{"id":"cb_question_bank:21e9dcf9-1432-4b6d-8d7b-5c3f7526992b","source_id":"cb_question_bank","source_item_id":"21e9dcf9-1432-4b6d-8d7b-5c3f7526992b","external_id":"f44d3920-a7ed-4c31-815c-11dbfe887d43","ibn":null,"question_id":"7e3f8363","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.B.","skill":"Linear functions","difficulty":"M","score_band":4,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"In the xy -plane, the graph of the linear function f contains the points left parenthesis 0 comma 3 right parenthesis and left parenthesis 7 comma 31 right parenthesis . Which equation defines f , where y equals f left parenthesis x right parenthesis ?","stimulus":"","stimulus_html":null,"stem_html":"<p>In the <em>xy</em>-plane, the graph of the linear function <math alttext=\"f\"><mi>f</mi>\n</math> contains the points <math alttext=\"left parenthesis 0 comma 3 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mrow><mn>3</mn></mrow></mrow></mfenced></math> and <math alttext=\"left parenthesis 7 comma 31 right parenthesis\"><mfenced><mrow><mrow><mn>7</mn></mrow><mo>,</mo><mrow><mn>31</mn></mrow></mrow></mfenced></math>. Which equation defines <math alttext=\"f\"><mi>f</mi>\n</math>, where <math alttext=\"y equals f left parenthesis x right parenthesis\"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math>?</p>","rationale_html":"<p style=\"text-align: left;\">Choice C is correct. In the&nbsp;<em>xy</em>-plane, an equation of the graph of 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> represents the slope and <math alttext=\"left parenthesis 0 comma b right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mi>b</mi></mrow></mfenced></math> represents the <em>y</em>-intercept 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>. It’s given that the graph of the linear function <math alttext=\"f\"><mi>f</mi>\n</math>, where <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&nbsp;<em>xy</em>-plane contains the point <math alttext=\"left parenthesis 0 comma 3 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>3</mn></mrow></mfenced></math>. Thus, <math alttext=\"b equals 3\"><mrow>\n\t<mi>b</mi>\n\t<mo>=</mo>\n\t<mn>3</mn>\n</mrow>\n</math>. The slope of the graph of a line containing any two points <math alttext=\"left parenthesis x 1 comma y 1 right parenthesis\"><mfenced><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><msub><mi>y</mi><mn>1</mn></msub></mrow></mfenced></math> and <math alttext=\"left parenthesis x 2 comma y 2 right parenthesis\"><mfenced><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><msub><mi>y</mi><mn>2</mn></msub></mrow></mfenced></math> can be found using the slope formula, <math alttext=\"m equals StartFraction y 2 minus y 1 Over x 2 minus x 1 EndFraction\"><mi>m</mi><mo>=</mo><mfrac><mrow><msub><mi>y</mi><mn>2</mn></msub><mo>-</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>-</mo><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></math>. Since it’s given that the graph of the linear function <math alttext=\"f\"><mi>f</mi>\n</math> contains the points <math alttext=\"left parenthesis 0 comma 3 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>3</mn></mrow></mfenced></math> and&nbsp;<math alttext=\"left parenthesis 7 comma 31 right parenthesis\"><mfenced><mrow><mn>7</mn><mo>,</mo><mn>31</mn></mrow></mfenced></math>, it follows that the slope of the graph of the line containing these points is <math alttext=\"m equals StartFraction 31 minus 3 Over 7 minus 0 EndFraction\"><mi>m</mi><mo>=</mo><mfrac><mrow><mn>31</mn><mo>-</mo><mn>3</mn></mrow><mrow><mn>7</mn><mo>-</mo><mn>0</mn></mrow></mfrac></math>, or <math alttext=\"m equals 4\"><mrow>\n\t<mi>m</mi>\n\t<mo>=</mo>\n\t<mn>4</mn>\n</mrow>\n</math>. Substituting <math alttext=\"4\"><mn>4</mn>\n</math> for <math alttext=\"m\"><mi>m</mi>\n</math> and <math alttext=\"3\"><mn>3</mn>\n</math> for <math alttext=\"b\"><mi>b</mi>\n</math> in <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 4 x plus 3\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>3</mn></math>.</p>\n<p style=\"text-align: left;\">Choice A is incorrect. This function represents a graph with a slope of <math alttext=\"28\"><mn>28</mn>\n</math> and a <em>y</em>-intercept of <math alttext=\"left parenthesis 0 comma 34 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>34</mn></mrow></mfenced></math>.</p>\n<p style=\"text-align: left;\">Choice B is incorrect. This function represents a graph with a slope of <math alttext=\"3\"><mn>3</mn>\n</math> and a <em>y</em>-intercept of <math alttext=\"left parenthesis 0 comma 38 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>38</mn></mrow></mfenced></math>.</p>\n<p style=\"text-align: left;\">Choice D is incorrect. This function represents a graph with a slope of <math alttext=\"7\"><mn>7</mn>\n</math> and a <em>y</em>-intercept of <math alttext=\"left parenthesis 0 comma 3 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>3</mn></mrow></mfenced></math>.</p>","correct_answer":["C"],"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 28 x plus 34\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow><mn>28</mn></mrow><mi>x</mi><mo>+</mo><mrow><mn>34</mn></mrow></math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"f left parenthesis x right parenthesis equals 3 x plus 38\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow><mn>3</mn></mrow><mi>x</mi><mo>+</mo><mrow><mn>38</mn></mrow></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"f left parenthesis x right parenthesis equals 4 x plus 3\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow><mn>4</mn></mrow><mi>x</mi><mo>+</mo><mrow><mn>3</mn></mrow></math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"f left parenthesis x right parenthesis equals 7 x plus 3\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow><mn>7</mn></mrow><mi>x</mi><mo>+</mo><mrow><mn>3</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}}