{"id":"cb_question_bank:d42b9470-8f30-456f-b5f8-8b54a8feb609","source_id":"cb_question_bank","source_item_id":"d42b9470-8f30-456f-b5f8-8b54a8feb609","external_id":"59249867-1d27-418b-aac8-59cddb26069b","ibn":null,"question_id":"04bbce67","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":"f left parenthesis x right parenthesis equals left parenthesis x plus 7 right parenthesis squared plus 4 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":"<p style=\"text-align: center;\"><math alttext=\"f left parenthesis x right parenthesis equals left parenthesis x plus 7 right parenthesis squared plus 4\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow>\n\t<msup>\n\t\t<mfenced>\n\t\t\t<mrow>\n\t\t\t\t<mi>x</mi>\n\t\t\t\t<mo>+</mo>\n\t\t\t\t<mn>7</mn>\n\t\t\t</mrow>\n\t\t</mfenced>\n\t\t<mn>2</mn>\n\t</msup>\n\t<mo>+</mo>\n\t<mn>4</mn>\n</mrow>\n</math></p>\n<p>The function <math alttext=\"f\"><mi>f</mi>\n</math> is defined by the given equation. For what value of <math alttext=\"x\"><mi>x</mi>\n</math> does&nbsp;<math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> reach its minimum?</p>","rationale_html":"<p>The correct answer is <math alttext=\"negative 7\"><mo>-</mo><mn>7</mn>\n</math>. For a quadratic function defined by an equation of the form &nbsp;<math alttext=\"f left parenthesis x right parenthesis equals a left parenthesis x minus h right parenthesis squared plus k\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>h</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mi>k</mi></math>, where <math alttext=\"a\"><mi>a</mi>\n</math>, <math alttext=\"h\"><mi>h</mi>\n</math>, and <math alttext=\"k\"><mi>k</mi>\n</math> are constants and <math alttext=\"a greater than 0\"><mi>a</mi><mo>&gt;</mo><mn>0</mn></math>, the function reaches its minimum when <math alttext=\"x equals h\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mi>h</mi>\n</mrow>\n</math>. In the given function, <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=\"h equals negative 7\"><mrow>\n\t<mi>h</mi>\n\t<mo>=</mo>\n\t<mo>-</mo><mn>7</mn>\n</mrow>\n</math>, and <math alttext=\"k equals 4\"><mrow>\n\t<mi>k</mi>\n\t<mo>=</mo>\n\t<mn>4</mn>\n</mrow>\n</math>. Therefore, the value of <math alttext=\"x\"><mi>x</mi>\n</math> for which <math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> reaches its minimum is <math alttext=\"negative 7\"><mo>-</mo><mn>7</mn>\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}}