Graphing Tools: Reflections and the Absolute Value Transformation
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You may want to review:
 Graphs of Functions
 Basic Models You Must Know
 Graphing Tools: Vertical and Horizontal Translations
 Graphing Tools: Vertical and Horizontal Scaling
There are things that you can DO to an equation of the form $\,y=f(x)\,$ that will change the graph in a variety of ways.
For example, you can move the graph up or down, left or right, reflect about the $\,x\,$ or $\,y\,$ axes, stretch or shrink vertically or horizontally.
An understanding of these transformations makes it easy to graph a wide variety of functions, by starting with a ‘basic model’ and then applying a sequence of transformations to change it to the desired function.
In this discussion, we will explore reflecting about the $x$axis and the $y$axis, and the absolute value transformation.
When you finish studying this lesson, you should be able to do a problem like this:
GRAPH: $\,y=\ln(x)\,$

Start with the graph of $\,y=\ln(x)\,.$ (This is the ‘basic model’.)

Replace every $\,x\,$ by $\,x\,,$ giving the new equation $\,y = \ln(x)\,.$ This reflects the graph about the $y$axis.

Take the absolute value of the previous $y$values, giving the new equation $\,y = \ln(x)\,.$ This takes any part of the graph below the $x$axis and reflects it about the $x$axis. Any part of the graph on or above the $x$axis remains the same.

Multiply the previous $y$values by $\,1\,,$ giving the new equation $\,y = \ln(x)\,.$ This reflects the graph about the $x$axis.
Here are ideas that are needed to understand graphical transformations.
Ideas Regarding Functions and the Graph of a Function

A function is a rule: it takes an input, and gives a unique output.

If $\,x\,$ is the input to a function $\,f\,,$ then the unique output is called $\,f(x)\,$ (which is read as ‘$\,f\,$ of $\,x\,$’).

The graph of a function is a picture of all of its (input,output) pairs. We put the inputs along the horizontal axis (the $x$axis), and the outputs along the vertical axis (the $y$axis).

Thus, the graph of a function $\,f\,$ is a picture of all points of the form: $$\bigl(x, \overset{\text{$y$value}}{\overbrace{ f(x)}} \bigr)$$
Here, $\,x\,$ is the input, and $\,f(x)\,$ is the corresponding output.

The equation $\,y=f(x)\,$ is an equation in two variables, $\,x\,$ and $\,y\,.$ A solution is a choice for $\,x\,$ and a choice for $\,y\,$ that makes the equation true. Of course, in order for this equation to be true, $\,y\,$ must equal $\,f(x)\,.$
Thus, solutions to the equation $\,y=f(x)\,$ are points of the form: $$\bigl(x, \overset{\text{$y$value}}{\overbrace{ f(x)}} \bigr)$$

Compare the previous two ideas!
To ‘graph the function $\,f\,$’ means to show all points of the form $\,\bigl(x,f(x)\bigr)\,.$
To ‘graph the equation $\,y=f(x)\,$’ means to show all points of the form $\,\bigl(x,f(x)\bigr)\,.$
These two requests mean exactly the same thing!
Ideas Regarding Reflecting About the $x$axis

Points on the graph of $\,y=f(x)\,$ are of the form $\,\bigl(x,f(x)\bigr)\,.$
Points on the graph of $\,y=f(x)\,$ are of the form $\,\bigl(x,f(x)\bigr)\,.$
Thus, the graph of $\,y=f(x)\,$ is found by taking the graph of $\,y=f(x)\,,$ and multiplying the $y$values by $\,1\,.$ This reflects the graph about the $x$axis.

Transformations involving $\,y\,$ work the way you would expect them to work—they are intuitive.

Here is the thought process you should use when you are given the graph of $\,y=f(x)\,$ and asked about the graph of $\,y=f(x)\,$:
$$ \begin{align} \cssId{s40}{\text{original equation:}} &\quad \cssId{s41}{y=f(x)}\cr \cssId{s42}{\text{new equation:}} &\quad \cssId{s43}{y=f(x)} \end{align} $$interpretation of new equation:
$$ \cssId{s45}{\overset{\text{the new $y$values}}{\overbrace{ \strut\ \ y\ \ }}} \cssId{s46}{\overset{\text{are}}{\overbrace{ \strut\ \ =\ \ }}}\ \cssId{s47}{\overset{\text{1 times}}{\overbrace{ \strut \ \ \ \ }}}\ \cssId{s48}{\overset{\text{the previous $y$values}}{\overbrace{ \strut\ \ f(x)\ \ }}} $$ 
In reflection about the $x$axis, a point $\,(a,b)\,$ on the graph of $\,y=f(x)\,$ moves to a point $\,(a,b)\,$ on the graph of $\,y=f(x)\,.$
Ideas Regarding Reflecting About the $y$Axis

Points on the graph of $\,y=f(x)\,$ are of the form $\,\bigl(x,f(x)\bigr)\,.$
Points on the graph of $\,y=f(x)\,$ are of the form $\,\bigl(x,f(x)\bigr)\,.$

How can we locate these desired points $\,\bigl(x,f(x)\bigr)\,$?

