Answer
$(\pm4,0),(0,-4)$,
symmetric with respect to the y-axis
Work Step by Step
Step 1. For x-intercept(s), let $y=0$, we have $|x|=4\longrightarrow x=\pm4\longrightarrow (\pm4,0)$, for y-intercept(s), let $x=0$, we have $y=-4\longrightarrow (0,-4)$,
Step 2. To test for x-axis symmetry, replace $(x,y)$ with $(x,-y)$, we have $-y=|x|-4$ which is different from the original equation, thus it is not symmetric with respect to the x-axis,
Step 3. To test for y-axis symmetry, replace $(x,y)$ with $(-x,y)$, we have $y=|-x|-4$ which is the same as the original equation, thus it is symmetric with respect to the y-axis,
Step 4. To test for origin symmetry, replace $(x,y)$ with $(-x,-y)$, we have $-y=|-x|-4$ which is different from the original equation, thus it is not symmetric with respect to the origin.
