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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: dekoyl on December 11, 2008, 12:14:08 am

Title: Parametric domain/range
Post by: dekoyl on December 11, 2008, 12:14:08 am: I don't completely understand Essential's explanation :(
My question is: when we find the domain or range of a cartesian equation using the parametric form, why do we use the range of the trigonometric function? For example:

$x=3sin(t) y=4cos(t), t\in [-\frac{\pi}{2},\frac{\pi}{2}]$
The domain of the graph (ellipses) when sketched would be [-3,3] because the range of x = 3sin(t) is [-3,3] for $t\in [-\frac{\pi}{2},\frac{\pi}{2}]$ . And same goes for the range of the graph (range of y = 4cos(t) is [0,4] for $t\in [-\frac{\pi}{2},\frac{\pi}{2}]$

Thanks :-[
Title: Re: Parametric domain/range
Post by: unknown id on December 11, 2008, 12:24:24 am: ^ This is because the domain of x(t) [ie. the defined t values] determines the values for which x(t) is defined (its range); that is, it sets up the domain (ie. the defined values of x) of the graph of y(x), where y is in terms of x.

$\therefore$ domain t sets up the range of x(t)
$\implies$ range of x(t) sets up the domain of y(x)
Title: Re: Parametric domain/range
Post by: shinny on December 11, 2008, 12:25:28 am: The domain of the actual graph is basically what values x can equal. The only restriction we have on this is in regards to its relation with t. Using its relation with t, we know that the range of x(t) is [-3,3], so these are the actual values that x can equal. The same applies for y in regards to the range of the actual graph. I'll elaborate further if you still don't understand.
Title: Re: Parametric domain/range
Post by: dekoyl on December 11, 2008, 12:43:29 am: Quote from: shinjitsuzx on December 11, 2008, 12:25:28 am
The domain of the actual graph is basically what values x can equal.
Yep.
Quote from: shinjitsuzx on December 11, 2008, 12:25:28 am
The only restriction we have on this is in regards to its relation with t.
Yep.
Quote from: shinjitsuzx on December 11, 2008, 12:25:28 am
Using its relation with t, we know that the range of x(t) is [-3,3], so these are the actual values that x can equal.
Not so yep :(

Could you possibly elaborate, shinny? =[
Why is the domain of x(t) or y(t) never used? Why only the range?

Thanks unknown id and shinny for trying to help a slow boy out :-[
Title: Re: Parametric domain/range
Post by: Collin Li on December 11, 2008, 12:53:30 am: Quote from: unknown id on December 11, 2008, 12:24:24 am
^ This is because the domain of x(t) [ie. the defined t values] determines the values for which x(t) is defined (its range); that is, it sets up the domain (ie. the defined values of x) of the graph of y(x), where y is in terms of x.

$\therefore$ domain t sets up the range of x(t)
$\implies$ range of x(t) sets up the domain of y(x)

This explanation is gold. Try to understand it.

Basically, you use the restrictions on $t$ to find out how $x$ and $y$ are restricted - since you have $x(t)$ and $y(t)$ .

Then, the restrictions on $x(t)$ - the range of $x(t)$ - become the domain of $y(x)$ .

The restrictions on $y(t)$ - the range of $y(t)$ - become the range of $y(x)$ .
Title: Re: Parametric domain/range
Post by: shinny on December 11, 2008, 12:56:24 am: Might be easier if I just list them out systematically
1. The domain of a typical $y=f(x)$ graph is the values that x can take
2. The range of a typical $y=f(x)$ graph is the values y can take
3. The parametrics state that x=g(t), y=h(t)
4. The values that x can take (i.e. the domain!? link this back to statement 1) is equal to the values that g(t) can take, because x=g(t)
5. The values that g(t) can take is basically the range of g(t)
6. Therefore the domain of x is the range of g(t)
Apply a similar train of thought for y to find the range

I might have to write up an example or something to make it even more obvious if you don't get this =\
Title: Re: Parametric domain/range
Post by: dekoyl on December 11, 2008, 12:59:15 am: F''' yeah. Thanks everyone I get it :D

$ctrl+C, ctrl+V, ctrl+P, enter$
Title: Re: Parametric domain/range
Post by: dekoyl on December 11, 2008, 12:07:14 pm: Are parametric stuff usually in the non-calculator able exam?

Sorry double post =P
Title: Re: Parametric domain/range
Post by: shinny on December 11, 2008, 12:09:22 pm: It can be in either exam as it'd be part of a larger vector calculus question.