The function f(x) = |x - 1| + 5 is translated 4 units up and 2 units right. Write the equation of the new function.

Step-by-step solution

1. 1

   Read the vertex from the given form. In $f(x)=|x-h|+k$ the vertex is $(h,k)$, so $f(x)=|x-1|+5$ has its corner at

   $(1,5)$

   Both translations will move this single point, and the arms of the V keep slopes $\pm 1$ throughout.

2. 2

   Shift right 2 by substituting $x\to x-2$.

   $\left|(x-2)-1\right|+5=|x-3|+5$

   Notice the inside constant changed from $1$ to $3$: shifting right increases the value of $h$, even though the operation performed was a subtraction inside the bars.

3. 3

   Shift up 4 by adding 4 outside.

   $|x-3|+5+4=|x-3|+9$

4. 4

   Write the answer and confirm the vertex moved correctly.

   $f(x)=|x-3|+9,\qquad \text{vertex }(3,9)$

   From $(1,5)$ to $(3,9)$ is right $2$ and up $4$ $\checkmark$ — the vertex displacement is the cleanest verification available.

5. 5

   Sanity-check a non-vertex point. The original passes through $(0,6)$ since $|0-1|+5=6$. Translating that point right $2$ and up $4$ predicts $(2,10)$, and the new formula gives $|2-3|+9=1+9=10\;\checkmark$. Note also that the whole graph now sits at or above $y=9$, so this function has no $x$-intercepts at all.