Which of the following functions describe the graph shown in the figure below?
The graph shows four panels P, Q, R, S. Panel P shows a V-shaped absolute value graph shifted left. Panel Q shows an absolute value graph opening upward and shifted right. Panel R shows an absolute value graph with vertex below the x-axis. Panel S shows an absolute value graph shifted right with vertex on the x-axis.
A. y = |x| + 1, y = -|x| - 1
B. y = |x - 1| - 1, y = |x| - 1
C. y = |x + 1| - 1, y = |x| - 1
D. y = |x - 1| - 1, y = |x + 1| - 1
GATE 2018 · General Aptitude · Absolute Value · medium
Answer: The two functions y = |x - 1| - 1 and y = |x| - 1 match the given graph. Answer: B.
Recall the vertex form and how shifts work: A positive h inside shifts the vertex right; a negative k outside shifts it down. For option B: y = |x - 1| - 1 has vertex at (1, -1) and y = |x| - 1 has vertex at (0, -1).
Eliminate options A, C, D: Option A: y = |x| + 1 has vertex at (0, 1) — vertex is ABOVE the x-axis, inconsistent with graph showing vertex below. Option C: y = |x + 1| - 1 has vertex at (-1, -1) — shifted LEFT; graph shows rightward shift. Option D: both functions have vertices at (1,-1) and (-1,-1) — one is left-shifted; graph does not match.
Verify option B: y = |x - 1| - 1 touches the x-axis at x = 0 and x = 2 with vertex at (1, -1). y = |x| - 1 is symmetric about the y-axis with vertex at (0, -1) and touches the x-axis at x = +/-1. Both match the shapes visible in the given graph panels.