The function f(x) = |x + 2| is translated 3 units down. Write the new function.

Step-by-step solution

1. 1

   Distinguish vertical from horizontal shifts. A vertical shift changes the output, so the constant is added or subtracted outside the function: $f(x)-k$ moves the graph down $k$ units. This is the intuitive direction, unlike horizontal shifts where the sign appears reversed inside.

2. 2

   Locate the current vertex. $\left|x+2\right|$ is zero when $x+2=0$, i.e. $x=-2$, so the vertex is at $(-2,0)$.

3. 3

   Subtract 3 from the whole function.

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

   The $-3$ stays outside the bars. Writing $\left|x+2-3\right|=\left|x-1\right|$ would instead be a horizontal shift of 3 to the right — a completely different graph.

4. 4

   Track the vertex. The new vertex is $(-2,-3)$: the $x$-coordinate is unchanged and the $y$-coordinate has dropped by 3 ✓.

5. 5

   Note what the shift changes qualitatively. The original graph touched the $x$-axis at exactly one point; the shifted graph dips below it and now crosses at two points, where $\left|x+2\right|=3$, that is $x=1$ and $x=-5$. Checking $f(1)=\left|3\right|-3=0$ ✓ and $f(-5)=\left|-3\right|-3=0$ ✓ confirms the new function.