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
$\,\bigl(x,
\overset{\text{yvalue}}{\overbrace{
f(x)}}
\bigr) \,$.
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
$\,\bigl(x,
\overset{\text{yvalue}}{\overbrace{
f(x)}}
\bigr) \,$.

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
$\,y=f(x)\,$:
$$
\begin{align}
\cssId{s40}{\text{original equation:}} &\quad \cssId{s41}{y=f(x)}\cr\cr
\cssId{s42}{\text{new equation:}} &\quad \cssId{s43}{y=f(x)}
\end{align}
$$
$$
\begin{gather}
\cssId{s44}{\text{interpretation of new equation:}}\cr\cr
\cssId{s45}{\overset{\text{the new yvalues}}{\overbrace{
\strut\ \ y\ \ }}}
\cssId{s46}{\overset{\text{are}}{\overbrace{
\strut\ \ =\ \ }}}
\cssId{s47}{\overset{\quad\text{1 times}\quad}{\overbrace{
\strut \ \ \ \ }}}
\cssId{s48}{\overset{\qquad\text{the previous yvalues}\quad}{\overbrace{
\strut\ \ f(x)\ \ }}}
\end{gather}
$$

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 $\,y=f(x)\,$:
$$
\begin{align}
\cssId{s62}{\text{original equation:}} &\quad \cssId{s63}{y=f(x)}\cr\cr
\cssId{s64}{\text{new equation:}} &\quad \cssId{s65}{y=f(x)}
\end{align}
$$
$$
\begin{gather}
\cssId{s66}{\text{interpretation of new equation:}}\cr\cr
\cssId{s67}{y = f(
\overset{\text{replace x by x}}{\overbrace{
\ \ x\ \ }}}
)
\end{gather}
$$
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 $\,y=f(x)\,$:
$$
\begin{align}
\cssId{s82}{\text{original equation:}} &\quad \cssId{s83}{y=f(x)}\cr\cr
\cssId{s84}{\text{new equation:}} &\quad \cssId{s85}{y=f(x)}
\end{align}
$$
$$
\begin{gather}
\cssId{s86}{\text{interpretation of new equation:}}\cr\cr
\cssId{s87}{\overset{\text{the new yvalues}}{\overbrace{
\strut\ \ y\ \ }}}
\cssId{s88}{\overset{\text{are}}{\overbrace{
\strut\ \ =\ \ }}}
\cssId{s89}{\overset{\quad\text{the absolute value of the previous yvalues}\quad}{\overbrace{
\strut\ \ f(x)\ \ }}}
\end{gather}
$$

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:
$(a,b)$