f(x) = +bx+c The function f is defined by ax^2, where a, b, and c are constants. The graph of = f(x) y in the xy-plane passes through the points (−3,0) (7,0) and. If a is an integer greater than 1, a+b which of the following could be the value of?
","rationale_html":"= Choice A is correct. It’s given that the graph of y f^xh in the xy-plane passes 7,0 -3,0 through the points ^ h and ^ h. It follows that when the value of x is either - 7 or 3, the value of f^xh is 0. It’s also given that the function f is defined by = 2+ + f^xh ax 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^xh a^x uh^x vh, where the value of f^xh is 0 when x is either u or v. Since - the value of f^xh is 0 when the value of x is either 7 or 3, and the value of f^xh 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^xh a^x uh^x vh yields = -7 - -3 = -7 +3 f^xh a^x h^x ^ hh, or f^xh a^x h^x h. Distributing the right-hand side = 2-7 +3 -21 = 2-4 -21 of this equation yields f^xh a^x x x h, or f^xh ax ax a. Since = 2+ + =- it’s given that f^xh ax 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. -3 #-3 2 + Therefore, a ^ h. It follows that the value of a b is less than or equal to -3 2 - - ^ h, 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.
","correct_answer":["A"],"has_media":false,"has_mathml":false,"has_table":false,"source_created_at":null,"source_updated_at":null,"options":[{"label":"A","content_html":"−6 −3
","ord":0},{"label":"B","content_html":"4
","ord":1},{"label":"C","content_html":"","ord":2},{"label":"D","content_html":"5 ST
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