Stretching and Compressing Graphs

I seem to be having an inordinate amount of difficult with understanding the concept of horizontal stretching and compressing graphs. Vertical stretching and compressing makes sense.

[math]y=2f(x^2)[/math]

If I understand it, this is simply multiplying the output of the function by two. All that happens is that the [math]y[/math] value is double whatever it would be as a result of [math]y=f(x^2)[/math]. This means that an input of 1 gives a [math]y[/math] value of 2, an input of 2 gives 8. This would stretch the graph away from the x-axis due to the y value increasing faster than it otherwise would.

The part that confuses me is when you multiply the [math]x[/math] value while it's inside of the function. I was unable to figure out how to graph Horizontal stretching on a calculator so I just manipulated the functions to show the graph that I wanted for this post.

Below is the graph of [math]y=f(x^2)[/math].

This is simple enough, what confuses me is what happens when I give [math]x[/math] a coefficient greater than 1.

[math]y=f(2x^2)[/math]

Now I realize that to graph this, all that I have to do is multiply the [math]x[/math] coordinate by [math]1/2[/math] and keep the [math]y[/math] coordinate the same. [math](1,1)[/math] becomes [math](.5,1)[/math] and [math](2,4)[/math] becomes [math](1,4)[/math] etc. The problem is, I don't understand why this happens? Why am I not doubling the [math]x^2[/math] like I intuitively would?

EDIT: I think I may have discovered what part of my problem is, [math]y=f(x^2)[/math] does not result in the graph that I have posted above as the function referenced by [math]f(x^2)[/math] is not given. I think what the book is asking is, what value of [math]x[/math], when multiplied by 2, gives the same output as the function would have if not multiplied by two. Is this correct? If so, it would explain what is happening as a function [math]f(x) = x^2[/math] (the actual function for the first graph) would indeed result in the value of [math]x[/math] needing to be .5 as [math]y=f(2x)[/math] when [math]f(2x)[/math] would be referring to [math]f(x) = x^2[/math]. .5 would be multiplied by 2 and then passed to the function which would square it to give the value of 1. Making the coordinates [math](.5,1)[/math].

Edited January 28, 200916 yr by Scientia
Fixin stuffs.