## Line Graph: Absolute Value Function f(x) = |2x - 3| + 1 ### Overview The image shows a line graph of the absolute value function f(x) = |2x - 3| + 1. The graph has a V-shape with a vertex at x = 1.5. Two points are marked on the graph: one at x = 2 and another at x = 5. The question asks whether the derivative of f(x) at x = 2 is larger than, equal to, or smaller than the derivative at x = 5. ### Components/Axes - **X-axis**: Labeled "x", ranging from -5 to 10 in increments of 1. - **Y-axis**: Labeled "f(x)", ranging from 0 to 10 in increments of 1. - **Legend**: Text "f(x) = |2x - 3| + 1" in blue, positioned near the top-left of the graph. - **Grid**: Light gray grid lines span the entire graph area. - **Question Text**: Positioned to the right of the graph, with choices (A) larger than, (B) equal to, (C) smaller than. - **Ground Truth**: "B" (equal to) is explicitly stated as the correct answer. ### Detailed Analysis 1. **Function Behavior**: - The function f(x) = |2x - 3| + 1 has a vertex at x = 1.5 (where 2x - 3 = 0). - For x > 1.5, the function simplifies to f(x) = 2x - 2 (slope = 2). - For x < 1.5, the function simplifies to f(x) = -2x + 4 (slope = -2). 2. **Marked Points**: - At x = 2: The function value is f(2) = |2(2) - 3| + 1 = 2. The derivative is 2 (since x > 1.5). - At x = 5: The function value is f(5) = |2(5) - 3| + 1 = 8. The derivative is also 2 (since x > 1.5). 3. **Derivative Comparison**: - Both x = 2 and x = 5 lie in the region x > 1.5, where the derivative is constant at 2. ### Key Observations - The graph is symmetric about x = 1.5, with equal slopes of ±2 on either side of the vertex. - The derivatives at x = 2 and x = 5 are identical because both points are in the same linear region (x > 1.5). - The red-box reasoning incorrectly assumes the derivative at x = 2 is "steeper" (larger in absolute value) than at x = 5, but this ignores the absolute value function's piecewise linearity. ### Interpretation The absolute value function's derivative is piecewise constant, changing only at the vertex (x = 1.5). Since both x = 2 and x = 5 are in the same linear segment (right side of the vertex), their derivatives are equal. The red-box reasoning misinterprets the slope's sign (negative vs. positive) as a comparison of magnitude, but the question explicitly asks about the derivative's value, not its absolute value. The correct answer (B) reflects the mathematical reality of the function's behavior.