When writing a function whose graph represents a combination of transformations, is the order not important?
depends on the transformation.
e.g. suppose we have f: (x,y) ----> (x,-y) , a reflection in the x-axis , and
g: (x,y) ---> (-x,-y) , a reflection in the origin
if we do fg
(x,y) ---> (x,-y) ---> (-x,y)
and if we do gf
(x,y) = (-x,-y) ---> (-x,y) , we get the same result, so for that combination the order does not matter,
but,
if u: (x,y) --- (x+2, y-1), a translation
and v: (x,y) ---> (-x,-y) , a reflection in the origin
uv ---(x,y) ---> (x+2,y-1) ---> (-x-2, -y+1)
vu ---(x,y) ---> (-x,-y) ---> (-x+2, y+1) , which is totally different
So in the last case, the order does matter
When writing a function with a graph that represents a combination of transformations, the order of the transformations is indeed important. The reason for this is that the order in which the transformations are applied affects the final result of the function.
To understand this concept, let's consider an example. Let's say we have a function y = f(x) and we want to apply two transformations: a translation and a reflection.
If we apply the translation first, the graph will be shifted horizontally or vertically before the reflection. On the other hand, if we apply the reflection first, the graph will be reflected and then shifted.
For instance, if we have the function f(x) = x^2 and we want to reflect it about the y-axis and then translate it 3 units up, we can follow the following steps:
1. Reflect about the y-axis: Replace x with -x. The function becomes f(x) = (-x)^2 = x^2.
2. Translate the graph 3 units up: Add 3 to the function. The function becomes f(x) = x^2 + 3.
However, if we want to apply the translation first, followed by the reflection, we would need to swap the order of the transformations.
So, when dealing with a combination of transformations, it is crucial to consider the order in which they are applied, as it will affect the final result.