Multiple transformations/stretches preformed on one function

1) say you had this function:
y=2f(4x-1)+3 where f(x)=x^2

Which order would you apply the transformations/stretches in? Does it matter? Is there some equivalent to bodmas?

If you have reflections too, would it matter what order they're preformed in?

Sort of yes. You decide what acts on the 'x' first using your rules of bodmas. so clearly the x is multiplied by 4 before you subtract the 1. Which means shrinking by a factor of a 4 in the x direction before you shift it to right by 1. Next for the 2f +3. Treat 'f' as your 'x'. Bodmas tells us 2 acts first on the f so you have to stretch it by a scale factor of 2 in the 'y' direction before you move it up by 3.

Edit: in case my explanation wasn't clear anything inside the function's brackets affect the x direction and outside it affects the y direction.

I know what all the different transformations and stretches do individually, I'm just unsure how you would tackle a function, which is being
stretched and transformed. I remember seeing something like say $f(2x+20)=f[2(x+10)]$, so in fact, you don't move to the left by 20 units, you move to the left by 10 units and $2f(x)+10=2[f(x)+5]$.

I'm presuming X stretches only effect x transformations and Y stretches might not actually have an effect on the transformation, because the way I've "factorized" it doesn't look right. If someone does know how you tackle questions like this, could you explain?

It might be difficult to believe but f(2x+20) and f(2(x+10)) look the same. Moving to the left by 10 and then shrinking by 2 has the same affect as shrinking by 2 and then moving to left by 20. To see what I mean try this with the sin function. Try sin(2x+20) and sin(2(x +10)). They're both the same. Simiarly with the 2f(x) +10 example you gave. Both functions look the same.

To get by this confusion my strategy is usually to expand the brackets and work with that. Hasn't gone wrong so far for me. I'm guessing this is for A-level/ GCSE standard?

A-level, I can see how the X stretches an transformations work because factorizing it like that is correct, as for the Y stretch and transformation, would I just stretch in the Y direction by a scale factor of 2 then shift it up in the y direction by 10 units, because factorizing it like: $2f(x)+10=2[f(x)+5]$ seems incorrect

I drew out some tables different functions and now I get what you mean kinda, I see that it doesn't matter if you do or don't factorise as bidmas will always lead to you getting it correct (doing stuff in the correct order). As if both the transformation and stretches are inside the brackets you shrink then transform and if the stretch is outside the brackets, you transform and then stretch. Thanks for the help.

Yep glad you get it now. Pro-tip, Wolfram will plot functions for you so you can try this with an assortment of functions and see the results for yourself. I also find it easier to work with non factorised forms in examples such as yours.