transformation qs help!

[kiimberly]

I'm currently using the essential book for transformations and the examples really suck. The question is:

-dilation by a factor of 2 from x axis
-reflection in the x axis
-translation 3 units in the positive direction of the x-axis
-translation 4 units in the negative direction of the y-axis

y=x^2

I get how to work it out by just looking it at and applying the transformation ( like in maths quest) but i'm a tad confused when it comes to mapping.

(x,y)-->(x,2y)--->(x,-2y)--->(x+3,-2(y-4))

then y'=-2(y-4)
and x'=x+3

then you transpose to find x and y and sub it back into the original equation. But do you expand the -2(y-4) because in the essentials book they don't expand?? i'm so confused

[Stick]

I'm not sure what you mean by this. Does it want you to find the equation with these translations?

Well, first of all, a dilation by a factor of 2 makes it y=2x^2
A reflection in the x axis inverts the graph to y=-2x^2
Then we have the translation points for x and y. So 3 in the positive direction of the x axis makes the equation y=-2(x-3)^2 (Alternatively: y=2(3-x)^2)
4 in the negative direction of the y axis makes the equation y=-2(x-3)^2-4 (Alternatively: y=2(3-x)^2-4)

Unless I've interpreted your question wrong, there's no real 'working out' for these styles of questions. I hope this helps. :S

[kiimberly]

sorry for the vagueness of the question, but yeah your suppose to find the equation. I get how to find the equation by looking at it, but when it comes to using mappings, i just get confused :L

[Greatness]

Yes, once you get the y' and x' sub it back into the original equation and rearrange for y and you will get the new equation.

[Phy124]

I used to do this (although in my head):












Alternatively, using your method:

You wrote;

(x,y)-->(x,2y)--->(x,-2y)--->(x+3,-2(y-4))

It should be:



Which leads to:















Lastly, for future reference, with transformation matrices:



- a produces a dilation by a factor of "a" in the x-direction (*Note: a negative if front of the "a" will produce a reflection about the y-axis)
- b produced a dilation by a factor of "b" in the y-direction (*Note: a negative if front of the "b" will produce a reflection about the x-axis)
- c translates the graph "c" units in the x-direction
- d translates the graph "d" units in the y-direction

We have:
- a dilation by a factor of 2 from the x-axis
- a reflection in the x-axis
- a translation 3 units in the positive direction of the x-axis
- a translation 4 units in the negative direction of the y-axis

So our matrix will be:






etc. (same as above)

[Planck's constant]

I'm currently using the essential book for transformations and the examples really suck. The question is:

-dilation by a factor of 2 from x axis
-reflection in the x axis
-translation 3 units in the positive direction of the x-axis
-translation 4 units in the negative direction of the y-axis

y=x^2

I get how to work it out by just looking it at and applying the transformation ( like in maths quest) but i'm a tad confused when it comes to mapping.

(x,y)-->(x,2y)--->(x,-2y)--->(x+3,-2(y-4))

then y'=-2(y-4)
and x'=x+3

then you transpose to find x and y and sub it back into the original equation. But do you expand the -2(y-4) because in the essentials book they don't expand?? i'm so confused

You have made one mistake:

(x,y)-->(x,2y)--->(x,-2y)--->(x+3,-2(y-4))   should read    (x,y)-->(x,2y)--->(x,-2y)--->(x+3,-2y-4)

Everything else is spot on, and the way these problems are meant to be done

[dinosaur93]

~My♥Little♥Pony~, where on earth did you use LaTeX, The last time I check, Its still not working on atarnotes, yeah?

[b^3]

~My♥Little♥Pony~, where on earth did you use LaTeX, The last time I check, Its still not working on atarnotes, yeah?

He just linked to images that the latex generator spits out. -> http://www.codecogs.com/latex/eqneditor.php
e.g.

Code: [Select]

[img][url=http://latex.codecogs.com/gif.latex?f(x)=x]http://latex.codecogs.com/gif.latex?f(x)=x[/url]^{2}[/img]Results in

[Phy124]

~My♥Little♥Pony~, where on earth did you use LaTeX, The last time I check, Its still not working on atarnotes, yeah?

Yes, currently the site which supports AN's latex code is down, so I use this and then copy the link to the image produced. Alternatively you can just add your latex code to the end of latex.codecogs.com/gif.latex? e.g. x² = latex.codecogs.com/gif.latex?x^2 and wrap these in img tags

Code: [Select]

[img][/img]
edit: Beaten, thanks b^3 

[kiimberly]

I'm currently using the essential book for transformations and the examples really suck. The question is:

-dilation by a factor of 2 from x axis
-reflection in the x axis
-translation 3 units in the positive direction of the x-axis
-translation 4 units in the negative direction of the y-axis

y=x^2

I get how to work it out by just looking it at and applying the transformation ( like in maths quest) but i'm a tad confused when it comes to mapping.

(x,y)-->(x,2y)--->(x,-2y)--->(x+3,-2(y-4))

then y'=-2(y-4)
and x'=x+3

then you transpose to find x and y and sub it back into the original equation. But do you expand the -2(y-4) because in the essentials book they don't expand?? i'm so confused

You have made one mistake:

(x,y)-->(x,2y)--->(x,-2y)--->(x+3,-2(y-4))   should read    (x,y)-->(x,2y)--->(x,-2y)--->(x+3,-2y-4)

Everything else is spot on, and the way these problems are meant to be done

thanks, but i thought that you were meant to use brackets? In the essentials book example they used brackets

[Planck's constant]

I'm currently using the essential book for transformations and the examples really suck. The question is:

-dilation by a factor of 2 from x axis
-reflection in the x axis
-translation 3 units in the positive direction of the x-axis
-translation 4 units in the negative direction of the y-axis

y=x^2

I get how to work it out by just looking it at and applying the transformation ( like in maths quest) but i'm a tad confused when it comes to mapping.

(x,y)-->(x,2y)--->(x,-2y)--->(x+3,-2(y-4))

then y'=-2(y-4)
and x'=x+3

then you transpose to find x and y and sub it back into the original equation. But do you expand the -2(y-4) because in the essentials book they don't expand?? i'm so confused

You have made one mistake:

(x,y)-->(x,2y)--->(x,-2y)--->(x+3,-2(y-4))   should read    (x,y)-->(x,2y)--->(x,-2y)--->(x+3,-2y-4)

Everything else is spot on, and the way these problems are meant to be done

thanks, but i thought that you were meant to use brackets? In the essentials book example they used brackets

No, you simply interpret the mapping intstructions literally in this type of problem.

You may have got yourself confused with another type of problem.
To put things in perspective, there are two basic types of transformation problems in VCE Maths

Type I problem : You are given an equation, and some mapping intstructions and you are asked to find the image of the given equation under the mapping instructions. This type of problem (Type I) you do exactly as you did it (except for your brackets - not needed)

Type II problem : You are given the equation of the IMAGE and you are required to fine the transformation steps which mapped some STANDARD function (eg sinx or x^2) to the given image. In this type of problem you have to re-arrange the equation of the image to 'look like' the standard function, which means that brackets must be introduced. You then follow a process opposite to the Type I problem, ie solve for (x', y') to find what the mapping of (x, y) -> (x', y') HAD BEEN.

Everything else eg, matrix methods etc are simply fancier ways of solving those basic Type I and Type II problems

[kiimberly]

okay thank you! i get it now