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

Step-by-step solution

1. 1

   Start from the parent function. The graph of $f(x)=|x|$ is a V with vertex at the origin $(0,0)$ and arms of slope $\pm 1$. Every absolute-value graph in this family is this same V placed somewhere else.

2. 2

   Shift 2 units right. Replace the input $x$ with $x-2$:

   $f(x-2)=|x-2|$

   The minus sign produces a shift to the right. A quick way to see why: the corner occurs when the inside is zero, and $x-2=0$ happens at $x=2$, two units right of the original corner.

3. 3

   Shift 4 units up. Add $4$ to the whole expression:

   $|x-2|+4$

4. 4

   Read off the general pattern. The result fits the standard vertex form

   $f(x)=|x-h|+k\quad\text{with } h=2,\;k=4,\quad\text{vertex }(2,4)$

   so $h$ lives inside with a minus and $k$ lives outside with a plus — the single rule behind every problem of this type.

5. 5

   Verify with points. The parent point $(0,0)$ should land at $(2,4)$: the new formula gives $|2-2|+4=4\;\checkmark$. The parent point $(-3,3)$ should land at $(-1,7)$: the formula gives $|-1-2|+4=3+4=7\;\checkmark$. Both arms keep slopes $\pm 1$, confirming that a translation changes position but never shape.