The following line graph shows f(x). Identify the correct expression for f(x) from the options below. A. f(x) = -1 - |x - 1| B. f(x) = -1 + |x - 1| C. f(x) = -2 + |x - 1| D. f(x) = -2 - |x - 1|

GATE 2016 · General Aptitude · Line Graph · medium

Answer: The correct expression is f(x) = -2 + |x - 1|, which is option C.

1. Identify key features from the graph: Vertex is at (1, -2): minimum point, so b > 0 (opens upward). The V-shape has slopes of magnitude 1, so b = 1. Thus f(x) = -2 + |x - 1|.
2. Eliminate wrong options: A: f(1) = -1 - 0 = -1 (wrong, vertex y = -2). B: f(1) = -1 + 0 = -1 (wrong). C: f(1) = -2 + 0 = -2 (correct). D: f(1) = -2 - 0 = -2 but opens downward (wrong direction). Only C gives vertex (1,-2) opening upward.
3. Verify option C with another point: At x = 3: f(3) = -2 + |3-1| = -2 + 2 = 0. At x = -1: f(-1) = -2 + |-1-1| = -2 + 2 = 0. The graph should pass through (3, 0) and (-1, 0), confirming x-intercepts at distance 2 from vertex.