{"id":"cb_question_bank:40ca55c6-6b20-40d8-a992-9e4f8e904589","source_id":"cb_question_bank","source_item_id":"40ca55c6-6b20-40d8-a992-9e4f8e904589","external_id":"de664bb9-acef-44d5-af54-80d0da2e25bb","ibn":null,"question_id":"18e35375","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"H","score_band":6,"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 left parenthesis x minus 14 right parenthesis left parenthesis x plus 19 right parenthesis The function f is defined by the given equation. For what value of x does f left parenthesis x right parenthesis reach its minimum?","stimulus":"","stimulus_html":null,"stem_html":"

fx=\n\t\n\t\t\n\t\t\tx\n\t\t\t-\n\t\t\t14\n\t\t\n\t\n\t\n\t\t\n\t\t\tx\n\t\t\t+\n\t\t\t19\n\t\t\n\t\n\n

\n

The function f\n is defined by the given equation. For what value of x\n does fx reach its minimum?

","rationale_html":"

Choice D is correct. It's given that fx=x-14x+19, which can be rewritten as fx=x2+5x-266. Since the coefficient of the x2-term is positive, the graph of y=fx in the xy-plane opens upward and reaches its minimum value at its vertex. The x-coordinate of the vertex is the value of x\n such that fx reaches its minimum. For an equation in the form fx=ax2+bx+c, where a\n, b\n, and c\n are constants, the x-coordinate of the vertex is -b2a. For the equation fx=x2+5x-266, a=1, b=5, and c=-266. It follows that the x-coordinate of the vertex is -521, or -52. Therefore, fx reaches its minimum when the value of x\n is -52.

\n

Alternate approach: The value of x\n for the vertex of a parabola is the x-value of the midpoint between the two x-intercepts of the parabola. Since it’s given that fx=x-14x+19, it follows that the two x-intercepts of the graph of y=fx in the xy-plane occur when x=14 and x=-19, or at the points 14,0 and -19,0. The midpoint between two points, x1,y1 and x2,y2, is x1+x22,y1+y22. Therefore, the midpoint between 14,0 and -19,0 is 14+-192,0+02, or -52,0. It follows that fx reaches its minimum when the value of x\n is -52.

\n

Choice A is incorrect. This is the y-coordinate of the y-intercept of the graph of y=fx in the xy-plane.

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Choice B is incorrect. This is one of the x-coordinates of the x-intercepts of the graph of y=fx in the xy-plane.

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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":"2023-08-02T20:25:59.828000+00:00","source_updated_at":"2023-08-02T20:25:59.828000+00:00","options":[{"label":"A","content_html":"

-266\n

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

-19\n

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

\n\t-\n\t\n\t\t33\n\t\t2\n\t\n\n

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

\n\t-\n\t\n\t\t5\n\t\t2\n\t\n\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}}