The following are a view of parametric
equations given the range of t from 0 to 2 pi. We will look at the pair
of functions noted below as the power is squared, cubed, and raised higher.
We will also pay particular attention to a and b, when a is equal to b,
when a is greater than b, and when a is less than b for each power indicated.
Notice in the figures given below, the
values of a and b are the same and the power of the function is 1 (odd power).
The figure is a circle and the values of a and b correspond with the length
of the radii.
The following figures are for the pair
of equations given below. Here the values of a are either greater than b
or less than b, but not equal to b. Notice the figures resulting are that
of an ellipse. The value given for a
corresponds as an x value and the value for
as a y value.
The pair of equations below are squared
(even power). Now the result becomes a straight line where once again the
a value corresponds with x and the b value corresponds with y.
Now look at the pairs of equations below.
Here is the cubed case (odd power). We get a closed figure with more of a concave
curve to the shape. I think by now we can settle on the idea that the a
values match with x and the b values match with y.
Here is the case with the 4th power (even power).
Notice that we get a curved line.
The following are equations for you to
try. What kind of shape do you think will result for each and where will
each value correspond to the x and y axis?
Scroll below to see if your guesses are
After exploring graphs of parametric equations
of the form there are several
conclusions which can be stated:
the resulting graph will be a closed figure.
Power of 1: If a is equal to b, then the
figure is a circle. If a is greater than or less than b then the figure
is an ellipse.
EVEN POWER: the resulting graph will be a segment or curve connecting from
the x and y axis for the values which
correspond with a (x-intercept) and b (y-intercept).
Power of 2: The graph will be a sement
with the the x and y intercepts matching with the numerical value of a and
INCREASE OF POWER: As the powers increase above 2, the figure(s) take on a curve.
The graphs become
increasingly more curved as the value of
the power continues to increase
a and b
The values of a correspond with the x-intercepts
for each graph.
The values for b correspond with the y-intercepts