Parametric Curves

By Courtney Cody

In this assignment, we will investigate the following parametric equations:

for

for different values of a and b. Next, we will investigate the graphs when the and components of the parametric equations are raised to a power, such as

for

and

for

etc.

LetÕs begin by graphing the general case when a = b = 1:

This is the graph of a circle with a radius of 1.  Hence, the equations  describe the unit circle!

Now letÕs graph the case where a=b (a and b are positive real numbers) for various values on same coordinate plane:

It is obvious that when a and b are positive real numbers and a=b, then  are the equations of a circle with a radius of a (=b)

What if we negate the values of a and b in the previous example?  How would this affect our graph?

By looking at the graph it appears as though nothing has changed.  Is this true?  Even though the graph looks the same as when a and b are positive, we can no longer say that the radius is a or  b since a radius cannot be negative since it is a measure of length.  Thus, in the case where a and b are negative and a=b, the radius is the absolute value of a and b.  Can we generalize what the graphs of  will look like when a=b for all real numbers a, b?  Sure we can! When a=b the parametric equation is a circle centered at the origin with a radius of .

What would the parametric curves look like for the case when a<b for positive real numbers a and b?

This graph resembles the graph of the circle except stretched vertically.  In particular, the graph is an ellipse centered at the origin.

What if a>b?

Again, it looks like our circle has been stretched—horizontally this time—into an ellipse.

Therefore, when a and b are not equal the graph is an ellipse centered about the origin.  Notice that when a=2 and b=1 the ellipse stretches from -2 to 2 on the x-axis and from -1 to 1 on the y-axis.  Similarly, when a=3 and b=2 the ellipse stretches from -3 to 3 on the x-axis and from -2 to 2 on the y-axis.

In general, we can say that when a<b the equations  form an ellipse with a major axis of length 2b and a minor axis of length 2a.  When a>b, the graph is an ellipse with a major axis of length 2a and a minor axis of length 2b.

Now we will investigate the effect that increasing the powers of the  and components of the parametric equations.  LetÕs begin by graphing the first five powers on the same coordinate place and weÕll let a=b=1.

From the graph, we can see that some of the parametric equations lie in all four quadrants while some lie only in the first quadrant of the coordinate place.  By looking at the equations and the graph, we see a pattern that explains this observation.  When the powers of the  and components of the parametric equations are odd, the graphs of the corresponding parametric equations lie in all four quadrants of the coordinate plane.  Also, when the powers of the  and components of the parametric equations are even, the graphs of the corresponding parametric equations lie just the first quadrant of the coordinate plane.  We can also see that the x- and y-intercepts are .  Furthermore, as the powers of  and  increase, the graphs of the parametric equations curve more and more becoming closer to the origin.

Next we will look at the different parametric equations when , or  and .

If we recall from the beginning of this investigation, we found that the parametric equations  when  form an ellipse with a major axis of 2b and a minor axis of 2a.  When we graph of the same plane the parametric equations with increasing powers of the  and components, we see that all of the graphs of the equations where the power is greater than 1 fit inside of the ellipse.  Additionally, the same observations as in the previous example still hold.  We can also see that the x-intercepts are  and the y-intercepts are .

Next we will look at the different parametric equations when , or  and .

This graph is very similar to the previous example except that the parametric equations  when  form an ellipse with a major axis of 2a and a minor axis of 2b and the x-intercepts are  and the y-intercepts are .  All other observations we have made thus far still hold.

Thus far, we have only considered positive values of a and b.  What if only one of a or b is negative.  WeÕll start by examining the example where  and .

It is obvious from looking at the graph that the only change we see is that the parametric equations with even powers of  and  changed from lying only in the first quadrant to lying only in the second quadrant.  All other previous observations still hold.

Now what if  and ?

It is obvious from looking at the graph that the only change we see yet again is that the parametric equations with even powers of  and  changed to lying only in the fourth quadrant.  All other previous observations still hold.

Now letÕs suppose that both of a and b are negative.  Specifically, weÕll let  and ?

Once again, we can see that the parametric equations with even powers of  and  changed to lying only in the third quadrant this time and all other previous observations still hold.

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In Conclusion, for increasing powers of  and  in the parametric equations

á      The graphs of the parametric equations with even powers of  and  lie in only one quadrant while the graphs of the parametric equations with odd powers lie in all four quadrants.

á      When  and , the graphs of the parametric equations with even powers of  and  lie in the first quadrant.

á      When  and , the graphs of the parametric equations with even powers of  and  lie in the second quadrant.

á      When  and , the graphs of the parametric equations with even powers of  and  lie in the third quadrant.

á      When  and , the graphs of the parametric equations with even powers of  and  lie in the fourth quadrant.

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