GRAPHING TOOLS:
REFLECTIONS and the ABSOLUTE VALUE TRANSFORMATION
LESSON READ-THROUGH
by Dr. Carol JVF Burns (website creator)
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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: $\,y=-|\ln(-x)|\,$

Here are ideas that are needed to understand graphical transformations.

IDEAS REGARDING FUNCTIONS AND THE GRAPH OF A FUNCTION

IDEAS REGARDING REFLECTING ABOUT THE $\,x$-AXIS
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
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:
$(a,|b|)$
Master the ideas from this section
by practicing the exercise at the bottom of this page.


When you're done practicing, move on to:
Graphical Transformations: All Mixed Up


On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9 10 11
12 13 14 15 16 17 18 19 20 21 22
23 24 25 26 27 28 29 30 31 32 33
34 35 36 37 38 39 40 41      
AVAILABLE MASTERED IN PROGRESS

(MAX is 41; there are 41 different problem types.)