Shortcut for drawing $f(-|x|)$

Question

I am trying to draw the graph of any function of type $f(-|x|)$. I know that $f(|x|)$ means reflecting the graph to the right of the y-axis in the y-axis. Ignore the left hand side part of the graph , and $f(-x)$ means reflecting the graph in y- axis.

Hence , i thought that $f(-|x|)$ can be interpreted such that firstly find the proper graph for $f(|x|)$ and after that reflect the graph of $f(|x|$ in y-axis to find $f(-|x|)$. However , my thought did not work. For example if $f(x)=x^2+4x+4$ , then $f(-|x|)=x^2-4|x|+4$ is not the same as what i thought.

Hence , i want to learn drawing the graph of $f(-|x|)$. Is there any rule for it ?

Answer 1

$ f(|x|) $ does mean graphing $f(x)$ for $x>0$ and then applying the reflection argument. However, values of $f(-|x|)$ cannot always be deduced from $f(|x|)$ since your function can be neither even nor odd.

To graph $f(-|x|)$, graph $f(-x)$ for $x>0$. Notice that only the left-hand side of the $2D$ space will be taken care of. After that, apply reflection.

Answer 2

You're close, but you just got the order of the transformations wrong. First, $g(x):=f(-x)$ is $f$ reflected across the $y$-axis. Then $h(x):=g(|x|)=f(-|x|)$ is the right side of the $y$-axis of $g$ reflected to the left side. In conclusion, we simply get the left part of $f$ copied and reflected to the right.

Formally: $$ f(-|x|) = \begin{cases}f(x) & x\le 0 \\ f(-x) & x>0\end{cases} $$