The function f(x) = |x| + 10 is translated 8 units left and 2 units down. Write the new function.

Step-by-step solution

1. 1

   Separate the two kinds of shift. Horizontal and vertical translations obey different rules, and mixing them up is the standard mistake:

   $\text{shift } h \text{ right: } x\mapsto x-h\ \text{(inside)},\qquad \text{shift } k \text{ up: } +k\ \text{(outside)}$

   Inside the function the sign is opposite to the direction of travel; outside it matches.

2. 2

   Apply the horizontal shift. Moving $8$ units left means $h=-8$, so $x$ is replaced by $x-(-8)=x+8$:

   $|x|+10\ \longrightarrow\ |x+8|+10$

   The $+8$ inside moving the graph left is exactly the counter-intuitive part: the vertex was at $x=0$ and is now at $x=-8$.

3. 3

   Apply the vertical shift. Moving $2$ units down means subtracting $2$ from the whole output:

   $|x+8|+10-2=|x+8|+8$

4. 4

   Track the vertex to confirm. The original vertex sits at $(0,10)$. Shifting $8$ left and $2$ down should land it at $(-8,8)$. From the new formula, $|x+8|$ is smallest when $x=-8$, where the value is $0+8=8$ - vertex $(-8,8)$ ✓.

5. 5

   Check a second point. In the original, $f(0)=10$. That point $(0,10)$ should move to $(-8,8)$, already confirmed; and the original point $(8,18)$ should move to $(0,16)$. The new function gives $|0+8|+8=16$ ✓.