Parametric Equations




Brandt Hacker


A parametric equation is one that defines the coordinates for x and y using some other parameter.  For instance, if we set x = f(t) and y = g(t), then the parametric equation would be the relationship between t and functions f and g graphed in a plane.  For the following problems the value of t will be limited by the inequality . 


In assignment 10 we will look to see how various values of a and b change the parametric equation of the of the form seen below.  The graph below sets both a and b equal to 1.



In the graph above it appears that the zeros occur at 1 and -1, but these are merely the x-values and y-values that correspond to some time t.  Because our parameters were set for , the x-intercepts actually represent the times at which our function y = sin(t) = 0.


We will now look at what happens to the graph as we vary a and b.



Below are graphs of different parametric equations of the same form with varying values of a and b.





There are a variety of interesting relationships as we change the values of a and b in the parametric equation.  One such relationship can be seen as we observe what happens when a=b.





It appears from the three graphs above that whenever a=b, the resulting parametric equation is a circle.


LetŐs now look at another interesting set of parametric equations.








In the images above we see that the graph is symmetric over the x-axis when the coefficient of b is divisible by the coefficient of a.  The other trend that we see developing is that the number of enclosed regions in the image is equal to b/a.  In other words, in the equations above, a = 4 and b = 20, since 20/4 = 5, the graph should have 5 enclosed regions.  It does.  Notice that this was not the case when a = 4 and b = 6, the second graph in this set.


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