(Original post by Ihategcse)
I'll try an explain the basics.
1. [f(x)+1] is a movement of +1 on the y axis.
Think of it like this if the value is outside the bracket as in outside the (x) then it acts on the Y AXIS.
[f(x)+2] would act on the Y axis, as would [f(x)+500]. So anything outside the brackets acts on the y axis. So, if there is a graph of y=x^2, a movement of [f(x)+1] means you move all the y value up by one so the value (0,0) now becomes (0,1).
If it was [f(x)-1] it would become (0,-1) because it is a movement of one.. but downwards.
Recap: If it is outside the (x) then it acts on the Y axis. If it is (-) then it moves down, if it is (+) it moves up.
2. [f(x+1)] is a movement of +1 on the X AXIS
If the value is INSIDE like so: (x+1) it acts on the x axis. Just remember if it is inside the bracket with the 'x' it acts on the 'x' because they are 'together.' Going back to the graph of y=x^2. If we were to say transform the graph like so: [f(x+1)] we would move it to the LEFT. Its best remembered if you think about graph sketching, if you have the values (x+1) (x-2) (x+5), the points you would draw on the x axis would be: -1, +2, -5. Kind of like the opposite of what is written in the bracket. So, apply the same principal here, it is a movement of in essence -1. So that means the new coordinates of the original point (0,0) would now be (-1,0). However, if it said transform as such:
[f(x-1)] you move it +1 to the right. So remember, do the opposite when inside the brackets. So if this were to be applied to the graph of y=x^2, the coordinates (0,0) would now be (1,0)
3. If the transformations is: [2f(x)]
If you remember what i previously said, if it is outside the brackets it acts on the Y AXIS. This holds true here as well. This means that we are basically multiplying all the y values by '2', the x values remain that same. So what does that mean? The graph gets longer. The the graph of x^2, when x=2, y also =2. (2,2). If we say transform it by [2f(x)] we would multiply the y value by 2, resulting in: (2,4).
4. If the transformation is [1/2f(x)] Then apply the same principals from (3). However, because it is '1/2' that means you multiply the Y value by '1/2' which is a half. So the coordinate (2,2) now becomes (2,1). As you may have noticed no changes in the 'x' coordinate.
5. If the transformation is [f(2x)]. We are now acting on the X AXIS, because it is inside the brackets. Remember how earlier we said if it was (x+1) we move it -1 opposite of what it suggests? Well, the same thing applies here. if it is [f(2x)] we DIVIDE it by 2. Or basically halve it. So the coordinate (2,2) divide the x value by 2 and you get (1,2).
6. If the transformation is [f(1/2x)] We are gain acting on the x axis. Similar principal to (5) because it says 1/2x we double it/multiply it by 2. SO going back to our coordinate (2,2) it now becomes (4,2) as the x coordinate has been doubled.
7. If the transformation is [f(-x)] we flip the graph, so in the x^2 graph now is negative. Instead of going up, it goes down. If the graph was already going down it would flip up. The minimum point (0,0) of the x^2 graph now becomes the maximum point. Or think of it as a multiplication of all the x values by -1. so (2,2) would be (-2,2)
8. If the transformation is [-f(x)]. We multiply all the y values by -1. So the coordinate (2,2) now would be (2,-2)
If you have any questions please feel free to ask. I have tried my best lol :P.