OG详解-OG5 数学1 Q26

Choice A is correct. It’s given that the graph of y = f(x) in the xy-plane passes through the points (7, 0) and (-3, 0). It follows that when the value of x is either 7 or -3, the value of f(x) is 0. It’s also given that the function f is defined byf(x) = ax² +bx+ c, where a, b, and c are constants. It follows that the function f is a quadratic function and, therefore, may be written in factored form as f(x) = a(x-u)(x-v), where the value of f(x) is 0 when x is either u or v. Since the value of f(x) is 0 when the value of x is either 7 or -3, and the value of f(x) is 0 when the value of x is either u or v, it follows that u and v are equal to 7 and -3. Substituting 7 for u and -3 for v in the equation f(x) = a(x-u)(x-v) yields f(x) = a(x-7)(x-(-3)), or f(x) = a(x-7)(x+3). Distributing the right-hand side  of this equation yields f(x) = a(x² -7x+3x-21), or f(x) = ax² -4ax-21a. Since it’s given that f(x) = ax²+bx+ c, it follows that b =-4a. Adding a to each side of this equation yields a+b = -3a. Since a+b =-3a, if a is an integer, the value of a+b must be a multiple of 3. If a is an integer greater than 1, it follows that a≥ 2.  Therefore, -3a≤-3(2). It follows that the value of a+b is less than or equal to -3(2), or -6. Of the given choices, only -6 is a multiple of 3 that’s less than or equal to -6.
Choice B is incorrect. This is the value of a + b if a is equal to, not greater than, 1. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.