# Stretching and Compressing Graphs

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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.

$y=2f(x^2)$

If I understand it, this is simply multiplying the output of the function by two. All that happens is that the $y$ value is double whatever it would be as a result of $y=f(x^2)$. This means that an input of 1 gives a $y$ 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 $x$ 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 $y=f(x^2)$.

This is simple enough, what confuses me is what happens when I give $x$ a coefficient greater than 1.

$y=f(2x^2)$

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

EDIT: I think I may have discovered what part of my problem is, $y=f(x^2)$ does not result in the graph that I have posted above as the function referenced by $f(x^2)$ is not given. I think what the book is asking is, what value of $x$, 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 $f(x) = x^2$ (the actual function for the first graph) would indeed result in the value of $x$ needing to be .5 as $y=f(2x)$ when $f(2x)$ would be referring to $f(x) = x^2$. .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 $(.5,1)$.

Edited by Scientia
Fixin stuffs.

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