Urgent Transformations Help Please!

[homosapiens]

I have my first methods sac tomorrow and after looking back over my notes and doing some questions i've become confused about transformations again!

Say my original equation is y=x^2... and the order of transformations is a horizontal translation of 2 units to the left, a dilation by scale factor 3 from the y-axis, then a reflection in the y-axis. When writing out an equation showing the transformations,
- does dilation from the y-axis mean multiplying the entire contents of the brackets by (1/3)? so g(x)=f(x+2) would become g(x)=f[(x+2)/3]?
- and would the y-axis reflection mean the whole (x+2) bracket is negative?

Thanks in advance (:

[brightsky]

So we want (x,y) --> (-3(x' - 2), y')

So x = -3(x' - 2) = -3x' + 6
x' = (6-x)/3
y = y'
So sub that into the original equation:
y = x^2 ---> becomes ---> y = ((6-x)/3)^2

I'm not sure if this is right so if anyone can verify, that would be great.

EDIT: Thanks illuminati

[illuminati]

I believe the answer is
g(x) = [(-x+2)^2]/3
First the horizontal translation left is obvious.
(x+2)^2
Then the dilation, it applies to the whole term brightsky
[(x+2)^2]/3
Reflection in the y-axis is just a negative in front of the x only.
so [(-x+2)^2]/3

I'm not completely sure myself...

[brightsky]

Hmm let's test a point, say (3,9) from the original equation.
First this point is moved 2 units to the left, so it becomes (1,9)
Then it's dilated by a factor of 3 from y-axis, so it goes back to (3,9)
Then a reflection, so it becomes (-3, 9)
Now sub this into my equation:
LHS = 9
RHS = (6+3)^2 / 9 = 9
I think the order of the linear transformations is really important and something that the OP must take heed of, if not already doing so, in the SAC. Horizontal translation of 2 units to the left doesn't automatically mean that it's, say, x + 2, if there are other transformations following it. The "test" I just did above illustrates this. The (3,9) was moved two places to the left, but the dilation away from the x-axis meant that the point returned to (3,9). However, feel free to disprove anything I've said thus far, because as mentioned, dilations, etc., isn't really that clear-cut to me.

EDITED.

[illuminati]

The thing with your explanation is that instead of a reflection in the y-axis, you've reflected it in the x-axis.

Furthermore, when you've dilated it from the y axis, it doesn't go from (1,9) to (3,9), since the new turning point of your parabola is at (-2,0).... So really it goes from (1,9) to (7,9)

[brightsky]

The thing with your explanation is that instead of a reflection in the y-axis, you've reflected it in the x-axis

Oh woops. Shall make emendations.

[illuminati]

Okay, although I've already edited my previous post.
Let's use the point (2,4) as a reference this time.
Horizontal translation of 2 left to (0,4)
But when you dilated that in the y axis by 1/3, it shouldn't remain as (0,4), but according to your explanation it does...

[brightsky]

Pretty sure it does remain as (0,4).

[illuminati]

EDIT: It is still a parabola, nvm

The thing with this question is, it gets a bit confusing since the dilation is after the horizontal translation.

[brightsky]

EDIT: It is still a parabola, nvm

The thing with this question is, it gets a bit confusing since the dilation is after the horizontal translation.

Yeah it does make things confusing. I think that 'algebraic method' that I used above is pretty effective because it means you can just resort to 'mindless', methodical algebra rather than doing any logical reasoning, but still arrive at the right answer (I hope!).

And I sort of see where you're coming from, just that it is a dilation from the y-axis, not from the turning point.

[illuminati]

Wording is death in methods. =(

[luken93]

y = x^2

horizontal translation of 2 units to the left: (x,y) -> (x'-2,y')

a dilation by scale factor 3 from the y-axis: (x-2,y) -> (3(x'-2),y')
x = 3(x' - 2)
x/3 = x' - 2
1/3(x+6) = x     <---------- Forgot this step  :buck2:

then a reflection in the y-axis: (3(x-2),y) -> (-3(x'-2),y) =

x = -3(x' - 2)
-x/3 = x' - 2
-1/3(x - 6)
1/3(6 - x)

y = (1/3(6-x))^2
y = ((6-x)^2)/9

[IMG]http://img688.imageshack.us/img688/5753/63068440.jpg[/img]

I do agree though, there needs to be a better system...

[Mao]

That's an interesting CAS you got there luken, what are you using?

Also, don't think you're plotting the right thing there mate (rectified). A dilation from the y axis implies implies the x coordinate of every point is modified with respect to the y axis. That then implies the turning point itself moves from x=-2 to x=-6. Let me demonstrate:

Original:

Translation first:
, TP(-2,0)
Dilation second:
, TP(-6,0)
Reflection last:
, TP(6,0)

alternatively (this is the incorrect interpretation):
Dilation first
, TP(0,0)
Translation second
, TP(-2,0)
Reflection last
, TP(2,0)


Generally, you are told to 'dilate before translate' when you are interpreting the steps of transformations from an equation. That is because in the general form of , we assume dilation occurs before translation. To use this interpretation, we must always factorize the argument of .

Alternatively, we can interpret the order of transformation with the translation step before dilation step. The general form is (note how the argument is not factorized). Both describe equivalent transformations, it really depend on which consistent set of assumptions/interpretation you are using

When we are applying transformations in a given order however, we have to be flexible to be able to apply the transformations in the given order. That is, step by step using substitution (see my workings). This is why using the 'dilation before translation' formula falls apart here.

[luken93]

That's an interesting CAS you got there luken, what are you using?

Microsoft Maths - Very easy to use

That makes more sense, however your "alternative" yielded the same as my answer - was I taking the wrong steps?

[Mao]

That's an interesting CAS you got there luken, what are you using?

Microsoft Maths - Very easy to use

That makes more sense, however your "alternative" yielded the same as my answer - was I taking the wrong steps?

aha, but the 'alternative' is wrong, because I dilated before translation, when the question stated translation before dilation!