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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
http://mathforum.org/dr.math/
```
Associated Topics:
High School Equations, Graphs, Translations
High School Functions

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