What order do you transform graphs in?

[jetwaterluffy]

Homework Statement

In my A2 maths class, we were doing revision on transformations of graphs, as in:

Homework Equations

with a graph f(x)
af(x) is a stretch scale factor a in the y-direction
f(bx) is a stretch scale factor 1/b in the x-direction
f(x)+c is a translation of c in the y- direction
f(x+d) is a translation of d in the negative x- direction
anyway, back to 1. Homework Statement
When a question came up where we had to do multiple transformations, and the order in which you do them mattered, most of my class got the question wrong, and when we asked the teachers how to do it, the 3 of them consulted, and they essentially said "try out different orders until you get the right answer", which seems awfully crude and time consuming to me, so I am wondering if there is a specific order in which you can do it to get the right answer. If not, why not? (please answer within the next week, as my exam is soon!)

The Attempt at a Solution

My hunch is you do:
stretches in y,
stretches in x,
translations in y,
translations in x, but that is only a hunch and I don't actually know, so please help!

 

[rock.freak667]

What exactly was the graph that you had to transform?

The way I would do it would be how your hunch said to do it.

 

[eumyang]

My hunch is you do:
stretches in y,
stretches in x,
translations in y,
translations in x, but that is only a hunch and I don't actually know, so please help!

No, that's not right. For multiple translations, I would follow the "order of operations", more or less. But things get confusing when there are multiple horizontal translations. For instance, if I have a transformed function
$a \cdot f(bx - c) + d$,
first rewrite as
$a \cdot f(b(x - c/b)) + d$,
because, within the parentheses, if you have a horizontal stretch/shrink and a translation, f(bx - c) does NOT have a horizontal translation of c units, but c/b units, due to the horizontal stretch/shrink.

Starting with
$f(x)$,
the order of transformations would be as follows:
$f(x - c/b)$
-> translation right/left c/b units,
$f(b(x - c/b)) = f(bx - c)$
-> horizontal stretch/shrink by a factor of b,
$a \cdot f(bx - c)$
-> vertical stretch/shrink by a factor of a, and
$a \cdot f(bx - c) + d$
-> translation up/down d units.


Hope I'm not giving too much away.

 

[jetwaterluffy]

What exactly was the graph that you had to transform?

The way I would do it would be how your hunch said to do it.

I'm looking for a general rule really, as I could be asked about it for any graph in the exam.

No, that's not right. For multiple translations, I would follow the "order of operations", more or less. But things get confusing when there are multiple horizontal translations. For instance, if I have a transformed function
$a \cdot f(bx - c) + d$,
first rewrite as
$a \cdot f(b(x - c/b)) + d$,
because, within the parentheses, if you have a horizontal stretch/shrink and a translation, f(bx - c) does NOT have a horizontal translation of c units, but c/b units, due to the horizontal stretch/shrink.

Starting with
$f(x)$,
the order of transformations would be as follows:
$f(x - c/b)$
-> translation right/left c/b units,
$f(b(x - c/b)) = f(bx - c)$
-> horizontal stretch/shrink by a factor of b,
$a \cdot f(bx - c)$
-> vertical stretch/shrink by a factor of a, and
$a \cdot f(bx - c) + d$
-> translation up/down d units.


Hope I'm not giving too much away.

Thanks, this is really helpful.