{"id":"cb_question_bank:f3d0fa6b-38a1-4377-a828-df1e49c5e0c0","source_id":"cb_question_bank","source_item_id":"f3d0fa6b-38a1-4377-a828-df1e49c5e0c0","external_id":"b76e3ad3-f70a-40d5-b897-a3b5e368da81","ibn":null,"question_id":"1853bb35","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":"For the function q , the value of q left parenthesis x right parenthesis decreases by 45 percent sign for every increase in the value of x by 1 . If q left parenthesis 0 right parenthesis equals 14 , which equation defines q ?","stimulus":"","stimulus_html":null,"stem_html":"<p>For the function <math alttext=\"q\"><mi>q</mi>\n</math>, the value of <math alttext=\"q left parenthesis x right parenthesis\"><mi>q</mi><mfenced><mi>x</mi></mfenced></math> decreases by <math alttext=\"45 percent sign\"><mn>45</mn>\n<mo>%</mo></math> for every increase in the value of <math alttext=\"x\"><mi>x</mi>\n</math> by <math alttext=\"1\"><mn>1</mn>\n</math>. If <math alttext=\"q left parenthesis 0 right parenthesis equals 14\"><mi>q</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mrow><mn>14</mn></mrow></math>, which equation defines <math alttext=\"q\"><mi>q</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">Choice C is correct. Since the value of&nbsp;<math alttext=\"q left parenthesis x right parenthesis\"><mi>q</mi><mfenced><mi>x</mi></mfenced></math> decreases by a fixed percentage,&nbsp;<math alttext=\"45 percent sign\"><mn>45</mn><mo>%</mo></math>, for every increase in the value of <math alttext=\"x\"><mi>x</mi>\n</math> by <math alttext=\"1\"><mn>1</mn>\n</math>, the function <math alttext=\"q\"><mi>q</mi>\n</math> is a decreasing exponential function. A decreasing exponential function can be written in the form <math alttext=\"q left parenthesis x right parenthesis equals a left parenthesis 1 minus StartFraction p Over 100 EndFraction right parenthesis Superscript x\"><mi>q</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>p</mi><mn>100</mn></mfrac></mrow></mfenced><mi>x</mi></msup></math>, where <math alttext=\"a\"><mi>a</mi>\n</math> is the value of&nbsp;<math alttext=\"q left parenthesis 0 right parenthesis\"><mi>q</mi><mfenced><mn>0</mn></mfenced></math> and the value of <math alttext=\"q left parenthesis x right parenthesis\"><mi>q</mi><mfenced><mi>x</mi></mfenced></math> decreases by <math alttext=\"p percent sign\"><mi>p</mi><mo>%</mo></math> for every increase in the value of <math alttext=\"x\"><mi>x</mi>\n</math> by <math alttext=\"1\"><mn>1</mn>\n</math>. If <math alttext=\"q left parenthesis 0 right parenthesis equals 14\"><mi>q</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>14</mn></math>, then <math alttext=\"a equals 14\"><mrow>\n\t<mi>a</mi>\n\t<mo>=</mo>\n\t<mn>14</mn>\n</mrow>\n</math>. Since the value of <math alttext=\"q left parenthesis x right parenthesis\"><mi>q</mi><mfenced><mi>x</mi></mfenced></math> decreases by&nbsp;<math alttext=\"45 percent sign\"><mn>45</mn><mo>%</mo></math> for every increase in the value of <math alttext=\"x\"><mi>x</mi>\n</math> by <math alttext=\"1\"><mn>1</mn>\n</math>, <math alttext=\"p equals 45\"><mrow>\n\t<mi>p</mi>\n\t<mo>=</mo>\n\t<mn>45</mn>\n</mrow>\n</math>. Substituting <math alttext=\"14\"><mn>14</mn>\n</math> for <math alttext=\"a\"><mi>a</mi>\n</math> and <math alttext=\"45\"><mn>45</mn>\n</math> for <math alttext=\"p\"><mi>p</mi>\n</math> in the equation&nbsp;<math alttext=\"q left parenthesis x right parenthesis equals a left parenthesis 1 minus StartFraction p Over 100 EndFraction right parenthesis Superscript x\"><mi>q</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>p</mi><mn>100</mn></mfrac></mrow></mfenced><mi>x</mi></msup></math> yields&nbsp;<math alttext=\"q left parenthesis x right parenthesis equals 14 left parenthesis 1 minus StartFraction 45 Over 100 EndFraction right parenthesis Superscript x\"><mi>q</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>14</mn><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mn>45</mn><mn>100</mn></mfrac></mrow></mfenced><mi>x</mi></msup></math>, which is equivalent to <math alttext=\"q left parenthesis x right parenthesis equals 14 left parenthesis 1 minus 0.45 right parenthesis Superscript x\"><mi>q</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>14</mn><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mn>0.45</mn></mrow></mfenced><mi>x</mi></msup></math>, or <math alttext=\"q left parenthesis x right parenthesis equals 14 left parenthesis 0.55 right parenthesis Superscript x\"><mi>q</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>14</mn><msup><mfenced><mn>0.55</mn></mfenced><mi>x</mi></msup></math>.</p>\n<p style=\"text-align: left;\">Choice A is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice B is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice D is incorrect. For this function, the value of&nbsp;<math alttext=\"q left parenthesis x right parenthesis\"><mi>q</mi><mfenced><mi>x</mi></mfenced></math> increases, rather than decreases, by&nbsp;<math alttext=\"45 percent sign\"><mn>45</mn><mo>%</mo></math> for every increase in the value of <math alttext=\"x\"><mi>x</mi>\n</math> by <math alttext=\"1\"><mn>1</mn>\n</math>.</p>","correct_answer":["C"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.825000+00:00","source_updated_at":"2023-08-02T20:25:59.825000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"q left parenthesis x right parenthesis equals 0.55 left parenthesis 14 right parenthesis Superscript x\"><mi>q</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow><mn>0.55</mn></mrow><msup><mfenced><mrow><mn>14</mn></mrow></mfenced><mi>x</mi></msup></math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"q left parenthesis x right parenthesis equals 1.45 left parenthesis 14 right parenthesis Superscript x\"><mi>q</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow><mn>1.45</mn></mrow><msup><mfenced><mrow><mn>14</mn></mrow></mfenced><mi>x</mi></msup></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"q left parenthesis x right parenthesis equals 14 left parenthesis 0.55 right parenthesis Superscript x\"><mi>q</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow><mn>14</mn></mrow><msup><mfenced><mrow><mn>0.55</mn></mrow></mfenced><mi>x</mi></msup></math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"q left parenthesis x right parenthesis equals 14 left parenthesis 1.45 right parenthesis Superscript x\"><mi>q</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow><mn>14</mn></mrow><msup><mfenced><mrow><mn>1.45</mn></mrow></mfenced><mi>x</mi></msup></math></p>","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}}