Advanced Math practice question

Answer: 48 or 48.0

Since 2 and 8 are the zeros, f(x) = a(x − 2)(x − 8). The graph is symmetric about x = (2 + 8)/2 = 5, so the extreme value occurs at x = 5: f(5) = a(5 − 2)(5 − 8) = a(3)(−3) = −9a. Because f has a minimum of −27, −9a = −27, so a = 3 (and a > 0, consistent with a minimum). Then f(0) = 3(0 − 2)(0 − 8) = 3(16) = 48. Common wrong answers: −48 comes from computing f(5) as a(3)(3) = 9a and getting a = −3 (dropping the sign of the factor 5 − 8); 16 comes from assuming a = 1 and ignoring the minimum-value condition entirely; −27 comes from confusing the minimum value with the y-intercept, i.e., assuming the minimum occurs at x = 0 rather than at the axis of symmetry x = 5.

Common wrong answers

If you answered 16

That's (0 − 2)(0 − 8) with a assumed to be 1. The minimum of −27 forces −9a = −27, so a = 3 and f(0) = 3·16 = 48.

If you answered -48

You computed f(5) as a(3)(3) = 9a, but 5 − 8 = −3, so f(5) = −9a = −27 and a = 3, giving f(0) = 48.

If you answered -27

−27 is the minimum value, which occurs at the axis of symmetry x = 5, not at x = 0. Use it to find a = 3, then f(0) = 3(−2)(−8) = 48.

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