Parametric to Cartesian

apple2357

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I am trying to convert this parametric equation to cartesian:

x= t2/4 , y= t3/16

By rearranging the first for t and substituting , i get y = 0.5x^(3/2)
If i plot this i only get half the graph compared to the parametric form,
should i express this as y= +-0.5x^(3/2)?

Or should i avoid this completely by raising both equations to the same power of t ? In this case i get y^2 = 0.25x^3

Whats the correct answer and approach?
 
I am trying to convert this parametric equation to cartesian:

x= t2/4 , y= t3/16

By rearranging the first for t and substituting , i get y = 0.5x^(3/2)
If i plot this i only get half the graph compared to the parametric form,
should i express this as y= +-0.5x^(3/2)?

Or should i avoid this completely by raising both equations to the same power of t ? In this case i get y^2 = 0.25x^3

Whats the correct answer and approach?

Both are correct; it depends on your goal.

If you want the form "y = ...", then you need the ±. If you don't want ±, then you need both variables raised to powers.

Presumably you recognize that you need the ± because you took a square root. The way to avoid that is to not take the square root.
 
I am trying to convert this parametric equation to cartesian:
x= t2/4 , y= t3/16
By rearranging the first for t and substituting , i get y = 0.5x^(3/2)
If i plot this i only get half the graph compared to the parametric form,
should i express this as y= +-0.5x^(3/2)?
Or should i avoid this completely by raising both equations to the same power of t ? In this case i get y^2 = 0.25x^3
You can see the graphs; GRAPH I GRAPH II Do they match?
 
While we are on the subject of parametrics and cartesians..

I am wondering why we bother with a parametric form given cartesians do such a good job? My only thought on this is for mechanics? For example Projectiles but are there any other cases when working with parametric equations are preferable?
 
While we are on the subject of parametrics and cartesians..

I am wondering why we bother with a parametric form given cartesians do such a good job? My only thought on this is for mechanics? For example Projectiles but are there any other cases when working with parametric equations are preferable?

Do Cartesian equations "do such a good job"? At what? You may be biased because that's all you saw for many years, so all the problems you are used to are those that work well with non-parametric forms.

If you think about it, you can probably see that it is not hard to write a parametric form that can't be solved to eliminate the parameter. Also, parametric form can always be a function, whereas most curves can't otherwise be written as functions. And there's a lot of calculus that is a lot easier with parameters.

Take a look here for a few ideas: https://en.wikipedia.org/wiki/Parametric_equation#Applications
 
Do Cartesian equations "do such a good job"? At what? You may be biased because that's all you saw for many years, so all the problems you are used to are those that work well with non-parametric forms.

If you think about it, you can probably see that it is not hard to write a parametric form that can't be solved to eliminate the parameter. Also, parametric form can always be a function, whereas most curves can't otherwise be written as functions. And there's a lot of calculus that is a lot easier with parameters.

Take a look here for a few ideas: https://en.wikipedia.org/wiki/Parametric_equation#Applications

Thanks!
 
Still thinking about parametric and cartesian equations, I have another query

Suppose i pick x=t, y=t , if i convert this to cartesian equation its clear it gives y=x, so a straight line through the origin.
But if i decide x=sint, y= sint , this is also y=x? So does this give a straight line in the same way? I guess so?
Or any other function for example, x=e^t, y=e^t ?
How are the second cases different the first one?
 
Still thinking about parametric and cartesian equations, I have another query

Suppose i pick x=t, y=t , if i convert this to cartesian equation its clear it gives y=x, so a straight line through the origin.
But if i decide x=sint, y= sint , this is also y=x? So does this give a straight line in the same way? I guess so?
Or any other function for example, x=e^t, y=e^t ?
How are the second cases different the first one?

The latter two parametric equations restrict the range of x and y, so they represent only part of the line y=x.

On the other hand, a parametrization like x=t^3, y=t^3 would represent the entire line, just traversed at a variable rate. That is the usual difference among parametrizations.
 
The latter two parametric equations restrict the range of x and y, so they represent only part of the line y=x.

On the other hand, a parametrization like x=t^3, y=t^3 would represent the entire line, just traversed at a variable rate. That is the usual difference among parametrizations.

Thanks that makes sense. So because sinx has a restricted range, the parametric equations can only go from -1 to 1.
I found it helpful to think about how the x coordinates are changing independently of the y coordinates and the rate at which these changes are happening.
 
Thanks that makes sense. So because sinx has a restricted range, the parametric equations can only go from -1 to 1.
I found it helpful to think about how the x coordinates are changing independently of the y coordinates and the rate at which these changes are happening.

A nice way to think of parametric (Cartesian) equations is to recall the Etch-A-Sketch toy, where you move x and y separately, each as its own function of time, to move a point.
 
While we are on the subject of parametrics and cartesians..

I am wondering why we bother with a parametric form given cartesians do such a good job? My only thought on this is for mechanics? For example Projectiles but are there any other cases when working with parametric equations are preferable?
The Cartesian equation for a circle with center at (0,0) and radius R is \(\displaystyle x^2+ y^2= R^2\). y is NOT a function of x nor is x a function of y. Parametric x= Rcos(t), y= Rsin(t) also describes that circle but now x and y are functions of t.
 
A nice way to think of parametric (Cartesian) equations is to recall the Etch-A-Sketch toy, where you move x and y separately, each as its own function of time, to move a point.


Ah yes, i remember the toy well. Great analogy.
 
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