The function f(x) = |x + 2| - 3 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

   Read the starting vertex. $f(x)=|x+2|-3$ opens upward with vertex $(-2,-3)$, and it crosses the $x$-axis twice, where $|x+2|=3$, that is at $x=-5$ and $x=1$.

2. 2

   Reflect in the $x$-axis: negate everything. Every output changes sign, so the whole right-hand side gets a minus sign in front:

   $-f(x)=-\left(|x+2|-3\right)=-|x+2|+3$

   The constant flips from $-3$ to $+3$. Distributing that minus over both terms is the entire difficulty of the problem.

3. 3

   Translate 2 units down. Subtract $2$ from the reflected function:

   $-|x+2|+3-2=-|x+2|+1$

4. 4

   Final equation and its shape.

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

   The vertex went from $(-2,-3)$ to $(-2,3)$ under reflection, then down $2$ to $(-2,1)$.

5. 5

   Verify with points and intercepts. Original $(0,-1)$ reflects to $(0,1)$ then shifts to $(0,-1)$; the formula gives $-|0+2|+1=-2+1=-1\;\checkmark$. The new $x$-intercepts satisfy $|x+2|=1$, so $x=-3$ and $x=-1$ — the two roots have moved inward from $\{-5,1\}$, consistent with a downward-opening V whose peak is only $1$ unit above the axis.