{"id":"cb_question_bank:29f0dfe3-c90d-47aa-8c1e-b878268a3b52","source_id":"cb_question_bank","source_item_id":"29f0dfe3-c90d-47aa-8c1e-b878268a3b52","external_id":"c1e6cb10-5989-4a1e-a246-78972ed7bf5a","ibn":null,"question_id":"4dd4efcf","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"H","score_band":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"f left parenthesis x right parenthesis equals a x squared plus 4 x plus c In the given quadratic function, a and c are constants. The graph of y equals f left parenthesis x right parenthesis in the xy -plane is a parabola that opens upward and has a vertex at the point left parenthesis h comma k right parenthesis , where h and k are constants. If k less than 0 and f left parenthesis negative 9 right parenthesis equals f left parenthesis 3 right parenthesis , which of the following must be true? c less than 0 a greater than or equals 1","stimulus":"","stimulus_html":null,"stem_html":"

fx=ax2+4x+c

\n

In the given quadratic function, a\n and c\n are constants. The graph of y=fx in the xy-plane is a parabola that opens upward and has a vertex at the point h,k, where h\n and k\n are constants. If k<0 and f-9=f3, which of the following must be true?

\n
    \n
  1. c<0
    2. \n
  2. a≥1
    3. \n
","rationale_html":"

Choice D is correct. It's given that the graph of y=fx in the xy-plane is a parabola with vertex h,k. If f-9=f3, then for the graph of y=fx, the point with an x-coordinate of -9\n and the point with an x-coordinate of 3\n have the same y-coordinate. In the xy-plane, a parabola is a symmetric graph such that when two points have the same y-coordinate, these points are equidistant from the vertex, and the x-coordinate of the vertex is halfway between the x-coordinates of these two points. Therefore, for the graph of y=fx, the points with x-coordinates -9\n and 3\n are equidistant from the vertex, h,k, and h\n is halfway between -9\n and 3\n. The value that is halfway between -9\n and 3\n is -9+32, or -3\n. Therefore, \n\th\n\t=\n\t-3\n\n. The equation defining f\n can also be written in vertex form, fx=ax-h2+k. Substituting -3\n for h\n in this equation yields fx=ax--32+k, or fx=ax+32+k. This equation is equivalent to fx=ax2+6x+9+k, or fx=ax2+6ax+9a+k. Since fx=ax2+4x+c, it follows that \n\t\n\t\t6\n\t\ta\n\t\n\t=\n\t4\n\n and \n\t\n\t\t\n\t\t\t9\n\t\t\ta\n\t\t\n\t\t+\n\t\tk\n\t\n\t=\n\tc\n\n. Dividing both sides of the equation \n\t\n\t\t6\n\t\ta\n\t\n\t=\n\t4\n\n by 6\n yields a=46, or \n\ta\n\t=\n\t\n\t\t2\n\t\t3\n\t\n\n. Since 23<1, it's not true that a≥1. Therefore, statement II isn't true. Substituting \n\t2\n\t3\n\n for a\n in the equation \n\t\n\t\t\n\t\t\t9\n\t\t\ta\n\t\t\n\t\t+\n\t\tk\n\t\n\t=\n\tc\n\n yields 923+k=c, or 6+k=c. Subtracting 6\n from both sides of this equation yields \n\tk\n\t=\n\t\n\t\tc\n\t\t-\n\t\t6\n\t\n\n. If k<0, then c-6<0, or c<6. Since c\n could be any value less than 6\n, it's not necessarily true that c<0. Therefore, statement I isn't necessarily true. Thus, neither I nor II must be true.

\n

Choice A is incorrect and may result from conceptual or calculation errors.

\n

Choice B is incorrect and may result from conceptual or calculation errors.

\n

Choice C is incorrect and may result from conceptual or calculation errors.

","correct_answer":["D"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.827000+00:00","source_updated_at":"2023-08-02T20:25:59.827000+00:00","options":[{"label":"A","content_html":"

I only

","ord":0},{"label":"B","content_html":"

II only

","ord":1},{"label":"C","content_html":"

I and II

","ord":2},{"label":"D","content_html":"

Neither I nor II

","ord":3}],"attribution":{"source_id":"cb_question_bank","source_name":"College Board digital SAT question bank (community dump)","rights_holder":"College Board","canonical_url":"https://satsuitequestionbank.collegeboard.org/","retrieved_from":"https://raw.githubusercontent.com/mdn522/sat-question-bank/main/data/cb-digital-questions.json","attribution":"SAT practice questions © College Board, from the official SAT Suite Question Bank. Retrieved via the community dump at github.com/mdn522/sat-question-bank.","disclaimer":"SAT® and PSAT/NMSQT® are trademarks registered by the College Board, which is not affiliated with, does not sponsor, and does not endorse this project.","license":"College Board copyright; source repository carries no licence. Local use only.","redistributable":false,"item_count":2017}}