


$y = f(x)$
$y = f(x)\,$
reflect about the $\,x\,$axis

$y = f(x)$
$y = f(x)\,$
reflect about the $\,y\,$axis 
$y = f(x)$
$y = f(x)\,$
the absolute value transformation;
any part of the graph above the $\,x\,$axis
stays the same;
any part of the graph below the $\,x\,$axis
‘flips up’ 
The lesson
Graphing Tools: Reflections and the Absolute Value Transformation
in the
Algebra II curriculum
gives a thorough discussion of reflecting about the $x$axis and the $y$axis,
and the absolute value transformation.
The key concepts are repeated here.
The exercises in this lesson duplicate those in
Graphing Tools: Reflections and the Absolute Value Transformation.
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
$\,y=f(x)\,$:
$$
\begin{align}
\text{original equation:} &\quad y=f(x)\cr\cr
\text{new equation:} &\quad y=f(x)
\end{align}
$$
$$
\begin{gather}
\text{interpretation of new equation:}\cr\cr
\overset{\text{the new yvalues}}{\overbrace{
\strut\ \ y\ \ }}
\overset{\text{are}}{\overbrace{
\strut\ \ =\ \ }}
\overset{\quad\text{1 times}\quad}{\overbrace{
\strut \ \ \ \ }}
\overset{\qquad\text{the previous yvalues}\quad}{\overbrace{
\strut\ \ f(x)\ \ }}
\end{gather}
$$

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
$\,\color{red}{\bigl(x\,,\,f(x)\bigr)}\,$
on the graph of
$\,\color{red}{y=f(x)}\,$.
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
$\,\color{purple}{x}$value must be multiplied by
$\,\color{purple}{1}\,$;
that is, each
$\,\color{purple}{x}$value must be sent to its opposite.
The
$\,\color{purple}{y}$value remains the same.
This causes the point to reflect about the $\,y$axis, and gives
the desired point
$\,\color{green}{\bigl(x,f(x)\bigr)}\,$.
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 $\,y=f(x)\,$:
$$
\begin{align}
\text{original equation:} &\quad y=f(x)\cr\cr
\text{new equation:} &\quad y=f(x)
\end{align}
$$
$$
\begin{gather}
\text{interpretation of new equation:}\cr\cr
y = f(
\overset{\text{replace x by x}}{\overbrace{
\ \ x\ \ }}
)
\end{gather}
$$
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 $\,y=f(x)\,$:
$$
\begin{align}
\text{original equation:} &\quad y=f(x)\cr\cr
\text{new equation:} &\quad y=f(x)
\end{align}
$$
$$
\begin{gather}
\text{interpretation of new equation:}\cr\cr
\overset{\text{the new yvalues}}{\overbrace{
\strut\ \ y\ \ }}
\overset{\text{are}}{\overbrace{
\strut\ \ =\ \ }}
\overset{\quad\text{the absolute value of the previous yvalues}\quad}{\overbrace{
\strut\ \ f(x)\ \ }}
\end{gather}
$$

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.
EXAMPLES:
Question:
Start with $\,y = \sqrt{x}\,$.
Reflect about the $\,x$axis.
What is the new equation?
Answer:
$y = \sqrt{x}\,$
Question:
Start with $\,y = {\text{e}}^x\,$.
Reflect about the $\,y$axis.
What is the new equation?
Answer:
$y = {\text{e}}^{x}$
Question:
Suppose $\,(a,b)\,$ is a point on the graph of $\,y = x^3\,$.
Then, what point is on the graph of $\,y = x^3\,$?
Answer:
$(a,b)$