Transforming $y=f(x)$ to $|y|=f(x)$

Question

I am learning transformation of graphs and I have studied some of the standard rules. But this part has me stuck. How do I transform a graph from $$y=f(x)$$ to $$|y|=f(x)$$.

For example, I was trying to draw the graph of $|y|=\ln({|x|})$

I started from

$$y=\ln{(x)}$$

Mirrored it about the $y$-axis to get the graph for,

$$y=\ln{(|x|)}$$

How do I transform further?

Any help would be really appreciated. Thank you.

Answer 1

If $|A|=B$ for some real numbers $A$ and $B$, then we can immediately conclude that $B$ has to be non-negative, i.e. $B\ge0$, because it's the absolute value of another number. Therefore, simply from the definition of the absolute value of a real number, we can say that $$|A|=B \quad \iff \quad B\ge0 \text{ and } A=\pm B.$$ Therefore, to plot the graph of $|y|=f(x)$ from the known graph of $y=f(x)$, you perform the following two steps:

1. Erase all parts of the graph of $y=f(x)$ that lie below the $x$-axis (that's the "$B\ge0$" part).
2. To the resulting graph, add its reflection over the $x$-axis (that's the "$A=\pm B$" part).