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Old 22nd May 2013, 06:35 PM   #1 (permalink)
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Math Qn on transformation

Hi Guys! Wld appreciate if you can help me solve this qn. Pardon me if it seems so easy.

The Curve C1 is given by the parametric eqns:

x = sin(t)cos(t) , y=2cos^2(t)

The Curve C2 is given by the parametric eqns:

x = sin(2t) , y = 0.5cos(2t)

Describe a sequence of three transformations which map C1 to C2.

Thanks for ur help!

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Old 22nd May 2013, 07:29 PM   #2 (permalink)
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Re: Math Qn on transformation

okay,
1. Scale 1/2 units along the x-axis.
2. Translate -1 unit in the positive y direction
3. Scale 0.5 units along the y axis.


Last edited by megaman123; 22nd May 2013 at 08:01 PM. Reason: Opps careless.
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Old 22nd May 2013, 07:45 PM   #3 (permalink)
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Re: Math Qn on transformation

Hi megaman, can you provide me with ur steps to derive ur ans?

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Old 22nd May 2013, 08:04 PM   #4 (permalink)
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Re: Math Qn on transformation

Original graph C1,
x=sin(t)cos(t)=0.5sin(2t)
y=2cos^2(t)=2cos^2(t)-1 +1=cos(2t)+1

So to get C2,
x=0.5sin(2t) -> x=sin(2t)
y=cos(2t)+1 -> y=0.5cos(2t)

I guess from here the transformation is pretty clear.

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Old 22nd May 2013, 08:27 PM   #5 (permalink)
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Re: Math Qn on transformation

I do it this way.

New x is TWICE of old x. Because sin2t = 2 sint cos t.
New y is HALF of (old y MINUS 1). Because cos2t = 2(cost)^2 - 1.

So the x-transformation is just scale horizontally by 2.
And the y-transformation is just translate vertically by -1, then scale vertically by 1/2.


Last edited by MadHat; 22nd May 2013 at 08:29 PM.
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Old 22nd May 2013, 08:31 PM   #6 (permalink)
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Re: Math Qn on transformation

Megaman, you sure your x-transformation is correct? Scale by 1/2 along x-axis?

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Old 22nd May 2013, 08:36 PM   #7 (permalink)
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Re: Math Qn on transformation

Yes, cos our transformation is like:
f(x) -> f(2x)
u scale 0.5 units of x instead of scaling it by 2.

See y=x^2 and y=(2x)^2:http://www.wolframalpha.com/input/?i...3D%282x%29%5E2

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Old 22nd May 2013, 08:57 PM   #8 (permalink)
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Re: Math Qn on transformation

For the x-transformation, I don't think we are transforming y = f(x) to y = f(2x). The parametric form cannot be converted directly to the cartesian form like that. I find it safer to directly compare the new x with the old x and avoid going through the cartesian form altogether.

I plotted the two parametric graphs online at desmos.com. Here is what I got:



Cool dude, it's not such an easy question after all. Quite tricky this one. I may be wrong, and megaman may still be right, but the parametric plots seem to be quite clear. I don't know. See what megaman say first.

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Old 22nd May 2013, 09:08 PM   #9 (permalink)
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Re: Math Qn on transformation

Originally Posted by MadHat View Post
For the x-transformation, I don't think we are transforming y = f(x) to y = f(2x). The parametric form cannot be converted directly to the cartesian form like that. I find it safer to directly compare the new x with the old x and avoid going through the cartesian form altogether.

I plotted the two parametric graphs online at desmos.com. Here is what I got:



Cool dude, it's not such an easy question after all. Quite tricky this one. I may be wrong, and megaman may still be right, but the parametric plots seem to be quite clear. I don't know. See what megaman say first.
Yep i am wrong on this one, should be a scale of 2. Cos you have direct scaling from 0.5(sint[2t]) to sin[2t].

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Old 22nd May 2013, 09:59 PM   #10 (permalink)
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Re: Math Qn on transformation

ok cool ..

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Old 22nd May 2013, 10:44 PM   #11 (permalink)
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Re: Math Qn on transformation

Thanks guys for the help!!

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