﻿ Graphing Tools: Reflections and the Absolute Value Transformation
GRAPHING TOOLS:
REFLECTIONS and the ABSOLUTE VALUE TRANSFORMATION
by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!

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\cr \cssId{s42}{\text{new equation:}} &\quad \cssId{s43}{y=-f(x)} \end{align} $$\begin{gather} \cssId{s44}{\text{interpretation of new equation:}}\cr\cr \cssId{s45}{\overset{\text{the new y-values}}{\overbrace{ \strut\ \ y\ \ }}} \cssId{s46}{\overset{\text{are}}{\overbrace{ \strut\ \ =\ \ }}} \cssId{s47}{\overset{\quad\text{-1 times}\quad}{\overbrace{ \strut \ \ -\ \ }}} \cssId{s48}{\overset{\qquad\text{the previous y-values}\quad}{\overbrace{ \strut\ \ f(x)\ \ }}} \end{gather}$$
• 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} $$\begin{gather} \cssId{s86}{\text{interpretation of new equation:}}\cr\cr \cssId{s87}{\overset{\text{the new y-values}}{\overbrace{ \strut\ \ y\ \ }}} \cssId{s88}{\overset{\text{are}}{\overbrace{ \strut\ \ =\ \ }}} \cssId{s89}{\overset{\quad\text{the absolute value of the previous y-values}\quad}{\overbrace{ \strut\ \ |f(x)|\ \ }}} \end{gather}$$
• 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?
$y = -\sqrt{x}\,$
Question:
Start with $\,y = {\text{e}}^x\,$.
Reflect about the $\,y$-axis.
What is the new equation?
$y = {\text{e}}^{-x}$
Question:
Suppose $\,(a,b)\,$ is a point on the graph of $\,y = x^3\,$.
Then, what point is on the graph of $\,y = |x^3|\,$?
$(a,|b|)$
Master the ideas from this section