{"id":"cb_question_bank:f0a35adc-1da3-4beb-9efe-f6362c4566cc","source_id":"cb_question_bank","source_item_id":"f0a35adc-1da3-4beb-9efe-f6362c4566cc","external_id":"65743096-f9d1-4ab0-9a10-3acd3ea46114","ibn":null,"question_id":"4d7064a6","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 10 right parenthesis left parenthesis x plus 13 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\t10\n\t\t\n\t\n\t\n\t\t\n\t\t\tx\n\t\t\t+\n\t\t\t13\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-10x+13, which can be rewritten as  fx=x2+3x-130. 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+3x-130, a=1, b=3, and c=-130. It follows that the x-coordinate of the vertex is -321, or -32. Therefore, fx reaches its minimum when the value of x\n is -32.

\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-10x+13, it follows that the two x-intercepts of the graph of y=fx in the xy-plane occur when x=10 and x=-13, or at the points 10,0 and -13,0. The midpoint between two points, x1,y1 and x2,y2, is x1+x22,y1+y22. Therefore, the midpoint between 10,0 and -13,0 is 10+-132,0+02, or -32,0. It follows that fx reaches its minimum when the value of x\n is -32.

\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":"

-130\n

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

-13\n

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

\n\t-\n\t\n\t\t23\n\t\t2\n\t\n\n

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

\n\t-\n\t\n\t\t3\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}}