The function f is defined by f(x) = ax 2 + bx + c, where a, b, and c are constants. The graph of y = f(x) in the xy-plane passes through the points (7, 0) and (−3, 0). If a is an integer greater than 1, which of the following could be the value of a + b ?

Question

Accepted Answer

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 by f(x) = ax 2 + 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(x 2 − 4x − 21), or f(x) = ax 2 − 4ax − 21a. Since it’s given that f(x) = ax 2 + 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.

Want a deeper, personalized explanation?

Related questions