The following two sets of parametric functions both represent the same ellipse. Explain the difference between the graphs. x = 3 cos t and y = 8 sin t x = 3 cos 4t and y = 8 sin 4t (4 points)

Answer 1

The difference between the two sets of parametric functions lies in the frequency at which the cosine and sine functions oscillate.

In the first set of parametric functions, x = 3 cos t and y = 8 sin t, the cosine and sine functions have a frequency of 1. This means that the ellipse will complete one full revolution after the parameter t has ranged from 0 to 2π.

In the second set of parametric functions, x = 3 cos 4t and y = 8 sin 4t, the cosine and sine functions have a frequency of 4. This means that the ellipse will complete one full revolution after the parameter t has ranged from 0 to 2π/4 = π/2. In other words, the ellipse will complete four full revolutions in the same interval that the first set of parametric functions completes one revolution.

As a result, the second set of parametric functions will generate a more "squished" ellipse compared to the first set of parametric functions, as the frequency at which the functions oscillate causes the ellipse to be stretched or compressed along the x and y axes.