The function f(x) = |x + 2| - 1 is reflected in the x-axis and then translated 2 units down. Write the equation of the new function.

Step-by-step solution

1. 1

   Identify the starting graph. $f(x)=|x+2|-1$ is a V opening upward with vertex at $(-2,-1)$, since $|x+2|=|x-(-2)|$ puts $h=-2$ and the trailing $-1$ puts $k=-1$.

2. 2

   Reflect in the $x$-axis by negating the whole output. This is the step that trips people: the minus sign must apply to the entire function, constant included, not just the absolute-value term:

   $-f(x)=-\left(|x+2|-1\right)=-|x+2|+1$

   Writing $-|x+2|-1$ at this stage would be the classic error — the $-1$ becomes $+1$ under reflection, because the point $(-2,-1)$ flips to $(-2,1)$.

3. 3

   Translate 2 units down by subtracting 2.

   $-|x+2|+1-2=-|x+2|-1$

4. 4

   State the result and its features.

   $f(x)=-|x+2|-1,\qquad \text{vertex }(-2,-1),\ \text{opens downward}$

   The vertex has returned to its original coordinates by coincidence — the reflection lifted it from $-1$ to $+1$ and the shift dropped it back by $2$ — but the graph is genuinely different, now an upside-down V.

5. 5

   Check with points. Original $(0,f(0))=(0,1)$; reflecting gives $(0,-1)$ and shifting down $2$ gives $(0,-3)$. The final formula: $-|0+2|-1=-2-1=-3\;\checkmark$. Original vertex $(-2,-1)\to(-2,1)\to(-2,-1)$, and the formula gives $-|{-2}+2|-1=-1\;\checkmark$. Since the maximum value is $-1$, the graph never reaches the $x$-axis.