2003 AMC 12A Problems/Problem 19

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Problem

Solution

If we take the parabola ax^2 + bx + c and reflect it over the x - axis, we have the parabola -ax^2 - bx - c. Without loss of generality, let us say that the parabola is translated 5 units to the left, and the reflection to the right. Then:

f(x) = a(x+5)^2 + b(x+5) + c = ax^2 + 10ax + 25a + bx +5b +c = ax^2 + (10a+b)x + 25a + 5b + c. g(x) = -a(x-5)^2 - b(x-5) - c = -ax^2 + 10ax -bx - 25a + 5b - c

Adding them up:

(f+g)(x) = 20ax + 10b which is a line with slope 20a. We know this is not horizontal, as a is not equal to 0. This is true because if a = 0, then the graph of ax^2 + bx +c would not be a parabola as stated in the problem.

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