audio read-through Reflecting About Axes, and the Absolute Value Transformation

 

reflecting about the x-axis
$y = f(x)$

$y = -f(x)$
Reflect about the $x$-axis
reflecting about the y-axis
$y = f(x)$

$y = f(-x)$
Reflect about the $y$-axis
the absolute value transformation
$y = f(x)$

$y = |f(x)|$
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

Ideas Regarding Reflecting About the $y$-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