Associated Topics || Dr. Math Home || Search Dr. Math

### 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)

```
Associated Topics:
High School Functions

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search