A translation moves every point on a graph the same distance. For a graph like , track the corner of the V: a horizontal shift changes the input inside the absolute value, while a vertical shift changes the output outside it. The tempting trap is the inside sign. A right shift uses subtraction, not addition.
Hints
- Hint 1
The vertex is the corner of the V in , at . After the rightward shift, where should that corner be? The expression inside the absolute value must equal there.
- Hint 2
A horizontal change goes inside the absolute value; a vertical change goes outside it. Check the inside sign by finding the input that makes it , then decide what to add outside for the upward shift.
Step-by-step
Track the corner of the V
Step 1Write the right shift
No Desmos needed. The shifts give the rule directly. The vertex is the V’s corner, at for . After shifting right by , that corner needs to be at . The expression inside the absolute value must be at the new corner, so use , since . The right-shifted rule is . Using would put the corner at , to the left.
- Step 2
Add the upward shift
Shifting up by adds outside the absolute value, raising every output: . Its corner is , matching both moves. Choice C.