Multiple choice

In the xy-plane, the point (-1, 2) is the minimum of the quadratic function f(x) = x2 + ax + b. What is the value of |a – 2b|?

  1. 4

  2. 3

  3. 2

  4. 1
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A Correct answer
Explanation

The minimum of f(x) = x^2 + ax + b occurs at x = -a/2. Given the vertex is (-1, 2), -a/2 = -1, so a = 2. Since f(-1) = 2, (-1)^2 + 2(-1) + b = 2, so 1 - 2 + b = 2, which means b = 3. Then |a - 2b| = |2 - 2(3)| = |2 - 6| = 4.

AI explanation

The vertex form of a quadratic is f(x) = (x - h)2 + k, where the vertex is given by the point (h, k). Matching the given vertex (-1, 2) to this form yields f(x) = (x + 1)2 + 2, which expands to x2 + 2x + 3. By comparing this to f(x) = x2 + ax + b, we find a = 2 and b = 3. Calculating the absolute difference gives |2 - 6| = 4.

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