Multiple choice

A quadratic function f(x) attains a maximum of 3 at x = 1. The value of the function at x = 0 is 1. What is the value of f(x) at x = 10?

  1. -119

  2. -159

  3. -110

  4. -180

  5. -105
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B Correct answer
Explanation

Vertex form: f(x) = a(x-h)^2 + k. Vertex is (1, 3), so f(x) = a(x-1)^2 + 3. Given f(0) = 1, so a(0-1)^2 + 3 = 1, a + 3 = 1, a = -2. f(x) = -2(x-1)^2 + 3. f(10) = -2(10-1)^2 + 3 = -2(81) + 3 = -162 + 3 = -159.

AI explanation

Because the quadratic function reaches its maximum of 3 at x = 1, its vertex form is f(x) = a(x - 1)2 + 3. We are given that f(0) = 1, so substituting 0 for x gives 1 = a(0 - 1)2 + 3, which simplifies to a = -2. The complete function is f(x) = -2(x - 1)2 + 3, and evaluating this at x = 10 gives f(10) = -2(10 - 1)2 + 3. Calculating -2(81) + 3 yields -159.

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