{"id":"cb_question_bank:386182b1-5c1a-4e03-a741-3956cae54e04","source_id":"cb_question_bank","source_item_id":"386182b1-5c1a-4e03-a741-3956cae54e04","external_id":"b2aba242-20bd-454b-aaca-55c42bdd0501","ibn":null,"question_id":"ee857afb","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"H","score_band":6,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"y equals x squared minus 14 x plus 22 The given equation relates the variables x and y . For what value of x does the value of y reach its minimum?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"y equals x squared minus 14 x plus 22\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<msup>\n\t\t\t<mi>x</mi>\n\t\t\t<mn>2</mn>\n\t\t</msup>\n\t\t<mo>-</mo>\n\t\t<mrow>\n\t\t\t<mn>14</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mn>22</mn>\n\t</mrow>\n</mrow>\n</math></p>\n<p style=\"text-align: left;\">The given equation relates the variables <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"y\"><mi>y</mi>\n</math>. For what value of <math alttext=\"x\"><mi>x</mi>\n</math> does the value of <math alttext=\"y\"><mi>y</mi>\n</math> reach its minimum?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"7\"><mn>7</mn>\n</math>. When an equation is of the form <math alttext=\"y equals a x squared plus b x plus c\"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></math>, where <math alttext=\"a\"><mi>a</mi>\n</math>, <math alttext=\"b\"><mi>b</mi>\n</math>, and <math alttext=\"c\"><mi>c</mi>\n</math> are constants, the value of <math alttext=\"y\"><mi>y</mi>\n</math> reaches its minimum when <math alttext=\"x equals minus StartFraction b Over 2 a EndFraction\"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>b</mi><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></math>. Since the given equation is of the form&nbsp;<math alttext=\"y equals a x squared plus b x plus c\"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></math>, it follows that <math alttext=\"a equals 1\"><mrow>\n\t<mi>a</mi>\n\t<mo>=</mo>\n\t<mn>1</mn>\n</mrow>\n</math>, <math alttext=\"b equals negative 14\"><mrow>\n\t<mi>b</mi>\n\t<mo>=</mo>\n\t<mo>-</mo><mn>14</mn>\n</mrow>\n</math>, and <math alttext=\"c equals 22\"><mrow>\n\t<mi>c</mi>\n\t<mo>=</mo>\n\t<mn>22</mn>\n</mrow>\n</math>. Therefore, the value of <math alttext=\"y\"><mi>y</mi>\n</math> reaches its minimum when <math alttext=\"x equals minus StartFraction left parenthesis negative 14 right parenthesis Over 2 left parenthesis 1 right parenthesis EndFraction\"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mfenced><mrow><mo>-</mo><mn>14</mn></mrow></mfenced><mrow><mn>2</mn><mfenced><mn>1</mn></mfenced></mrow></mfrac></math>, or <math alttext=\"x equals 7\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mn>7</mn>\n</mrow>\n</math>.</p>","correct_answer":["7"],"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":[],"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}}