Question

PLEASE HELP!!!!

Graph each pair of parametric equations.

1. x=3sin^3(t)
y=3cos^3(t)

2. x=7sin(t)+sin(7t)
y=7cos(t)+cos(7t)

3. x=2t
y=t+5, -2<=t<=3

4. x=2t-1
y=t^2+5, -4<=t<=4

5. x=6sin(t)
y=6cos(t), 0<=t<=2pi

Answer 1

I have no clue how to do this and my lesson explains it poorly

Answer 2

@SithsAndGiggles

Answer 3

Parametric equations like these can be considered to be rules for both the \(x\) and \(y\) coordinate of points, where each coordinate is dependent on the parameter \(t\).

To graph, you would take a range of \(t\) values and plug them into each equation \(x(t)\) and \(y(t)\), then plot the point you get.

For example, if you had
\[\begin{cases}x(t)=2t\\
y(t)=t^2\end{cases}\]
then you could take all the values of \(0\le t\le4\), for instance, and you'd have a bunch of points like so:
\[\begin{matrix}
\underline{t}&&\underline{x}&&\underline{y}&&\underline{(x,y)}\\
0&&2(0)=0&&0^2=0&&(0,0)\\
1&&2(1)=2&&1^2=1&&(2,1)\\
2&&2(2)=4&&2^2=4&&(4,4)\\
3&&2(3)=6&&3^2=9&&(6,9)\\
4&&2(4)=8&&4^2=16&&(8,16)
\end{matrix}\]
which gives a rough plot like so:

Notice that with some equations, you can represent them in rectangular coordinates, which would involve being able to express one coordinate explicitly in terms of another. This involves a method called "eliminating the parameter", which, as the name suggests, involves getting rid of \(t\) and writing solely in terms of \(x\) and \(y\).

Keeping with this example, you could solve for \(t\) in the first equation so you have \(t=\dfrac{1}{2}x\), then substitute into the second one, giving you \(y=\left(\dfrac{1}{2}x\right)^2=\dfrac{1}{4}x^2\). If you can graph a parabola, you can graph a parametric representation of that parabola.

Eliminating the parameter isn't always easy or possible, but you can always rely on the method of plugging in \(t\) values.