audio read-through Graphing Tools: Reflections and the Absolute Value Transformation

Click here for a printable version of the discussion below.

You may want to review:

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

reflection about the x-axis

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: $(a,|b|)$

Concept Practice