__Parametric
Equations__

__By __

__Tonya
DeGeorge__

For this assignment, we are investigating the following:

for

We want to know how the system changes as we change the values of

aandb.

What do we get whena=b?

To begin this investigation, letÕs look at a simple value of

aandb: where both are equal to one. We get the following diagram:

It seems that when the values of

aandbare equal, we will get a circle. And in this case, we get a circle with a radius of one. However, does this change as we change the values ofaandb? Suppose we look at different values ofaandb, but still keep them equal. LetÕs try the following:

¯ Positive values

o Very small positive values

o ÒIntermediateÓ positive
values

o Very large positive values

¯ Negative values

o Very small negative values

o ÒIntermediateÓ negative
values

o Very large negative values

(Note: when

a=b= 0, the parametric equations equal zero and therefore this will not be investigated in this lesson).So letÕs begin by looking at very small positive values. Although the number 1 is a small positive value, what would happen if we look at values of

aandbsmaller than one, but still greater than zero? Would the graph change? LetÕs trya=b= 0.5:

It appears that the circle
has shrunk, but now it has a radius of ½ - which is the value we gave *a* and *b*. So does this mean
that *a* and *b* affect the size of the circle? To check this, letÕs look at all different values of *a* and *b*. To investigate
this, I chose the following values of a and b to look at:

a) *a* = *b* = 0.2

b)
*a* = *b* = 4

c)
*a* = *b* = 20

d)
*a* = *b* = 100

e)
*a* = *b* = -0.2

f)
*a* =
*b* = -4

g)
*a* =
*b* = -20

h)
*a* =
*b* = -100

Take a look at the graphs below (please take note:
the scales of the graphs had to be changed in order to see the entire picture):

__Positive values of a and b:__

a) *a* = *b* = 0.2 b) *a* = *b*
= 4

c)* a* = *b* = 20
d) *a* = *b* = 100

e) *a* = *b* = -0.2 f)
*a* = *b* = -4

g) *a* = *b* = -20
h) *a* = *b* = -100

Conclusions (for whena=b):It is clear from the graphs that when

a=b, we will get a circle. The values ofaandbdetermine the size of the circle that is drawn (with the center at the origin). We can also see that the radius of each circle is equal to the absolute value ofaandb. For example, whenaandbwere -20, the radius of the circle was |-20| = 20.

Radius of Circle = |*a*| = |*b*|

What do we get whena<b?Whenever we want to compare two values, there are different cases to consider:

á
*a* very small, *b* very small (i.e. *a* = 0.1, *b* = 0.5)

á
*a* very small, *b* ÒintermediateÓ value (i.e. * a* = 0.1, *b* = 4)

á
*a* very small, *b* very large (i.e. *a* = 0.1, *b* = 100)

á
*a* ÒintermediateÓ value, *b* very small **(exclude because a must be less than b)**

á
*a *ÒintermediateÓ value, *b* ÒintermediateÓ value (i.e. *a* = 4, *b* = 10)

á
*a* ÒintermediateÓ value, *b* very large (i.e. *a* = 4, *b* = 100)

á
*a* very large, *b* very small **(exclude because a must be less than b)**

á
*a* very large, *b *Òintermediate value **(exclude
because a must be less than b)**

á
*a* very large, *b* very large (i.e. *a* = 100, *b* = 200)

So letÕs continue with the investigation. (Please note: some of the scales of the graphs had to be changed in order to see the entire picture. The original graph (before any scale changes) are in the bottom left corner of each graph).

For:a= 0.1,b= 0.5Fora= 0.1,b= 4:

Fora= 0.1,b= 100:Fora= 4,b= 10:

__For a = 4, b = 100__:

Conclusions (for whena<b):Clearly, we can see that we no longer get a circle. In fact, due to the values of

aandb, the new shape appears to be an ellipse. From observation, we can see that the size of the ellipse is directly related to the values ofaandb. For instance, whena= 100 andb= 200, we can see that the length of the ellipse along they-axis is from -200 to 200 and the length of the ellipse along thex-axis is from -100 to 100. So it appears that the value ofacontrols the length along thex-axis and the value ofbcontrols the length of they-axis.

What do we get whena>b?Again, whenever we want to compare two values, there are different cases to consider:

á
*a* very small, *b* very small (i.e. *a* = 0.5, *b* = 0.1)

á
*a* very small, *b* ÒintermediateÓ value **(exclude
because b must be less than a)**

á
*a* very small, *b* very large **(exclude because b must be less than a)**

á
*a* ÒintermediateÓ value, *b* very small (i.e. *a* = 10, *b *= 0.1)

á
*a *ÒintermediateÓ value, *b* ÒintermediateÓ value (i.e. *a* = 10, *b* = 4)

á
*a* ÒintermediateÓ value, *b* very large **(exclude because b must be less than a)**

á
*a* very large, *b* very small (i.e. *a* = 100, *b* = 0.1)

á
*a* very large, *b *Òintermediate value (i.e. *a* = 100, *b* = 10)

á
*a* very large, *b* very large (i.e. *a* = 200, *b* = 100)

So letÕs continue with the investigation. (Please note: some of the scales of the graphs had to be changed in order to see the entire picture. The original graph (before any scale changes) are shown on the bottom (or left corner) of each graph).

Fora= 0.5,b= 0.1:Fora= 10,b= 0.1:

Fora= 10,b= 4:Fora= 100,b= 0.1:

Fora= 100,b= 10:Fora= 200,b= 100:

Conclusions (for whena>b):Again, we can see that we no longer get a circle. From observation, we can see that the size of the ellipse is directly related, again, to the values of

aandb. For instance, whena= 200 andb= 100, we can see that the length of the ellipse along they-axis is from -100 to 100 and the length of the ellipse along thex-axis is from -200 to 200. So this confirms that the value ofacontrols the length along thex-axis and the value ofbcontrols the length of they-axis.

Overall conclusions:From this investigation, we have found the following:

¯ When the value of *a* is equal to the value of *b*, we get a circle with the center at
the origin.

o The size of the circle is
dependent upon the value of *a *and *b*.

o 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*:

o *a *determines
the length of the ellipse along the *x*-axis
(or 2*a* = length along *x*-axis)

o *b *determines
the length of the ellipse along the *y*-axis
(or 2*b* = length along *y*-axis)