Parametric equations questions

[Dapperdub]

Hi everybody, i just joined. Here's my first post

I have a few questions.

When looking at curve C and by the eye test, it doesn't pass the VLT, even though x=f (t ) and y=g (t) are functions,...then overall with curve C not passing the VLT, is it still a function since f (t) and g (t) are?

I feel like curve C is still a function because it just shows the path of the parametric equations based on coordinates that are functions. Is my understanding correct on this assumptios?

2nd question
Can parametric equations take in f (t) and g (t) as relations or will there be an error of some sort?

3rd
Is there any relation or similarites (maybe in graphs) between compositing functions vs parametric equations?

Thanks in advance!

 

[RUber]

Hi Dapperdub, welcome to PF.
In regards to curve C, the curve plotted in the x-y plane cannot be defined simply by a "function" y= f(x). The vertical line test is to determine if y is a proper function of x. In this case they are functions of t, so if you were to plot in the x,y,t plane, you should see a unique (x,y) value for each distinct t input, so C is a function of t.

I am not sure how to address your second and third questions. Can you provide a little more detail about an application or example you are considering?

 

[Dapperdub]

Hi Dapperdub, welcome to PF.
In regards to curve C, the curve plotted in the x-y plane cannot be defined simply by a "function" y= f(x). The vertical line test is to determine if y is a proper function of x. In this case they are functions of t, so if you were to plot in the x,y,t plane, you should see a unique (x,y) value for each distinct t input, so C is a function of t.

I am not sure how to address your second and third questions. Can you provide a little more detail about an application or example you are considering?

Wow!...perfect answer for my 1st question! I figured it so and when you brought up the x, y, t plane it made it all the more clearer! Thank you!

As for clarity on my second question…let’s see if I can rephrase it.

Let’s say f(t) is an equation of a horizontal parabola and g(t) is some function …what happens in terms of its parametric graph? Does it produce two paths in curve C since there are 2 y values for the same x from the horizontal parabola f(t)?

And for question 3

Im just wondering …..It just seems like parametric equations is similar to compositing functions (i.e. f(g(x))). Are there any similarities or correlations of the 2 concepts at all?

 

[RUber]

So, you are looking at f(t) which is not a function, such as f(t) = t and -t, this is not a parabola, but for simplicity's sake it gives the same effect of having two outputs.
and if x=f(t), y = g(t) were your parametrization scheme, then yes, you would have two outputs for each t.
You will generally not see this, since it makes it very difficult to find a unique solution. In these cases, you would separate f(t) into f_+ and f_- and solve each one individually.

For the third point, yes. When possible, thinking of parametric equations as composed functions is helpful.
You might have a parametrized curve C with points x(t) and y(t), but there might be a function y = f(x). In which case y(t) can be written as f(x(t)).

There are times when these different perspectives are more useful than others, and the best advice I have seen given is to work a lot of problems and gain some intuition.

 

[Dapperdub]

So, you are looking at f(t) which is not a function, such as f(t) = t and -t, this is not a parabola, but for simplicity's sake it gives the same effect of having two outputs.
and if x=f(t), y = g(t) were your parametrization scheme, then yes, you would have two outputs for each t.
You will generally not see this, since it makes it very difficult to find a unique solution. In these cases, you would separate f(t) into f_+ and f_- and solve each one individually.

For the third point, yes. When possible, thinking of parametric equations as composed functions is helpful.
You might have a parametrized curve C with points x(t) and y(t), but there might be a function y = f(x). In which case y(t) can be written as f(x(t)).

There are times when these different perspectives are more useful than others, and the best advice I have seen given is to work a lot of problems and gain some intuition.

You're right some information is useful than others, but yet it still bothered me. I just want to make sure I understand as much angles as i possibly can about the concept of parametric.

Thank you for your insight! I can rest for a bit until the next question arises...