{"id":"cb_question_bank:74ba1cb8-d88c-4d44-9f89-6aae884d66e2","source_id":"cb_question_bank","source_item_id":"74ba1cb8-d88c-4d44-9f89-6aae884d66e2","external_id":"17ec22bd-b74b-4395-bda1-506081819636","ibn":null,"question_id":"92f812bb","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":"In the xy -plane, a parabola has vertex left parenthesis 9 comma negative 14 right parenthesis and intersects the x -axis at two points. If the equation of the parabola is written in the form y equals a x squared plus b x plus c , where a , b , and c are constants, which of the following could be the value of a plus b plus c ?","stimulus":"","stimulus_html":null,"stem_html":"

In the xy-plane, a parabola has vertex (9,-14) and intersects the x-axis at two points. If the equation of the parabola is written in the form \n\ty\n\t=\n\t\n\t\t\n\t\t\ta\n\t\t\t\n\t\t\t\tx\n\t\t\t\t2\n\t\t\t\n\t\t\n\t\t+\n\t\t\n\t\t\tb\n\t\t\tx\n\t\t\n\t\t+\n\t\tc\n\t\n\n, where a\n, b\n, and c\n are constants, which of the following could be the value of \n\ta\n\t+\n\tb\n\t+\n\tc\n\n?

","rationale_html":"

Choice D is correct. The equation of a parabola in the xy-plane can be written in the form y=ax-h2+k, where a\n is a constant and h,k is the vertex of the parabola. If a\n is positive, the parabola will open upward, and if a\n is negative, the parabola will open downward. It’s given that the parabola has vertex 9,-14. Substituting 9\n for h\n and -14\n for k\n in the equation y=ax-h2+k gives y=ax-92-14, which can be rewritten as y=ax-9x-9-14, or y=ax2-18x+81-14. Distributing the factor of a\n on the right-hand side of this equation yields y=ax2-18ax+81a-14. Therefore, the equation of the parabola, y=ax2-18ax+81a-14, can be written in the form y=ax2+bx+c, where a=a, b=-18a, and c=81a-14. Substituting \n\t-\n\t18\n\ta\n\n for b\n and \n\t\n\t\t81\n\t\ta\n\t\n\t-\n\t14\n\n for c\n in the expression a+b+c yields a+-18a+81a-14, or 64a-14. Since the vertex of the parabola, 9,-14, is below the x-axis, and it’s given that the parabola intersects the x-axis at two points, the parabola must open upward. Therefore, the constant a\n must have a positive value. Setting the expression \n\t\n\t\t64\n\t\ta\n\t\n\t-\n\t14\n\n equal to the value in choice D yields 64a-14=-12. Adding 14\n to both sides of this equation yields \n\t\n\t\t64\n\t\ta\n\t\n\t=\n\t2\n\n. Dividing both sides of this equation by 64\n yields a=264, which is a positive value. Therefore, if the equation of the parabola is written in the form y=ax2+bx+c, where a\n, b\n, and c\n are constants, the value of a+b+c could be -12.

\n

Choice A is incorrect. If the equation of a parabola with a vertex at 9,-14 is written in the form y=ax2+bx+c, where a\n, b\n, and c\n are constants and a+b+c=-23, then the value of a\n will be negative, which means the parabola will open downward, not upward, and will intersect the x-axis at zero points, not two points. 

\n

Choice B is incorrect. If the equation of a parabola with a vertex at 9,-14 is written in the form y=ax2+bx+c, where a\n, b\n, and c\n are constants and a+b+c=-19, then the value of a\n will be negative, which means the parabola will open downward, not upward, and will intersect the x-axis at zero points, not two points.

\n

Choice C is incorrect. If the equation of a parabola with a vertex at 9,-14 is written in the form y=ax2+bx+c, where a\n, b\n, and c\n are constants and a+b+c=-14, then the value of a\n will be 0\n, which is inconsistent with the equation of a parabola.

","correct_answer":["D"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.828000+00:00","source_updated_at":"2023-08-02T20:25:59.828000+00:00","options":[{"label":"A","content_html":"

-23\n

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

-19\n

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

-14\n

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

-12\n

","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}}