The function f is defined by f x = a x 2 + b x + c , where a

Question

The function f is defined by f x = a x 2 + b x + 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 ?

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 = a x 2 + b x + 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 - 7 x + 3 x - 21 , or f x = a x 2 - 4 a x - 21 a . Since it's given that f x = a x 2 + b x + c , it follows that b = - 4 a . Adding a to each side of this equation yields a + b = - 3 a . Since a + b = - 3 a , 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 &#8805; 2 . Therefore, - 3 a &#8804; - 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.

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