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)
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.