The function f(x) = |x - 4| is translated 6 units up and 2 units right. Write the equation of the new function.

Step-by-step solution

1. 1

   Note the starting position. With no constant outside the bars, $f(x)=|x-4|$ has vertex

   $(4,0)$

   sitting right on the $x$-axis, with the two arms rising at slopes $-1$ and $+1$.

2. 2

   Translate 2 units right. Replace $x$ by $x-2$ inside the function:

   $\left|(x-2)-4\right|=|x-6|$

   The two subtractions accumulate into a single shift of $6$, so the corner is now at $x=6$.

3. 3

   Translate 6 units up. Add $6$ to the output:

   $|x-6|+6$

   Because the original had no vertical constant, this shift alone supplies the $+6$.

4. 4

   Give the final equation and vertex.

   $f(x)=|x-6|+6,\qquad \text{vertex }(6,6)$

5. 5

   Check and note a consequence. Original point $(4,0)$ should map to $(6,6)$, and $|6-6|+6=6\;\checkmark$; original $(0,4)$ should map to $(2,10)$, and $|2-6|+6=4+6=10\;\checkmark$. Since the minimum value is now $6>0$, the translated graph never touches the $x$-axis, whereas the original touched it exactly once — a real change in behaviour caused purely by the vertical shift.