Revise: Absolute Value Functions

y = |x−h|+k: upward V, vertex (h,k).

y=|x−3|+2 → vertex (3,2).

Sign of a: direction. |a|: narrowness.

y=4|x| is narrower than y=|x|.

|x−p|=q → x=p+q or x=p−q (q≥0).

|x−3|=5 → x=8 or x=−2.

|x−p|<q: one interval. |x−p|>q: two intervals.

|x−3|<5 → −2<x<8.

y=|x−h|+k opens upward, vertex (h,k), symmetric about x=h.

y=|x+4|−1 → vertex (−4,−1).

y=−|x−h|+k opens downward, vertex is a maximum.

y=−|x−5|+4 → vertex (5,4), a maximum.

Sign of a sets direction; |a| sets narrowness. Vertex of y=a|x| stays at (0,0).

y=−0.3|x| opens down, wider than y=|x|.

y=a|x−h|+k combines all transformations: vertex (h,k), direction/width from a.

y=−3|x−4|+2 → vertex (4,2), a maximum.

|x−p|=q splits into x−p=q or x−p=−q, when q≥0.

|x+2|=7 → x=5 or x=−9.

"Less than" → one band. "Greater than" → two rays.

|x+1|>4 → x<−5 or x>3.