The parent function f(x) = |x| is reflected in the x-axis and then translated 6 units down. Write the equation of the new function.

Step-by-step solution

1. 1

   Begin with the parent V. $f(x)=|x|$ has vertex $(0,0)$, opens upward, and has arms of slope $-1$ on the left and $+1$ on the right. Its range is $[0,\infty)$.

2. 2

   Reflect in the $x$-axis. Reflecting across the $x$-axis replaces each output $y$ by $-y$:

   $-f(x)=-|x|$

   The V now opens downward, arms of slope $+1$ then $-1$, with range $(-\infty,0]$. (A reflection across the $y$-axis would instead replace $x$ by $-x$ and change nothing here, since $|-x|=|x|$ — worth noting because the problem says only "reflected".)

3. 3

   Translate 6 units down. Subtract $6$ from the output:

   $-|x|-6$

4. 4

   Describe the resulting graph.

   $f(x)=-|x|-6,\qquad \text{vertex }(0,-6),\ \text{range }(-\infty,-6]$

   The maximum value is $-6$, attained only at $x=0$.

5. 5

   Check points and note there are no roots. At $x=0$: $-|0|-6=-6\;\checkmark$ (the vertex). At $x=3$: $-3-6=-9$, and the parent point $(3,3)$ indeed reflects to $(3,-3)$ then drops to $(3,-9)\;\checkmark$. Because the largest output is $-6<0$, the equation $-|x|-6=0$ has no solution — this graph never meets the $x$-axis.