You use "followed by" when the order matters
If only dilations and reflections occur, the order doesn't matter, eg:
(x,y) -> (ax,y) -> (-ax,y) is the same as (x,y) -> (-x,y) -> (-ax,y), so you'd say "a reflection in the y-axis and a dilation by a factor of a parallel to the x-axis"
If you have a translation involved then the order does matter:
(x,y) -> (ax,y) -> (ax+ac,y) is the same as (x,y) -> (x+c,y) -> (ax+ac,y)
When describing it, you can do it either way around, but it's a little different in either case:
(x,y) -> (ax,y) -> (ax+ac,y)
"dilation by a factor of a parallel to the x-axis followed by a translation of ac units in the positive direction of the x-axis"
As opposed to
(x,y) -> (x+c,y) -> (ax+ac,y)
"translation of c units in the positive direction of the x-axis followed by a dilation by a factor of a parallel to the x-axis"
You can do either way as long as it's a correct description, though one is usually somewhat easier than the other (e.g. you might pick one alternative in the interest of avoiding fractions or something like that, which is fine since it usually looks nicer)
If they asked you to describe a transformation of
)
to
)
, you'd be best off solving for x:
This tells you that the transformation was:
(x,y) -> (ax+c,y) = (x',y')
It's easy to read the transformation off that:
"dilation by a factor of a parallel to the x-axis followed by a translation of c units in the positive direction of the x-axis"
EDIT: I guess I should use your transformation as a more comprehensive example:
)
is transformed to
+c)
Rearranging gives
)
So we have
and
and
This tells you the transformation:
(x,y) -> (x/b,ay+c)
"dilation by a factor of 1/b parallel to the x-axis and a factor of a parallel to the y-axis, followed by a translation of c units in the positive direction of the y-axis"
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