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ABS(f(x)) and f(ABS(x)) Graphs

Date: 08/27/2013 at 14:44:55
From: Pavan Patel
Subject: Determining the graphs of functions

A graph of f(x) has a hump up to y = 1 for x between 0 and 2. To the left
of the origin, it slowly tails off to negative infinity.

I have to sketch the graphs of the functions
   (a) f(x + 1)
   (b) f(-x)
   (c) abs(f(x))
   (d) f(abs(x))

I am confused as to how I am supposed to plot f(abs(x)): even after I
looked up its solution, I am unsure as to how they got that answer.



Date: 08/27/2013 at 15:31:24
From: Doctor Peterson
Subject: Re: Determining the graphs of functions

Hi, Pavan.

Let's draw your graph like this:

                +1    *
                |  *    *
                |*        *
    +-----+-----*-----+-----*
   -2    -1    *|      1     2
             *  |
    *           +-1
  
(a) f(x + 1) will be a horizontal shift, 1 unit to the left:
  
                      *1
                   *  | *
                 *    |   *
    +-----+-----*-----+-----*-----+
   -3    -2    *-1    |     1     2
             *        |
    *                 +-1

To check this, what will f(x + 1) be for x = 0? f(0 + 1) = f(1), which
from the given graph is 1. And that's what the shifted graph shows.

For more on this, see the prior Dr. Math conversation

  Graph with f(x)
    http://mathforum.org/library/drmath/view/54509.html  

(b) f(-x) will be a reflection in the y axis, because x is replaced by
-x for any point on the graph:

        *     +1
      *    *  |
    *        *|
  *-----+-----*-----+-----+
 -2    -1     |*    1     2
              |  *
              +-1         *

To check this, what will f(-x) be for x = -1? f(-(-1)) = f(1), which from
the given graph is 1. And that's what the reflected graph shows.

For more on this, see

  Order of Transformations of a Function
    http://mathforum.org/library/drmath/view/68503.html 

(c) |f(x)| will reflect any negative parts of the graph up over the
x-axis, making them positive:


  *           +1    *
           *  |  *    *
             *|*        *
  +-----+-----*-----+-----*
 -2    -1     |      1     2
              |
              +-1

To check this, what will |f(x)| be for x = -2? |f(-2)|, which from the
given graph is |-1| = 1. And that's what the new graph shows.

For more on this, see

  Graphing Absolute Values
    http://mathforum.org/library/drmath/view/60968.html  

  Graphing the Absolute Value/Square Root of a Function
    http://mathforum.org/library/drmath/view/61112.html  

(d) f(|x|) will ignore values of f for negative x (since the argument of f
will never be negative, and replace it with a reflection of the right
side, as explained in the fifth page above:

        *     +1    *
      *    *  |  *    *
    *        *|*        *
  *-----+-----*-----+-----*
 -2    -1     |      1     2
              |
              +-1

To check this, what will f(|x|) be for x = -1? f(|-1|) = f(1), which from
the given graph is 1. And that's what the new graph shows.

For more on this, see

  Graphing f(2x) and f(|x|)
    http://mathforum.org/library/drmath/view/64038.html 
    
I hope you notice how I checked my graphs; this is very good thing to do
both to make sure you're right and to get a better understanding of what
these transformations do, and why. Ideally, you should check more than one
point, especially in the absolute value examples.

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



Date: 08/27/2013 at 15:47:14
From: Pavan Patel
Subject: Thank you (Determining the graphs of functions)

Thank you! Your explanation was really thorough and helpful!
Associated Topics:
High School Functions

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