# When transforming a graph, does it matter which order you list the transformations?

I.e. Say if it asks Describe the geometrical transformation to take y=cosx to y=5cos(x+3)

And what would the transformation be with something a little more difficult like
Y=cosx to y=5cos(3x+2)
Thanks guys

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it definitely matters whether you first shift and then stretch the graph vertically, or the other way around. though in some cases it doesn't, such as in your first example where you stretch in one direction and shift in the other...
i think it would be

1) enlargement sf 1/3 along x axis
2) translation (-2, 0)
3) enlargement sf 5 along y axis

It defo matters what order you put it
Original post by kirstyy93
i think it would be

1) enlargement sf 1/3 along x axis
2) translation (-2, 0)
3) enlargement sf 5 along y axis

It defo matters what order you put it

Alright, thankyou.
But then, how do i decide which goes first?
Original post by DifferentiationKin
Alright, thankyou.
But then, how do i decide which goes first?

It's hard to explain, but everything that changes inside the brackets must always happen before anything that happens outside the brackets, if that makes sense

So like y=cosx translated to 5(3cosx+2) the change inside the brackets must happen first which is enlargement sf 1/3 followed by translation (-2,0) and then because the 5 is on the outside, it means all of it is multiplied by 5, so it must come last and so its an enlargement sf 5

I'm really sorry if that's a bad explanation, but it's kinda hard to write it out :/
Original post by DifferentiationKin
I.e. Say if it asks Describe the geometrical transformation to take y=cosx to y=5cos(x+3)

And what would the transformation be with something a little more difficult like
Y=cosx to y=5cos(3x+2)
Thanks guys

Any linear transformation of $y=\mathrm{f}(x)$ can be written as

$y=d(\mathrm{f}(b(x+a))+c)$

Where a,b,c and d are constants. The order we apply the transformation is given by a,b,c,d.

The curve is shifted 'a' along. If a>0 then it's to the left. If a<0 then it's to the right.

The curve is then squeezed/stretched in the x-direction by a factor of $\frac{1}{b}$. If b is negative, consider what happens if it is positive, then flip everything in the y-axis.

The curve is then moved up or down by an amount 'c'.

Finally the curve is squeezed or stretched in the y-direction by a factor of d. If d is negative, consider what happens if it is positive, then flip everything in the x-axis.

So 'a' and 'c' give us physical movements of the curve, i.e. right or left, up or down. 'b' and 'd' give us squeezes and stretches - in the x-direction and y-direction respectively.

Edit: so for y=cos(x) to y=5cos(3x+2), we write this as

$y=5(\mathrm{cos}(3(x+\frac{2}{3}))+0)$

i.e. a=$\frac{2}{3}$, b=3, c=0 and d=5.

This transformation of y=cos(x) is then relatively straightforward from the steps outlined above. It is moved $\frac{2}{3}$ along to the left, stretched in the x-direction by a factor of $\frac{1}{3}$, not moved up or down, and then stretched in the y-direction by a factor of 5.
(edited 12 years ago)
Original post by Zii
Any linear transformation of $y=\mathrm{f}(x)$ can be written as

$y=d(\mathrm{f}(b(x+a))+c)$

The order is a,b,c,d.

The curve is shifted 'a' along. If a>0 then it's to the left. If a<0 then it's to the right.

The curve is then 'squeezed' in the x-direction by a factor of b (or if 0<b<1, stretched). If b is negative, consider what happens if it is positive, then flip everything in the y-axis.

The curve is then moved up or down by an amount 'c'.

Finally the curve is squeezed or stretched in the y-direction by a factor of d.

I have just been told that it goes on the order of RST- Reflection, stretch, translation.

So...now im confused. Are you correct or them?
Ah man, thankyou. Looking at it the way you presented is easy. So just to recap it makes aboslutely NO difference in which order the transformations are applied, EVER?
BTW i MUCH apreciate your help man
Original post by DifferentiationKin
Ah man, thankyou. Looking at it the way you presented is easy. So just to recap it makes aboslutely NO difference in which order the transformations are applied, EVER?

