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Graphing the Absolute Value/Square Root of a Function

Date: 07/26/2002 at 09:32:33
From: Daniel Brostin
Subject: Abs value/square root of functions

Can you illustrate and discuss how taking the absolute value or the
square root of a function affects the graph of the function?  

Thanks for the help!

Date: 07/26/2002 at 13:04:48
From: Doctor Peterson
Subject: Re: Abs value/square root of functions

Hi, Daniel.

Think about what the absolute value does to a number. If the number is 
positive, its absolute value is the number itself: the absolute value 
function does not change it. If the number is negative, its absolute 
value is the negative of the number (which is positive): the absolute 
value function reflects the number in the origin, sending it to the 
point the same distance away but on the opposite side.

So what happens if you take the absolute value of a function? Wherever 
the function is positive (or zero), the graph stays right where it 
was. Wherever the function is negative, the graph is reflected up over 
the x-axis, so that its y coordinate is positive rather than negative:

    |      ___              |      ___
    |   __/   \             +   __/   \   /
    |  /       \            |\ /       \ /
    +-+---------+--  ---->  +-+---------+--
    |/           \          |
    +             \         |
    |                       |

Now how about the square root? In this case, if f(x) is negative, its 
square root does not exist (since we are presumably talking about real 
functions). You can just erase that part of the graph! If f(x) is 
positive, the square root will squash it or stretch it vertically, 
depending on its value. Look at the graph of y=sqrt(x), compared with 
the graph of y=x. You will see that if x<1, sqrt(x) is larger, while 
when x>1, sqrt(x) is smaller. So when f(x)<1, sqrt(f(x)) will be 
above f(x), and when f(x)>1, sqrt(f(x)) will be below f(x):

    |      ___              |      ___
   1+   __/   \            1+   __/   \
    |  /       \            |  /       \
    +-+---------+--  ---->  +-+---------+--
    |/           \          |
    +             \         |
    |                       |

I can't get enough detail in my graph to show the stretching. But you 
can put a dot on the graph where f(x)=1, so the graph of the square 
root will go through that point; then you can warp the given graph 
upward where it is below y=1 and downward where it is above y=1. The 
peaks and valleys of the new graph will lie at the same x coordinates 
as in the original graph, but will be at different heights.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Equations, Graphs, Translations
High School Functions

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