The function f(x) = |x - 4| - 3 is translated 6 units up and 2 units right. Write the equation of the new function.

Step-by-step solution

1. 1

   Locate the starting vertex. For $f(x)=|x-h|+k$ the corner sits at $(h,k)$. Reading $f(x)=|x-4|-3$ gives $h=4$ and $k=-3$, so the vertex is at

   $(4,-3)$

   Tracking the vertex is the fastest way to keep translations straight, because the V-shape itself never changes — only where its corner sits.

2. 2

   Apply the horizontal shift by replacing $x$ with $x-2$. Moving a graph $2$ units right means every input must be $2$ larger to produce the same output, so the substitution subtracts:

   $f(x-2)=\left|(x-2)-4\right|-3=|x-6|-3$

   The sign feels backwards to most students — right shifts subtract inside — which is exactly why it is worth doing as a substitution rather than from memory.

3. 3

   Apply the vertical shift by adding outside. Moving $6$ units up adds $6$ to the whole output:

   $|x-6|-3+6=|x-6|+3$

   Vertical shifts behave the intuitive way: up means $+$.

4. 4

   State the transformed function and its vertex.

   $f(x)=|x-6|+3,\qquad \text{vertex }(6,3)$

   The vertex moved from $(4,-3)$ to $(6,3)$: right $2$, up $6$ — precisely the instructions, which is the built-in check.

5. 5

   Verify with a test point. On the original graph, $f(4)=|4-4|-3=-3$ (the corner). The corresponding point after moving right $2$ and up $6$ should be $(6,3)$, and the new formula gives $|6-6|+3=3\;\checkmark$. Checking one more: original $f(0)=|{-4}|-3=1$ maps to $(2,7)$, and the new function gives $|2-6|+3=4+3=7\;\checkmark$.