Department of
Mathematics and Science Education 







Allyson Hallman 







A Very Unexciting Exploration of Parametric Equations 






Consider: x = a cos t for three fairly obvious cases. y = b sin t 






Case #1: 








Pictured we have
x = 2 cos t y = 2 sin t We have generated a circle centered at the origin with radius 2. Click here to view Clip where a and b vary from 5 to 5. 

These graphs are symmetric about both the x and yaxis. A negative a value reflects over the yaxis and a negative b value reflects over the xaxis. But due to the symmetry of these graphs these negative values have no effect on the graphs generated. And so we consider the absolute value of a and b without loss of generality. 






Case #2: 








WhatÕs going on? Looks like fun with ellipses. So we have ellipses centered at the origin and with major axis of length along the xaxis and with minor axis of length along the yaxis 






Case #2: 









Still ellipses. (whoopteedoo) So we have ellipses centered at the origin and with minor axis of length along the xaxis and with major axis of length along the yaxis 

Ok, what happens if square stuff? 






Consider: x = a (cos t)^{2} y = b (sin t)^{2} 







Case #1: 





Pictured we have
x = 3 (cos t)^{2} y = 3 (sin t)^{2} We have generated a line with slope 1 and yintercept of 3. Click here to view Clip where a and b vary from 5 to 5. 






Case #2: 














OH FUN. A bunch of lines. What does it all mean? The slope of the line is equal to , the yintercept is b, and the xintercept is a. 






Consider: x = a (cos t)^{3} y = b (sin t)^{3} 












Ok, well this is much less boring than what we had before. Our lines are curvy, forming these cool diamond like things that are symmetrical over the x and yaxis. xintercepts are (a, 0) and (a, 0) and our yintercepts are (0, b) and (0, b). 






I wonder what happens if increase the exponent more. 







Values of n take on integers from 1 to 10. As the exponents decrease from 10 to 2, we are approaching our straight line. Then we jump from the straight line out to the ellipse which had an exponent of 1. I wonder if we examine exponent values between 1 and 2 if we will fill in the space between the straight line and the ellipse 






Ha, we have some additional values of n = 1.05, 1.15, 1.25, 1.5, 1.75 And lo and behold they fall within the space between the straight line and the ellipse. Super, I am glad this one finally got a bit more interesting. 






Not to thrilled with this assignment? (yea, me neither) Go home. 
