Graph transformation

What is the graph transformation f(1-x)? I'm trying to graph f(x)=2ln(1-x), and I got stretch in y axis factor 2, shift left 1 unit and reflect in y axis. However, the answers say it is a shift right one unit. I checked on an online graph maker and they also show a shift right. Can anyone explain why the 1 causes a shift to the right and not the left? I thought it had something to do with the fact it has been reflected but was not sure...

Depends on the order, but there is no stretch in $f(1-x)$ from $f(x)$

Firstly it would be reflected in x=0 (the y-axis) - $f(x) \rightarrow f(-x)$

Then it would be shifted by vector [1,0] - $f(-x) \rightarrow f(-(x-1))=f(1-x)$

Oh thank you! I get it! So when considering a transformation that involves moving along the x axis, the form must be including a positive x value as you have shown, yes?

What do you mean?

Translating $f(x)$ by vector [a,0] would make it $f(x-a)$

So when we learn the rules like f(x-1) is shift to right etc... The x must be positive for the transformation to be valid; if x is negative we can't just reflect and then move by 1 unit to the right because of -1, it must be to the right because the x is negative so we factorise to change it to. Positive by taking out negative 1 and that would make the transformation go to the left as it would create a positive 1...

Okay it makes sense to me, I'm sorry it's complicated ahhaha

The best advice I can give you is to put brackets around the x, and do any transformations concerning it inside that bracket. I will redo the first example with brackets to clear up any confusion and hopefully make it clearer:

Reflection in the y-axis: $f[(x)] \rightarrow f[(-x)]=f[-x]$

Translate by vector [1,0]: $f[-(x)] \rightarrow f[-(x-1)]=f[1-x]$

Then for the banter we can choose to translate it by another vector [-8,0]: $f[1-(x)] \rightarrow f[1-(x+8)]=f[-7-x]$

And then stretch it in along the x-axis by scale factor 1/2: $f[-7-(x)] \rightarrow f[-7-(2x)]=f[-7-2x]$