{"id":"cb_question_bank:f51c8149-eb58-439d-8c48-043f8442e010","source_id":"cb_question_bank","source_item_id":"f51c8149-eb58-439d-8c48-043f8442e010","external_id":"0fbeb13e-6775-487f-8f08-a6bf51a5bad6","ibn":null,"question_id":"a8ae0d22","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":"Two variables, x and y , are related such that for each increase of 1 in the value of x , the value of y increases by a factor of 4 . When x equals 0 , y equals 200 . Which equation represents this relationship?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">Two variables, <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"y\"><mi>y</mi>\n</math>, are related such that for each increase of <math alttext=\"1\"><mn>1</mn>\n</math> in the value of <math alttext=\"x\"><mi>x</mi>\n</math>, the value of <math alttext=\"y\"><mi>y</mi>\n</math> increases by a factor of <math alttext=\"4\"><mn>4</mn>\n</math>. When <math alttext=\"x equals 0\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math>, <math alttext=\"y equals 200\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mn>200</mn>\n</mrow>\n</math>. Which equation represents this relationship?</p>","rationale_html":"<p style=\"text-align: left;\">Choice D is correct. Since the value of <math alttext=\"y\"><mi>y</mi>\n</math> increases by a constant factor, <math alttext=\"4\"><mn>4</mn>\n</math>, for each increase of <math alttext=\"1\"><mn>1</mn>\n</math> in the value of <math alttext=\"x\"><mi>x</mi>\n</math>, the relationship between <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"y\"><mi>y</mi>\n</math> is exponential. An exponential relationship between <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"y\"><mi>y</mi>\n</math> can be represented by an equation of the form <math alttext=\"y equals a left parenthesis b right parenthesis Superscript x\"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mfenced><mi>b</mi></mfenced><mi>x</mi></msup></math>, where <math alttext=\"a\"><mi>a</mi>\n</math> is the value of <math alttext=\"x\"><mi>x</mi>\n</math> when <math alttext=\"y equals 0\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math> and <math alttext=\"y\"><mi>y</mi>\n</math> increases by a factor of <math alttext=\"b\"><mi>b</mi>\n</math> for each increase of <math alttext=\"1\"><mn>1</mn>\n</math> in the value of <math alttext=\"x\"><mi>x</mi>\n</math>. Since <math alttext=\"y equals 200\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mn>200</mn>\n</mrow>\n</math> when <math alttext=\"x equals 0\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math>, <math alttext=\"a equals 200\"><mrow>\n\t<mi>a</mi>\n\t<mo>=</mo>\n\t<mn>200</mn>\n</mrow>\n</math>. Since <math alttext=\"y\"><mi>y</mi>\n</math> increases by a factor of <math alttext=\"4\"><mn>4</mn>\n</math> for each increase of <math alttext=\"1\"><mn>1</mn>\n</math> in the value of <math alttext=\"x\"><mi>x</mi>\n</math>, <math alttext=\"b equals 4\"><mrow>\n\t<mi>b</mi>\n\t<mo>=</mo>\n\t<mn>4</mn>\n</mrow>\n</math>. Substituting <math alttext=\"200\"><mn>200</mn>\n</math> for <math alttext=\"a\"><mi>a</mi>\n</math> and <math alttext=\"4\"><mn>4</mn>\n</math> for <math alttext=\"b\"><mi>b</mi>\n</math> in the equation <math alttext=\"y equals a left parenthesis b right parenthesis Superscript x\"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mfenced><mi>b</mi></mfenced><mi>x</mi></msup></math> yields <math alttext=\"y equals 200 left parenthesis 4 right parenthesis Superscript x\"><mi>y</mi><mo>=</mo><mn>200</mn><msup><mfenced><mn>4</mn></mfenced><mi>x</mi></msup></math>. Thus, the equation <math alttext=\"y equals 200 left parenthesis 4 right parenthesis Superscript x\"><mi>y</mi><mo>=</mo><mn>200</mn><msup><mfenced><mn>4</mn></mfenced><mi>x</mi></msup></math> represents the relationship between <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"y\"><mi>y</mi>\n</math>.</p>\n<p style=\"text-align: left;\">Choice A is incorrect and may result from conceptual errors.</p>\n<p style=\"text-align: left;\">Choice B is incorrect. This equation represents a relationship where for each increase of <math alttext=\"1\"><mn>1</mn>\n</math> in the value of <math alttext=\"x\"><mi>x</mi>\n</math>, the value of <math alttext=\"y\"><mi>y</mi>\n</math> increases by a factor of <math alttext=\"200\"><mn>200</mn>\n</math>, not <math alttext=\"4\"><mn>4</mn>\n</math>, and when <math alttext=\"x equals 0\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math>, <math alttext=\"y\"><mi>y</mi>\n</math> is equal to <math alttext=\"4\"><mn>4</mn>\n</math>, not <math alttext=\"200\"><mn>200</mn>\n</math>.</p>\n<p style=\"text-align: left;\">Choice C is incorrect and may result from conceptual errors.</p>","correct_answer":["D"],"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=\"y equals 4 left parenthesis x right parenthesis Superscript 200\"><mi>y</mi><mo>=</mo><mn>4</mn><msup><mfenced><mi>x</mi></mfenced><mn>200</mn></msup></math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"y equals 4 left parenthesis 200 right parenthesis Superscript x\"><mi>y</mi><mo>=</mo><mn>4</mn><msup><mfenced><mn>200</mn></mfenced><mi>x</mi></msup></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"y equals 200 left parenthesis x right parenthesis Superscript 4\"><mi>y</mi><mo>=</mo><mn>200</mn><msup><mfenced><mi>x</mi></mfenced><mn>4</mn></msup></math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"y equals 200 left parenthesis 4 right parenthesis Superscript x\"><mi>y</mi><mo>=</mo><mn>200</mn><msup><mfenced><mn>4</mn></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}}