Graphing Absolute Values

Date: 07/03/2002 at 03:06:47
From: Loralei Landon
Subject: Finding an equation of a funtion when given a graph and
domain only and confusion over how to represent absolute value
graphically when given only a sketch of a graph, domain, and plot
points.

Greetings,

I'm currently enrolled in pre-calculus algebra, and I am 
stuck on the following problem: 

  The graph of a function f with domain [-3,3] is shown in 
  the figure. Sketch the graph of the given equation.  

Basically, what I have is a grid, with the given function's graph.  I
am asked to sketch graphs of various equations such as y=f(x-2),
y=f(x)-2, etc.  I can do this with no problem. 

However, when I try to sketch the graph for the following |f(x)|, I 
don't know how to graphically represent it.  I need to give you a bit 
more information about the problem.  

The following points are plotted on the graph: 

  (3,0), (2,0), (1,1), (2,0), (3,-1), and (4,0). 

Given that information, I'm fairly sure that the range is (-1,2).  

Is there any way I can find the answer by creating an overall 
equation for the base graph f(x) given the graphical information that 
I do have, such as the plotted points, domain, and range?  In other 
words, given this information, can I reproduce or create what this 
absolute value graph should look like? 

I have tried several methods to create the graph, and I know that I
have missed or overlooked a key step somewhere. Currently my base
graph, if it were to be combined with the graph for y=|f(x)|, has a
big X dissecting it at the origin.  I have tried graphing this on my
calculator as well, and I just don't understand how to shift, reflect,
stretch, shrink or do whatever I'm supposed to do to this particular
problem based on the shape of the base graph.  I have searched my text
thoroughly for similar questions, and I have not had much luck.  Could
you please suggest to me how I could go about solving this problem?  

Thank You,
Loralei Landon

Date: 07/03/2002 at 14:43:34
From: Doctor Peterson
Subject: Re: Finding an equation of a funtion when given a graph and
domain only and confusion over how to represent absolute value
graphically when given only a sketch of a graph, domain, and plot
points.

Hi, Loralei.

Presumably you understand the ideas behind graphing y=f(x-2) and 
y=f(x)-2 given the graph of y=f(x). The first shifts the graph to the 
right, because x has to be 2 units larger before f(x-2) is what f(x) 
was. The second shifts the graph two units down, because you take the 
value of y and subtract 2 from it. This page illustrates these ideas, 
in case you are not quite sure:

      http://mathforum.org/library/drmath/view/54509.html 

The graph of |f(x)| is more like the second of those: you find the 
value of f(x), as shown on the graph you are given, and then take the 
absolute value of y. Think about what the absolute value does to a 
number. If it is positive, it leaves the number unchanged; |4| is 
just 4. So wherever f(x) is positive, the graph of |f(x)| is the same!

But the absolute value of a negative number is its opposite; |-4| is 
4, which you can think of as -(-4). So whereever f(x) is negative, 
you can think of |f(x)| as -f(x). What does that look like? For a 
given value of x, if y is negative, you replace y with -y, making it 
positive. Your point (3,-1) on the original graph will become (3,1) 
on the new graph. This is like reflecting the graph in the x axis.

In fact, you can imagine the absolute value as "folding" the graph 
along the x axis, so that the top half stays in place, while the 
bottom half is folded up onto the top half plane. (Imagine it being 
on transparent paper.)

      y=f(x)                 y=|f(x)|           
    |         /\           |\        /\     __ 
    |        /  \          | \___   /  \   /  
    |       /    \         |     \ /    \ /   
    +------/------\----    +------+------+----
    |  ___/        \       |
    | /             \__    |
    |/                     |
       <0     >0    <0
      flip | keep |flip

You don't want to find the equation for the graph you are shown, 
because that could be absolutely anything; there is no way to 
determine for certain the function represented by a given graph, and 
it would probably be very complicated. (Take mine above, for 
instance!) And knowing the equation wouldn't help you at all. The 
whole point of what you are learning is to make it easier for you to 
graph a function, by understanding how various transformations affect 
a graph. This way, you can build a complicated function out of simple 
ones, and see what it looks like without having to plot points. And 
if you used a calculator to do the graphing, you wouldn't be gaining 
any knowledge of how functions work, which can help in solving 
problems. Your "short cut", as often happens, would turn out to be 
the hard way to the goal.

If you have any further questions, feel free to write back. If you're 
still confused, we'll work it out eventually!

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

Date: 07/03/2002 at 18:10:29
From: Loralei Landon
Subject: Thank you (Finding an equation of a funtion when given a
graph and domain only and confusion over how to represent absolute
value graphically when given only a sketch of a graph, domain, and
plot points.)

Dear Dr. Peterson,

Thank you so much for your help!  I completely 
understand the concept now.  As I mentioned, I was well 
aware of how each transformation would appear graphically, 
but my text never represented a situation such as this 
using absolute value with a sketched graph, and while later 
that evening I found a few chapter problems that possibly 
could have illustrated this type of transformation, they 
were even-numbered, and therefore had no solution in the 
back of the book for me to ponder.  Again, I thank you so 
much. 

Thanks again!  Loralei Landon