{"id":"cb_question_bank:857d6a0c-f3b9-4dde-84f8-57727f7a50e2","source_id":"cb_question_bank","source_item_id":"857d6a0c-f3b9-4dde-84f8-57727f7a50e2","external_id":"b7b056ed-d7cc-4901-9a1b-f261a1455bee","ibn":null,"question_id":"bf883fde","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.B.","skill":"Linear functions","difficulty":"E","score_band":2,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"For the function f , the graph of y equals f left parenthesis x right parenthesis in the xy -plane has a slope of 3 and passes through the point left parenthesis 0 comma negative 8 right parenthesis . Which equation defines f ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">For the function <math alttext=\"f\"><mi>f</mi>\n</math>, the graph of&nbsp;<math alttext=\"y equals f left parenthesis x right parenthesis\"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> in the&nbsp;<em>xy</em>-plane has a slope of <math alttext=\"3\"><mn>3</mn>\n</math> and passes through the point&nbsp;<math alttext=\"left parenthesis 0 comma negative 8 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mo>-</mo><mrow><mn>8</mn></mrow></mrow></mfenced></math>. Which equation defines <math alttext=\"f\"><mi>f</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">Choice B is correct. An equation defining a linear function can be written in the form <math alttext=\"f left parenthesis x right parenthesis equals m x plus b\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>b</mi></math>, where <math alttext=\"m\"><mi>m</mi>\n</math> and <math alttext=\"b\"><mi>b</mi>\n</math> are constants, <math alttext=\"m\"><mi>m</mi>\n</math> is the slope of the graph of <math alttext=\"y equals f left parenthesis x right parenthesis\"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> in the <em>xy</em>-plane, and <math alttext=\"left parenthesis 0 comma b right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mi>b</mi></mrow></mfenced></math> is the <em>y</em>-intercept of the graph. It's given that for the function <math alttext=\"f\"><mi>f</mi>\n</math>, the graph of <math alttext=\"y equals f left parenthesis x right parenthesis\"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> in the <em>xy</em>-plane has a slope of <math alttext=\"3\"><mn>3</mn>\n</math>. Therefore, <math alttext=\"m equals 3\"><mi>m</mi><mo>=</mo><mn>3</mn></math>. It's also given that this graph passes through the point <math alttext=\"left parenthesis 0 comma negative 8 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mo>-</mo><mn>8</mn></mrow></mfenced></math>. Therefore, the <em>y</em>-intercept of the graph is <math alttext=\"left parenthesis 0 comma negative 8 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mo>-</mo><mn>8</mn></mrow></mfenced></math>, and it follows that <math alttext=\"b equals negative 8\"><mi>b</mi><mo>=</mo><mo>-</mo><mn>8</mn></math>. Substituting <math alttext=\"3\"><mn>3</mn>\n</math> for <math alttext=\"m\"><mi>m</mi>\n</math> and <math alttext=\"negative 8\"><mo>-</mo><mn>8</mn>\n</math> for <math alttext=\"b\"><mi>b</mi>\n</math> in the equation&nbsp;<math alttext=\"f left parenthesis x right parenthesis equals m x plus b\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>b</mi></math> yields <math alttext=\"f left parenthesis x right parenthesis equals 3 x minus 8\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>8</mn></math>. Thus, the equation that defines <math alttext=\"f\"><mi>f</mi>\n</math> is&nbsp;<math alttext=\"f left parenthesis x right parenthesis equals 3 x minus 8\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>8</mn></math>.</p>\n<p style=\"text-align: left;\">Choice A is incorrect. For this function, the graph of&nbsp;<math alttext=\"y equals f left parenthesis x right parenthesis\"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> in the <em>xy</em>-plane passes through the point <math alttext=\"left parenthesis 0 comma 0 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math>, not&nbsp;<math alttext=\"left parenthesis 0 comma negative 8 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mo>-</mo><mn>8</mn></mrow></mfenced></math>.</p>\n<p style=\"text-align: left;\">Choice C is incorrect. For this function, the graph of&nbsp;<math alttext=\"y equals f left parenthesis x right parenthesis\"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> in the <em>xy</em>-plane passes through the point <math alttext=\"left parenthesis 0 comma 5 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>5</mn></mrow></mfenced></math>, not <math alttext=\"left parenthesis 0 comma negative 8 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mo>-</mo><mn>8</mn></mrow></mfenced></math>.</p>\n<p style=\"text-align: left;\">Choice D is incorrect. For this function, the graph of&nbsp;<math alttext=\"y equals f left parenthesis x right parenthesis\"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> in the <em>xy</em>-plane passes through the point <math alttext=\"left parenthesis 0 comma 11 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>11</mn></mrow></mfenced></math>, not <math alttext=\"left parenthesis 0 comma negative 8 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mo>-</mo><mn>8</mn></mrow></mfenced></math>.</p>","correct_answer":["B"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.812000+00:00","source_updated_at":"2023-08-02T20:25:59.812000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"f left parenthesis x right parenthesis equals 3 x\"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mrow><mn>3</mn></mrow><mi>x</mi></math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"f left parenthesis x right parenthesis equals 3 x minus 8\"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mrow><mn>3</mn></mrow><mi>x</mi><mo>-</mo><mrow><mn>8</mn></mrow></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"f left parenthesis x right parenthesis equals 3 x plus 5\"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mrow><mn>3</mn></mrow><mi>x</mi><mo>+</mo><mrow><mn>5</mn></mrow></math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"f left parenthesis x right parenthesis equals 3 x plus 11\"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mrow><mn>3</mn></mrow><mi>x</mi><mo>+</mo><mrow><mn>11</mn></mrow></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}}