Pick a value of $\,x\,.$ First, go to the point $\,\bigl(x\,,\,f(x)\bigr)\,$ on the graph of $\,y=f(x)\,.$ This point has the $y$value that we want, but it has the wrong $x$value.

The $\,x$value of this point is $\,x\,,$ but the desired $x$value is just $\,x\,.$ Thus, the current $x$value must be multiplied by $\,1\,$; that is, each $x$value must be sent to its opposite. The $y$value remains the same. This causes the point to reflect about the $y$axis, and gives the desired point $\,\bigl(x,f(x)\bigr)\,.$
Thus, the graph of $\,y=f(x)\,$ is the same as the graph of $\,y=f(x)\,,$ except that it has been reflected about the $y$axis.

Here is the thought process you should use when you are given the graph of $\,y=f(x)\,$ and asked about the graph of $\,y=f(x)\,$:
$$ \begin{align} \cssId{s62}{\text{original equation:}} &\quad \cssId{s63}{y=f(x)}\cr \cssId{s64}{\text{new equation:}} &\quad \cssId{s65}{y=f(x)} \end{align} $$interpretation of new equation:
$$ \cssId{s67}{y = f( \overset{\text{replace $x$ by $x$}}{\overbrace{ \ \ x\ \ }}} ) $$ Replacing every $\,x\,$ by $\,x\,$ in an equation causes the graph to be reflected about the $y$axis. 
In reflection about the $y$axis, a point $\,(a,b)\,$ on the graph of $\,y=f(x)\,$ moves to a point $\,(a,b)\,$ on the graph of $\,y=f(x)\,.$
Ideas Regarding the Absolute Value Transformation

Points on the graph of $\,y=f(x)\,$ are of the form $\,\bigl(x,f(x)\bigr)\,.$
Points on the graph of $\,y=f(x)\,$ are of the form $\,\bigl(x,f(x)\bigr)\,.$

Thus, the graph of $\,y=f(x)\,$ is found by taking the graph of $\,y=f(x)\,$ and taking the absolute value of the $y$values.
Points with positive $y$values stay the same, since the absolute value of a positive number is itself. That is, points above the $x$axis don't change.
Points with $\,y=0\,$ stay the same, since the absolute value of zero is itself. That is, points on the $x$axis don't change.
Points with negative $y$values will change, since taking the absolute value of a negative number makes it positive. That is, any point below the $x$axis reflects about the $x$axis.
These actions are summarized by saying that ‘any part of the graph below the $x$axis flips up’.

Here is the thought process you should use when you are given the graph of $\,y=f(x)\,$ and asked about the graph of $\,y=f(x)\,$:
$$ \begin{align} \cssId{s82}{\text{original equation:}} &\quad \cssId{s83}{y=f(x)}\cr \cssId{s84}{\text{new equation:}} &\quad \cssId{s85}{y=f(x)} \end{align} $$interpretation of new equation:
$$ \cssId{s87}{\overset{\text{the new $y$values}}{\overbrace{ \strut\ \ y\ \ }}} \cssId{s88}{\overset{\text{are}}{\overbrace{ \strut\ \ =\ \ }}} \cssId{s89}{\overset{\text{the absolute value of the previous $y$values}}{\overbrace{ \strut\ \ f(x)\ \ }}} $$ 
In the absolute value transformation, a point $\,(a,b)\,$ on the graph of $\,y=f(x)\,$ moves to a point $\,(a,b)\,$ on the graph of $\,y=f(x)\,.$
Summary
Reflecting about the $x$axis:
going from $\,y = f(x)\,$ to $\,y = f(x)$
Reflecting about the $y$axis:
going from $\,y = f(x)\,$ to $\,y = f(x)$
Absolute Value Transformation:
going from $\,y = f(x)\,$ to $\,y = f(x)$
Any part of the graph on or above the
$x$axis
stays the same;
any part of the graph below the
$x$axis flips up.
Make Sure You See the Difference!
Make sure you see the difference between $\,y = f(x)\,$ and $\,y = f(x)\,$!
In the case of $\,y = f(x)\,,$ the minus sign is ‘on the outside’; we're dropping $\,x\,$ in the $\,f\,$ box, getting the corresponding output, and then multiplying by $\,1\,.$ This is reflection about the $x$axis.
In the case of $\,y = f(x)\,,$ the minus sign is ‘on the inside’; we're multiplying $\,x\,$ by $\,1\,$ before dropping it into the $\,f\,$ box. This is reflection about the $y$axis.