As long as you work out the correct stretch scale factors, and physical movements, no.
Original post by DifferentiationKin
BTW i MUCH apreciate your help man

re-read what I wrote. I made a slight error - - it is a stretch of a factor of 1/b.

Other than that I am right. I don't think what they've put actually contradicts anything, it's just another way of phrasing the same thing.
Horizontal and vertical transformations do not affect each other. Thus the only problematic questions are those with $f(ax+b)$ and $af(x) + b$. In the case of the former, the graph is first translated and then stretched. In case of the latter, the graph is first stretched and then translated.
(edited 12 years ago)
Original post by porkstein

Original post by porkstein
Horizontal transformations and vertical transformations do not affect each other. Thus the only problematic questions are those with $f(ax+b)$ and $af(x) + b$. In these cases, the graph is first stretched and then translated.

f(ax+b) is a translation then a stretch. Think of it this way:

When you change to f(x+b), you replace x with x+b so you translate by -b in the x direction

When you change to f(ax+b), you replace x with ax so you stretch by factor 1/a in the x direction

Like this: http://www.wolframalpha.com/input/?i=y%3D%282x%2B1%29^2%2C+y%3D%28x%2B1%29^2%2C++y%3Dx^2

If you were to do the stretch before the translation then you would get a different result.
Original post by porkstein
Horizontal transformations and vertical transformations do not affect each other. Thus the only problematic questions are those with $f(ax+b)$ and $af(x) + b$. In these cases, the graph is first stretched and then translated.

For the first one, I thought it was translated then stretched? And for the other one it doesn't matter as it would still be the same graph?

I'm confused (even worse, the exam's tomorrow )
Guys, there is SO many conflicting views here.
Seriously, does anybody KNOW the answer?
Original post by DifferentiationKin
Guys, there is SO many conflicting views here.
Seriously, does anybody KNOW the answer?

Some of the stuff here is correct, some of it is wrong. The only post I agree with entirely (apart from my own) is post 2.

Take a look at this example:

y= c*f(bx+a)+d

For this, you do the following;

Translate by -a in the x direction
Stretch by factor 1/b in the x direction
Stretch by factor c in the y direction
Translate by d in the y direction

There is a certain amount of room for changing the order here (depending on what you're changing), but I won't go through that as it will confuse you. Just learn it in this order.
Won't it be -a/b ?
Is there any resource online that covers this? There is so many different views here that i dont know what to believe without some proper verification.
Sorry if I'm adding more confusion here, but I always thought of it as:

x stretch, x shift; y stretch, y shift
Original post by ttoby
Some of the stuff here is correct, some of it is wrong. The only post I agree with entirely (apart from my own) is post 2.

Take a look at this example:

y= c*f(bx+a)+d

For this, you do the following;

Translate by -a in the x direction
Stretch by factor 1/b in the x direction
Stretch by factor c in the y direction
Translate by d in the y direction

There is a certain amount of room for changing the order here (depending on what you're changing), but I won't go through that as it will confuse you. Just learn it in this order.

Why have you got translate before stretch once, and stretch before translate for the second set? I would have done it the other way (stretch, translate, stretch, translate) as that actually follows some kind of ruling that makes sense.

I guess it would depend on whether it's bx + a or b(x+a)?
Original post by Pendulum

Original post by Pendulum
Why have you got translate before stretch once, and stretch before translate for the second set? I would have done it the other way (stretch, translate, stretch, translate) as that actually follows some kind of ruling that makes sense.

I guess it would depend on whether it's bx + a or b(x+a)?

I'll prove why my method works for f(bx+a).

Let g(x)=f(x+a).

Then to get g(x) we have to translate f(x) by -a in the x direction.

We have g(bx)=f(bx+a).

To get g(bx) we have to stretch g(x) in the x direction by factor 1/b. But we know g(x) was obtained by a translation.

Therefore, to get f(bx+a) we translate by -a in the x direction then stretch by 1/b in the x direction.

To answer your other question, yes it would be different for b(x+a).