{"id":"cb_question_bank:2dced346-b788-4d86-a8fb-ce069b5bc569","source_id":"cb_question_bank","source_item_id":"2dced346-b788-4d86-a8fb-ce069b5bc569","external_id":"2ae9a93c-e2a5-4e1e-9012-cdaf6592e09a","ibn":null,"question_id":"841ef26c","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 4 x squared plus 64 x plus 262 The function g is defined by g left parenthesis x right parenthesis equals f left parenthesis x plus 5 right parenthesis . For what value of x does g left parenthesis x right parenthesis reach its minimum?","stimulus":"","stimulus_html":null,"stem_html":"

fx=\n\t\n\t\t4\n\t\t\n\t\t\tx\n\t\t\t2\n\t\t\n\t\n\t+\n\t\n\t\t64\n\t\tx\n\t\n\t+\n\t262\n\n

\n

The function g\n is defined by gx=fx+5. For what value of x\n does gx reach its minimum?

","rationale_html":"

Choice A is correct. It's given that gx=fx+5. Since fx=4x2+64x+262, it follows that fx+5=4x+52+64x+5+262. Expanding the quantity x+52 in this equation yields fx+5=4x2+10x+25+64x+5+262. Distributing the 4\n and the 64\n yields fx+5=4x2+40x+100+64x+320+262. Combining like terms yields fx+5=4x2+104x+682. Therefore, gx=4x2+104x+682. For a quadratic function defined by an equation of the form gx=ax-h2+k, where a\n, h\n, and k\n are constants and a\n is positive, gx reaches its minimum, k\n, when the value of x\n is h\n. The equation gx=4x2+104x+682 can be rewritten in this form by completing the square. This equation is equivalent to gx=4x2+26+682, or gx=4x2+26x+169-169+682. This equation can be rewritten as gx=4x+132-169+682, or gx=4x+132-4169+682, which is equivalent to gx=4x+132+6. This equation is in the form gx=ax-h2+k, where \n\ta\n\t=\n\t4\n\n, \n\th\n\t=\n\t-13\n\n, and \n\tk\n\t=\n\t6\n\n. Therefore, gx reaches its minimum when the value of x\n is -13\n.

\n

Choice B is incorrect. This is the value of x\n for which fx, rather than gx, reaches its minimum.

\n

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

\n

Choice D is incorrect. This is the value of x\n for which fx-5, rather than fx+5, reaches its minimum.

","correct_answer":["A"],"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":"

-13\n

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

-8\n

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

-5\n

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

-3\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}}