Finding x and y Equations without t: Tips and Tricks

[devanlevin]

given x(t)=30t-t$$^{3}$$
y(t)=22t-4t$$^{2}$$
what i need to do in this excercise is fine an equation with x as a function of y x(y) or the opposite y(x)---- 1 function without any variable t.
i have managed up till now by finding an expression for t and plugging that in instead of t in the 2nd equation, problem here is the extra t's, sqared and cubed, tried dividing both by t-not much help.. any ideas?

 

[HallsofIvy]

given x(t)=30t-t$$^{3}$$
y(t)=22t-4t$$^{2}$$
what i need to do in this exercise is fine an equation with x as a function of y x(y) or the opposite y(x)---- 1 function without any variable t.
i have managed up till now by finding an expression for t and plugging that in instead of t in the 2nd equation, problem here is the extra t's, sqared and cubed, tried dividing both by t-not much help.. any ideas?

There won't be one way to do that because the two functions are not one-to-one: x(0)= 0 and x($\sqrt{30})= 0$. There is no single function that will give both of those.
Similarly, y(0)= y(11/2)= 0.

What you can do is, for example, is treat the second equation as a quadratic equation in t: 4t²- 22t+ y= 0 and use the quadratic formula to solve for t:
$$t= \frac{22\pm\sqrt{(22)^2- 4(4)y}}{8}= \frac{22\pm\sqrt{484- 16y}}{8}$$
It's that "$\pm$" that is the problem. Choose either "+" or "-" and put that value of t into the equation x= 30t- t³ to get x as a function of y.

 

[Дьявол]

Do you mean like composition y(t) $\circ$ x(t) = h(t), or y(x(t))=h(t).

For y(30t-t³)=22(30t-t³)-4(30t-t³)²

Can you continue out of here?

 

[devanlevin]

no, can't continue from there, i don't see how you got there either, from what i see you did was placed (30t-t^3) in place of "t", which doesn't seem right to me, i need an equation that will be y as a function of x, that i can plug in a given y/x and find the other.

 

[HallsofIvy]

Do you mean like composition y(t) $\circ$ x(t) = h(t), or y(x(t))=h(t).

For y(30t-t³)=22(30t-t³)-4(30t-t³)²

Can you continue out of here?

no, can't continue from there, i don't see how you got there either, from what i see you did was placed (30t-t^3) in place of "t", which doesn't seem right to me, i need an equation that will be y as a function of x, that i can plug in a given y/x and find the other.

No, that's not what you want to do: you want to convert parametric equations to a Cartesian equation, not find a composition. The simplest thing to do is to solve for t as a function of y, as I showed and then replace t by that expression in the formula for x.

 

[devanlevin]

so what would i do to solve that, can't see what youre doing from here