Answer:
y = |x -1| -1
Explanation:
You want an equation for the graph of an absolute value function with its vertex at (1, -1).
Transformation
The usual transformations we study are ...
- vertical scaling
- reflection across the x-axis (vertical reflection)
- horizontal scaling
- reflection across the y-axis (horizontal reflection)
- up/down shift (vertical translation)
- right/left shift (horizontal translation)
For parent function f(x) these are manifested by the constants a, b, h, k:
g(x) = a·f((x -h)/b) +k
where 'a' and 'b' are the vertical and horizontal expansion factors, respectively, and 'h' and 'k' are the horizontal and vertical translation amounts, respectively. When 'a' is negative, the graph is reflected vertically. When 'b' is negative, the graph is reflected horizontally.
Application
For all of the graphs show, a = ±1, b = 1, and (h, k) represents the coordinates of the vertex. When the graph opens upward, a=1. When the graph opens downward, a=-1.
If we use y=|x| as the parent function, we have ...
- a: y = |x|, no translation
- b: y = |x -1|, translation right 1 unit
- c: y = -|x| +1, vertical reflection and translation up 1 unit
- d: y = |x -1| -1, translation to (h, k) = (1, -1)
- e: y = |x +1|, translation left 1 unit
- f: y = -|x +1| +1, vertical reflection and translation to (h, k) = (-1, 1)