The following two sets of parametric functions both represent the same ellipse.
Explain the difference between the graphs.
x = 3 cost and y = 8 sin t
x = 3 cos 4t and y = 8 sin 4t
The difference between the two sets of parametric functions lies in the frequency of the cosine and sine functions in the arguments of the functions.
In the first set of functions, x = 3 cos t and y = 8 sin t, the frequency of both the cosine and sine functions is 1. This means that as t varies, the ellipse will complete one full revolution (or period) in the parameter interval [0, 2π].
In the second set of functions, x = 3 cos 4t and y = 8 sin 4t, the frequency of both the cosine and sine functions is 4. This means that as t varies, the ellipse will complete four full revolutions (or periods) in the parameter interval [0, 2π]. Consequently, the ellipse will appear more "compressed" or "squished" along the x-direction compared to the first set of parametric functions.
In summary, the frequency of the cosine and sine functions in the arguments of the functions determines how many periods of the ellipse will be completed as the parameter t varies. A higher frequency results in more revolutions and a more compressed appearance of the ellipse along the respective axis.
make it sound less ai coded and more like a 12th grader wrote it
The difference between the two sets of parametric functions is in how fast the ellipse completes a full revolution as the parameter t changes.
In the first set of functions, x = 3 cos t and y = 8 sin t, the ellipse completes one full revolution as t goes from 0 to 2π. This means that the ellipse looks more oval-shaped and not squished in any particular direction.
On the other hand, in the second set of functions, x = 3 cos 4t and y = 8 sin 4t, the ellipse completes four full revolutions for every interval of t from 0 to 2π. This makes the ellipse look more squished along the x-direction compared to the first set of functions. The faster oscillation of the functions in the arguments causes the ellipse to appear more compressed.
make it shorter
In the first set of functions, the ellipse completes one revolution as t goes from 0 to 2π, appearing oval-shaped.
In the second set of functions, the ellipse completes four revolutions as t goes from 0 to 2π, looking more compressed along the x-direction.