I've never posted on here before, so I apologise for anything that does not comply with guidelines. I am also very sorry that my equations are not in the correct formatting, I can't work out how to do it.
I am unable to contact my teachers and am DESPERATE to understand this.
For some reason, my mind has suddenly become unable to understand transformations of graphs, especially parabolas.
I think I understand the individual transformations on a graph, e.g. on $y=x^2$ , ie:
- vertical dilation of factor $A$: $y/A=x^2 \Rightarrow y=A(x^2)$
- horizontal dilation of factor $a$: $y=(x/a)^2$
- reflection in x-axis: $-y=x^2 \Rightarrow y=-(x^2)$
- reflection in y-axis: $y=(-x)^2 \Rightarrow y=x^2$ (bec. axis of symmetry is $x=0$)
- vertical translation of $k$: $y-k=x^2 \Rightarrow y=(x^2)+k$
- horizontal translation of $h$: $y=(x-h)^2$
but my problem arises when I try to apply multiple transformations one step at a time. I thought that, in theory, you should be able to "build off" of the previous transformed equation, but this does not appear to be true. For example:
Say we want to horizontally translate $y=x^2$ by 4 and THEN reflect in the y-axis. I figured you'd apply the translation: $y=(x-4)^2$, and THEN apply the reflection: $y=(-(x-4))^2$. But this simplifies to $y=(x-4)^2$, which is NOT the desired graph?
So after playing around with the equation, I discovered that the graph I was after would have an equation of $y=(-x-4)^2$. This lead me to believe that, when applying successive transformations, you "bypass" the previous transformations and apply it directly to x.
BUT THEN this theory of mine was proven wrong by the following set of transformations:
Say we have $y=x^2$, and horizontally translate it by 5 and THEN horizontally dilate it by a factor of 4. Using the logic that entailed applying the transformations directly to x, you'd get: $y=(x-5)^2 \Rightarrow y=((x/4)-5)^2$, which does NOT give the desired graph.
I'm so sorry this doesn't make much sense. I suppose essentially what I'm wondering is how are multiple transformations meant to be applied to a graph?. I know there is the equation $y=A(a(x-h))^2 +k$, but if I inputted the transformation values directly into this equation, wouldn't it essentially apply the translation $h$ to $x$ BEFORE the dilation, when in some cases it might be meant to be applied the other way around?
Ugh, I'm so sorry, I'm just so confused. Any advice would be GREATLY appreciated. If possible, could it please be explained in such a way that a numpty like me would understand -- ELI5 :)