Exploring Parametric Equations

by Laura Lowe

Problem Statement:

Investigate each of the following for 0 < t < 2p.

Describe each when a = b, a < b, and a > b.

 y = acos(t) y = bsin(t) y = a(cos(t))2 y = b(sin(t))2 y = a(cos(t))3 y = b(sin(t))3 y = a(cos(t))4 y = b(sin(t))4 y = a(cos(t))5 y = b(sin(t))5 etc.

First letŐs look at the graphs when a = b.  Let a = 1 = b to begin with.

We can see that when the exponent is even, our graph is only in the first quadrant and each graph intersects the axes at (1, 0) and (0, 1).  However, when the exponent is odd, the graph makes a full circuit of the four quadrants and each graph intersects the axes at (1, 0), (0, 1), (-1, 0) and (0, -1).  In fact, the graphs intersect only at these points.  We can also see that as the exponent increases, the graphs are pulled to the origin while the points on the axes remain fixed.

LetŐs see what happens when we change a and b. Let a = 2 = b

Again, the graphs with the even exponents are only in the first quadrant and now each graph intersects the axes at (2, 0) and (0, 2).  And when the exponent is odd, the graph makes a full circuit of the four quadrants and each graph intersects the axes at (2, 0), (0, 2), (-2, 0) and (0, -2).  It looks like a (and therefore b) dictate the intersections with axes.  So we expect the intersections to be (a, 0), (0, a), (-a, 0) and (0, -a).

What happens when we change a and b to 3?  What about if a and b are 0.5?  Click here to see the graphs.

Now letŐs investigate when a < b.  Let a = 1 and b = 2

The pattern with even and odd exponents holds, but now the x and y intercepts are different.  It appears that the x-intercepts are (a, 0) and (-a, 0) and the y-intercepts are (0, b) and (0, -b).

Does this hold when a >b?  Try a = 2 and b = 1.

Again, it appears our hypothesis is true.  Verify our hypothesis with different values for a and b.

Why does the a value change the x-intercepts and the b value change the y-intercepts?  If we think about trig, this makes sense because cosine is x (and is being multiplied by a) and sine is y (and is being multiplies by b).  So what about negative values of a and b?  Since a effects the x-value, we should expect a rotation around the y-axis.  And since b effects the y-value, we should expect a rotation around the x-axis.  For the equations with odd exponents, this will not cause a visible shift in the graph, but for even exponents we should see the graph shift to the second quadrant when a is negative and b is positive. If a is positive and b is positive, we should expect the even exponent graphs to shift to the fourth quadrant.  And if a and b are both negative, we should expect the even exponent graphs to shift to the third quadrant.  Click here to see some examples.