let the value of

avary froma= 0 toa= 6, andb= 1.

When

a= 0, the graph is a line segment atx =1fromy= -1 toy= 1. Again, whena= 1, the graph is of a circle, as was initially investigated. As the value ofaincreases froma= 0 toa= 6, the graph reveals the graph ofx = cos yfromy= -1 toy= 1 with a decreasing period; this decreasing period causes the graph of cosine to occur multiple times within the parameters ofy= -1 toy= 1.Now, let

a= 1 and the value ofbvary fromb= 0 tob= 6.

When

b= 0, the graph is a line segment aty= 0 fromx= -1 tox= 1. Again, whenb= 1, the graph is of a circle, as was initially investigated. As the value ofbincreases fromb= 0 tob= 6, the graph reveals the graph of , the graph reveals the graph ofy = sin xfromx= -1 tox= 1 with a decreasing period; this decreasing period causes the graph of sine to occur multiple times within the parameters ofx= -1 tox= 1.

What if the constants **a** and **b** were placed as coefficients of the equations?

**x = a cos (t)**

**y = b sin (t)**

Let

a = b =2, just as the previous investigation.

It appears that, by placing the constants

aandbas coefficients of the equations,aandbexpand the distance from the roots of the graph and the y-intercepts of the graph from its center at the origin. What happens whenaandbvary?To investigate, let the value of

avary froma= 0 toa= 4, andb= 1.

When

a= 0, the value of the parametric equation forxequals 0 whent= 0, and the value of the parametric equation foryequals 0 whent= 0, regardless of the value ofbbecause the sin (0) = 0. Hence, there is no graph whena= 0. Furthermore, as the value ofaincreases toa= 1, the graph approaches the initial graph of a circle. However, as the value ofaincreases beyonda= 1 toa= 4, the graph becomes an expanding ellipse with an expanding major axis along the x-axis and a constant minor axis along the y-axis fromy= -1 toy= 1.Now, let

a= 1, and the value ofbvary fromb= 0 tob= 3.

When

b= 0, the value of the parametric equation forxequals 1 whent= 0 because the cos (0) = 1, and the value of the parametric equation foryequals 0 because of the value ofb. Hence, the graph is a line segment fromx= -1 tox= 1 whenb= 0. Furthermore, as the value ofbincreases tob= 1, the graph approaches the initial graph of a circle. However, as the value ofbincreases beyondb= 1 tob= 3, the graph becomes an expanding ellipse with an expanding major axis along the y-axis and a constant minor axis along the x-axis fromx= -1 tox= 1.

Thus, the position of **a** and **b** determine the transformation of the initial graph of a circle. When **a** and **b** are coefficients of **t**, they affect the number of periods of the graphs of **x = cos y** and **y = sin x** within the parameters of the values of **a** and **b**. When **a** and **b** are coefficients of the equations, they affect the length of the major axes of the ellipses formed as the values of **a** and **b** increase and decrease.