Question

The following two sets of parametric functions both represents the same ellipse. Explain the difference between the graphs.

X = 3 cos t and y = 8 sin
X = 3 cos 4t and y = 8 sin 4t

Answer 1

[AI-generated answer — Bot GPT 3.5]

Both sets of parametric functions represent ellipses with major axis of 8 units and minor axis of 6 units, centered at the origin. However, the main difference between the two graphs lies in the speed at which the parameter t is changing.

In the first set of parametric functions, t is increasing at a constant rate. This results in a smooth, continuous elliptical shape.

In the second set of parametric functions, t is increasing 4 times as fast. This causes the ellipse to be "traced out" 4 times quicker, leading to a "crazier" looking graph that appears to oscillate more rapidly.

In summary, while both sets of parametric functions represent the same ellipse, the second set will appear more compressed and oscillate faster due to the faster rate at which the parameter t is changing.