**Stretching Parametric Curves**

By: Russell Lawless

Here we are seeing the effects of *a* and *b* when x = *a*cos(t) and y = *b*sin(t) when 0 ≤ t ≤ 2π. First before we go into explaining the effects, try out yourself on the link below and see what you notice.

What did you find? Hopefully you found that circles and ellipses were created from this.

Let's look at Parameter *a *first. Below we have *b* set equal to 3 while *a* is equal to 1, -1, 2, and 3 in the respective order.

a = 1 a = -1 a = 2 a = 3

So from here we can see that *a* does in fact affect the horizontal shift of the image from the function. It appears that the width of the ellipse (circle) is determined by the value of *a*. We also see that when *a *is equal to *b* then the image of a circle is created. Another thing that I noticed was that when *a *was 1 and -1 it appeared that the same image was produced. However, in actuality the image of where* a *= -1 is just the reflected image of* a* = 1 over the y - axis.

Now let's look at parameter *b*. Below we have *a* set equal to 3 while *b* is equal to 1, -1, 2, and 3 in the respective order.

a = 1

a = -1

a = 2

a = 3

So from here we can see that* b* does in fact affect the vertical shift of the image from the function. It appears that the height of the ellipse (circle) is determined by the value of* b*. We also see that when* b *is equal to* a *then the image of a circle is created. Another thing that I noticed was that when* b* was 1 and -1 it appeared that the same image was produced. However, in actuality the image of where* b *= -1 is just the reflected image of *b* = 1 over the x - axis.

Hopefully you noticed the effects of *a* and *b. *Parameter *a* affects the horizontal shrinking/stretching of the image that is produced from the function while parameter *b* affects the vertical shrinking/stretching of the image that is produced from the function. When *a* and *b * are set equal to each other the image of a circle is created. For some vocabulary to have remember this:

The distance from the origin to any point of the circle is called the *radius*. When you see an ellipsis the *minor axis* is the shorter axis of an ellipsis while the *major axis *is the longer axis of an ellipsis. Both of these axes are perpendicular to each other. Also the equation to form a generic ellipse is