Further
Investigation of Parametric Curves
By
Tonya
DeGeorge
In this investigation, we will be looking at the following equations:
We will begin this investigation by looking at various values of a and b. There are three cases to consider: a < b, a = b, a > b. This part of the investigation was done in a previous assignment. To see the results in full detail, please click here.
The conclusions from the last investigation showed the following:
¯ When the value of a is equal to the value of b, we get a circle with the center at the origin.
á The size of the circle is dependent upon the value of a and b.
á Radius of the circle = |a| = |b|
¯ When a < b or when a > b, the circle changes into an ellipse.
¯ The size of the ellipse depends
on the values of a
and b:
á
a
determines
the length of the ellipse along the x-axis (or 2a = length along x-axis)
á b determines the length of the ellipse along the y-axis (or 2b = length along y-axis)
We will continue this investigation by considering what happens to the graphs when we add in another term. For instance the equations then become:
Cases to consider here are for values of a and b, as well as for values of h. We have the following to investigate:
a < b |
a = b |
a > b |
h < 0 |
h < 0 |
h < 0 |
h = 0 |
h = 0 |
h = 0 |
h > 0 |
h > 0 |
h > 0 |
However, we already know what happens to the graphs when h = 0; we get a circle when a = b, and an ellipse when a < b or when a > b (see above).
So letÕs investigate the other situations.
When a < b, and when h < 0:
LetÕs use a = 2, b = 4, h = -2, we get:
When a < b, and when h > 0:
LetÕs use a = 2, b = 4, h = 2, we get:
When a = b, and when h < 0:
LetÕs use a = 4, b = 4, h = -2, we get:
When a = b, and when h > 0:
LetÕs use a = 4, b = 4, h = 2, we get:
When a > b, and when h < 0:
LetÕs use a = 4, b = 2, h = -2, we get:
When a > b, and when h > 0:
LetÕs use a = 4, b = 2, h = 2, we get:
From observing these graphs, it appears that the value of h changes the direction of the ellipse (or circle). For example, when h < 0, the ellipse appears to be ÒslantedÓ down (from left to right), ÒmovingÓ from quadrant II to quadrant IV. However, when h > 0, the ellipse appears to be Òslanted upÓ (from left to right), ÒmovingÓ from quadrant III to quadrant I.
Final Conclusions:
¯ When the value of a is equal to the value of b, we get a circle with the center at the
origin.
á
The size of the circle is dependent upon the value of a and b.
á
Radius of the circle = |a| = |b|
¯ When a
< b or
when a
> b,
the circle changes into an ellipse.
¯ The size of the ellipse depends on the values of a and b:
á
a
determines
the length of the ellipse along the x-axis (or 2a = length along x-axis)
á
b determines the length of
the ellipse along the y-axis (or 2b = length along y-axis)
¯ The value of h determines the ÒdirectionÓ of the ellipse
á
when h < 0, the ellipse ÒslantsÓ down from quadrant II to quadrant IV
á
when h > 0, the
ellipse ÒslantsÓ up from quadrant III to quadrant I