GATE CS 2016 Set 2 QGA-10 — General Aptitude

To find the correct function represented by the graph, we can analyze the properties of absolute value functions of the form $f(x) = a \pm |x - h|$:

1. The graph of $y = -|x - h|$ is an inverted V-shape (pointing upwards) with its vertex (peak) at $(h, 0)$.
2. Adding a constant $a$ shifts the vertex vertically to $(h, a)$, giving the function $f(x) = a - |x - h|$.

Looking at the options, we can test the coordinates of the peak and the intercepts:

• Vertex (Peak): The graph has its peak at the point $(1, 2)$. This suggests a function of the form $f(x) = 2 - |x - 1|$.
• $x$-intercepts: The graph crosses the $x$-axis at $x = -1$ and $x = 3$.

Let us verify these points using the function in Option C, $f(x) = 2 - |x - 1|$:

• At $x = 1$: $$f(1) = 2 - |1 - 1| = 2 \quad \text{(Matches the peak at } (1, 2)\text{)}$$
• At $x = 3$: $$f(3) = 2 - |3 - 1| = 2 - 2 = 0 \quad \text{(Matches the intercept at } (3, 0)\text{)}$$
• At $x = -1$: $$f(-1) = 2 - |-1 - 1| = 2 - 2 = 0 \quad \text{(Matches the intercept at } (-1, 0)\text{)}$$

Thus, the function represented by the graph is indeed $f(x) = 2 - |x - 1|$.