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 xintercepts are (a, 0) and (a,
0) and the yintercepts 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.
Click here for some additional graphs.
Why does the a value change the xintercepts and the b value change the yintercepts?
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 xvalue, we should expect a rotation around the yaxis.
And since b effects the yvalue, we should expect a rotation around the xaxis.
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.