Square Root Functions and Transformations

Date: 09/07/2009 at 02:33:59
From: Jon
Subject: square root functions and transformations

How can we tell by looking at a graph if a square root function is 
being horizontally compressed or vertically stretched?  

For example, y = 2 sqrt(x) is being vertically stretched when 
graphed, but y = sqrt(2x) is being horizontally compressed. 

I'm confused because they both graphically look like they are being 
horizontally compressed or vertically stretched.  However, my 
textbook makes a distinction for both cases.  Ultimately, I would 
like to know how to tell the difference by merely looking at a graph.

Date: 09/07/2009 at 05:34:31
From: Doctor Ali
Subject: Re: square root functions and transformations

Hi Jon!

Thanks for writing to Dr. Math.  You asked a very nice question.

>How can we tell by looking at a graph if a square root function is 
>being horizontally compressed or vertically stretched?  

The answer is, we can't!  Note that in the example you gave, we can write

  y = Sqrt(2x)

as

  y = Sqrt(2) Sqrt(x) ~= 1.4142 Sqrt(x)

So, if we just have the graph, is it horizontally compressed by 1/2 or
vertically stretched by 1.4142?  We can't tell.

Does that make sense?  Please write back if you still have any
difficulties.

- Doctor Ali, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 09/07/2009 at 12:45:37
From: Jon
Subject: Thank you (square root functions and transformations)

Thank you very much for the quick response.  I really do 
appreciate it..

Date: 09/10/2009 at 10:27:34
From: Doctor Peterson
Subject: Re: square root functions and transformations

Hi, Jon.

It may be important to point out that not all functions behave like 
the square root.  In general, horizontal compression and vertical 
stretch are NOT equivalent.  (Try doing the same thing with a general 
linear function, y=ax+b.)  In the case of the square root, it happens 
that we can describe the same transformation either way, making it 
an arbitrary choice, but that is not usually true.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/