{"id":"cb_question_bank:16bcfef1-990d-47cc-ba3c-1baa993f2ca6","source_id":"cb_question_bank","source_item_id":"16bcfef1-990d-47cc-ba3c-1baa993f2ca6","external_id":"5d6ca18c-fd03-4a2f-89ab-d392b028d18b","ibn":null,"question_id":"b8f13a3a","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"H","score_band":7,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"Function f is defined by f left parenthesis x right parenthesis equals minus a Superscript x Baseline plus b , where a and b are constants. In the xy -plane, the graph of y equals f left parenthesis x right parenthesis minus 12 has a y- intercept at left parenthesis 0 comma negative StartFraction 75 Over 7 EndFraction right parenthesis . The product of a and b is StartFraction 320 Over 7 EndFraction . What is the value of a ?","stimulus":"","stimulus_html":null,"stem_html":"<p>Function <math alttext=\"f\"><mi>f</mi>\n</math> is defined by <math alttext=\"f left parenthesis x right parenthesis equals minus a Superscript x Baseline plus b\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><msup><mi>a</mi><mi>x</mi></msup><mo>+</mo><mi>b</mi></math>, where <math alttext=\"a\"><mi>a</mi>\n</math> and <math alttext=\"b\"><mi>b</mi>\n</math> are constants. In the <em>xy</em>-plane, the graph of <math alttext=\"y equals f left parenthesis x right parenthesis minus 12\"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>-</mo><mrow><mn>12</mn></mrow></math> has a <em>y-</em>intercept at <math alttext=\"left parenthesis 0 comma negative StartFraction 75 Over 7 EndFraction right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mrow><mrow><mo>-</mo><mfrac><mn>75</mn><mn>7</mn></mfrac></mrow></mrow></mrow></mfenced></math>. The product of <math alttext=\"a\"><mi>a</mi>\n</math> and <math alttext=\"b\"><mi>b</mi>\n</math> is <math alttext=\"StartFraction 320 Over 7 EndFraction\"><mfrac>\n\t<mn>320</mn>\n\t<mn>7</mn>\n</mfrac>\n</math>. What is the value of <math alttext=\"a\"><mi>a</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"20\"><mn>20</mn>\n</math>. It’s given that <math alttext=\"f left parenthesis x right parenthesis equals minus a Superscript x Baseline plus b\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><msup><mi>a</mi><mi>x</mi></msup><mo>+</mo><mi>b</mi></math>. Substituting&nbsp;<math alttext=\"minus a Superscript x plus b\"><mo>-</mo><msup><mi>a</mi><mi>x</mi></msup><mo>+</mo><mi>b</mi></math> for <math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> in the equation <math alttext=\"y equals f left parenthesis x right parenthesis minus 12\"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>-</mo><mn>12</mn></math> yields <math alttext=\"y equals minus a Superscript x Baseline plus b minus 12\"><mi>y</mi><mo>=</mo><mo>-</mo><msup><mi>a</mi><mi>x</mi></msup><mo>+</mo><mi>b</mi><mo>-</mo><mn>12</mn></math>. It’s given that the <em>y</em>-intercept of the graph of <math alttext=\"y equals f left parenthesis x right parenthesis minus 12\"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>-</mo><mn>12</mn></math> is <math alttext=\"left parenthesis 0 comma negative StartFraction 75 Over 7 EndFraction right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mo>-</mo><mfrac><mn>75</mn><mn>7</mn></mfrac></mrow></mfenced></math>. Substituting <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"negative StartFraction 75 Over 7 EndFraction\"><mo>-</mo><mfrac><mn>75</mn><mn>7</mn></mfrac></math> for <math alttext=\"y\"><mi>y</mi>\n</math> in the equation <math alttext=\"y equals minus a Superscript x Baseline plus b minus 12\"><mi>y</mi><mo>=</mo><mo>-</mo><msup><mi>a</mi><mi>x</mi></msup><mo>+</mo><mi>b</mi><mo>-</mo><mn>12</mn></math> yields <math alttext=\"negative StartFraction 75 Over 7 EndFraction equals minus a Superscript 0 Baseline plus b minus 12\"><mo>-</mo><mfrac><mn>75</mn><mn>7</mn></mfrac><mo>=</mo><mo>-</mo><msup><mi>a</mi><mn>0</mn></msup><mo>+</mo><mi>b</mi><mo>-</mo><mn>12</mn></math>, which is equivalent to <math alttext=\"negative StartFraction 75 Over 7 EndFraction equals negative 1 plus b minus 12\"><mo>-</mo><mfrac><mn>75</mn><mn>7</mn></mfrac><mo>=</mo><mo>-</mo><mn>1</mn><mo>+</mo><mi>b</mi><mo>-</mo><mn>12</mn></math>, or <math alttext=\"negative StartFraction 75 Over 7 EndFraction equals b minus 13\"><mo>-</mo><mfrac><mn>75</mn><mn>7</mn></mfrac><mo>=</mo><mi>b</mi><mo>-</mo><mn>13</mn></math>. Adding <math alttext=\"13\"><mn>13</mn>\n</math> to both sides of this equation yields <math alttext=\"StartFraction 16 Over 7 EndFraction equals b\"><mfrac><mn>16</mn><mn>7</mn></mfrac><mo>=</mo><mi>b</mi></math>. It’s given that the product of <math alttext=\"a\"><mi>a</mi>\n</math> and <math alttext=\"b\"><mi>b</mi>\n</math> is <math alttext=\"StartFraction 320 Over 7 EndFraction\"><mfrac>\n\t<mn>320</mn>\n\t<mn>7</mn>\n</mfrac>\n</math>, or <math alttext=\"a b equals StartFraction 320 Over 7 EndFraction\"><mrow>\n\t<mrow>\n\t\t<mi>a</mi>\n\t\t<mi>b</mi>\n\t</mrow>\n\t<mo>=</mo>\n\t<mfrac>\n\t\t<mn>320</mn>\n\t\t<mn>7</mn>\n\t</mfrac>\n</mrow>\n</math>. Substituting <math alttext=\"StartFraction 16 Over 7 EndFraction\"><mfrac>\n\t<mn>16</mn>\n\t<mn>7</mn>\n</mfrac>\n</math> for <math alttext=\"b\"><mi>b</mi>\n</math> in this equation yields <math alttext=\"left parenthesis a right parenthesis left parenthesis StartFraction 16 Over 7 EndFraction right parenthesis equals StartFraction 320 Over 7 EndFraction\"><mfenced><mi>a</mi></mfenced><mfenced><mfrac><mn>16</mn><mn>7</mn></mfrac></mfenced><mo>=</mo><mfrac><mn>320</mn><mn>7</mn></mfrac></math>. Dividing both sides of this equation by <math alttext=\"StartFraction 16 Over 7 EndFraction\"><mfrac>\n\t<mn>16</mn>\n\t<mn>7</mn>\n</mfrac>\n</math> yields <math alttext=\"a equals 20\"><mrow>\n\t<mi>a</mi>\n\t<mo>=</mo>\n\t<mn>20</mn>\n</mrow>\n</math>.</p>","correct_answer":["20"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.827000+00:00","source_updated_at":"2023-08-02T20:25:59.827000+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}}