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# How to remember Transformations of Functions (GCSE)?

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1. I can't remember them so should I make a poster or something? Any help is appreciated Thanks!
2. (Original post by AxSirlotl)
I can't remember them so should I make a poster or something? Any help is appreciated Thanks!
Yes make some notes, cover them and say the transformations out loud. Then make more notes and do the same thing.

Some help:

af(x)
f(x) + a
-f(x)

The constant is outside the brackets so these all occur in the y-direction i.e. up and down.

f(ax)
f(x+a)
f(-x)

The constant is inside the brackets so these are all in the x-direction i.e. left and right. And for these it's good to remember that the direction/scale factor is "opposite". I.e. f(2x) is a strech by 1/2 and f(x + 2) is a translation 2 to the left even though +2 seems to suggest 2 to the right.

An example:

f(x - 3)

The constant is inside the brackets so this is going to be a left-to-right transformation. And I recognise it as a translation. -3 would "usually" mean 3 to the left but the constant is inside the brackets so we do the opposite. So this is a translation 3 units to the right.

Let me know if you need more help.
3. That should help for now, thanks!
4. Idk, I try to understand them instead of memorising them because everything seems to stick a lot better.
5. (Original post by AxSirlotl)
I can't remember them so should I make a poster or something? Any help is appreciated Thanks!
f(x+2) move 2 to the left (when it's inside brackets it's counter intuitive)
f(2x) x- value (when it's inside brackets it's counter intuitive)

f(x)+2 move 2 up
2f(x) multiply y value by 2

6. (Original post by AxSirlotl)
I can't remember them so should I make a poster or something? Any help is appreciated Thanks!
notnek has described the standard approach to this. I'll give you my approach, which I believe is superior, but you may find it confusing, as it clashes with what you have learnt. If you find it confusing, I recommend that you simply ignore it.

In my approach, we look at what happens to the *axes* (represented by the letters ), not what happens to the function. It is a more consistent way to look at transformations.

a) Translations:

Suppose that then we have:

Note that we have rearranged the second equation to indicate how the axis variable is changed. Both the x- and y-axes are now treated the same way (they are both additions)

In both these cases, we have added to the axis variable. We interpret this as moving the *axis* units in the +ve direction, while holding the function in place. This now makes sense: a +ve shift always moves something in +ve direction (and that thing is the axis). This now works consistently both for the and axes.

Since the axis has moved in the +ve direction, it looks like the function has moved in the -ve direction, but we no longer think of it like that: remember: a +ve shift always moves the axis +ve.

For -ve shifts, we apply the same reasoning, but now the axis moves -ve while the function is held in place; again we treat the axis variable consistently:

Again, note that a -ve number *always* corresponds to a -ve shift (of the axes).

b) Scale factors

Again the second equation now looks like the first, as far as the axis variable goes. We can now interpret both in the same way. And for scaling, we interpret this as saying that we have stretched the *axis* by a factor of while keeping the function unchanged. To look at a specific example, we may have:

where both indicate that the axis has been stretched by a factor of 2, making it look like the function was shrunk by a factor 1/2 relative to the now-larger axis; because we now treat the x- and y- axes consistently, we don't have to remember different rules for "inside and outside the brackets". Similarly with:

we see that the axis has been shrunk by 2, while keeping the function unchanged. Consequently, the function seems to have been stretched by a factor of 2.

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Updated: April 12, 2016
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