{"id":"cb_question_bank:3144e26c-9fd5-4ab7-9e60-aad1dbc9de3e","source_id":"cb_question_bank","source_item_id":"3144e26c-9fd5-4ab7-9e60-aad1dbc9de3e","external_id":"212a6362-e804-4f52-add2-1e4b1d20094c","ibn":null,"question_id":"d135f4bf","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"H","score_band":6,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The function f is defined by f left parenthesis x right parenthesis equals left parenthesis x minus 6 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis x plus 6 right parenthesis . In the xy -plane, the graph of y equals g left parenthesis x right parenthesis is the result of translating the graph of y equals f left parenthesis x right parenthesis up 4 units. What is the value of g left parenthesis 0 right parenthesis ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">The function <math alttext=\"f\"><mi>f</mi>\n</math> is defined by <math alttext=\"f left parenthesis x right parenthesis equals left parenthesis x minus 6 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis x plus 6 right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow>\n\t<mfenced>\n\t\t<mrow>\n\t\t\t<mi>x</mi>\n\t\t\t<mo>-</mo>\n\t\t\t<mn>6</mn>\n\t\t</mrow>\n\t</mfenced>\n\t<mfenced>\n\t\t<mrow>\n\t\t\t<mi>x</mi>\n\t\t\t<mo>-</mo>\n\t\t\t<mn>2</mn>\n\t\t</mrow>\n\t</mfenced>\n\t<mfenced>\n\t\t<mrow>\n\t\t\t<mi>x</mi>\n\t\t\t<mo>+</mo>\n\t\t\t<mn>6</mn>\n\t\t</mrow>\n\t</mfenced>\n</mrow>\n</math>. In the <em>xy</em>-plane, the graph of <math alttext=\"y equals g left parenthesis x right parenthesis\"><mi>y</mi><mo>=</mo><mi>g</mi><mfenced><mi>x</mi></mfenced></math> is the result of translating 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> up <math alttext=\"4\"><mn>4</mn>\n</math> units. What is the value of <math alttext=\"g left parenthesis 0 right parenthesis\"><mi>g</mi><mfenced><mn>0</mn></mfenced></math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"76\"><mn>76</mn>\n</math>. It's given that the graph of <math alttext=\"y equals g left parenthesis x right parenthesis\"><mi>y</mi><mo>=</mo><mi>g</mi><mfenced><mi>x</mi></mfenced></math>&nbsp;is the result of translating 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> up <math alttext=\"4\"><mn>4</mn>\n</math> units in the <em>xy</em>-plane. It follows that the graph of&nbsp;<math alttext=\"y equals g left parenthesis x right parenthesis\"><mi>y</mi><mo>=</mo><mi>g</mi><mfenced><mi>x</mi></mfenced></math> is the same as the graph of&nbsp;<math alttext=\"y equals f left parenthesis x right parenthesis plus 4\"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>+</mo><mn>4</mn></math>. Substituting <math alttext=\"g left parenthesis x right parenthesis\"><mi>g</mi><mfenced><mi>x</mi></mfenced></math> for <math alttext=\"y\"><mi>y</mi>\n</math> in the equation <math alttext=\"y equals f left parenthesis x right parenthesis plus 4\"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>+</mo><mn>4</mn></math> yields&nbsp;<math alttext=\"g left parenthesis x right parenthesis equals f left parenthesis x right parenthesis plus 4\"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>+</mo><mn>4</mn></math>. It's given that <math alttext=\"f left parenthesis x right parenthesis equals left parenthesis x minus 6 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis x plus 6 right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>6</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></mfenced></math>. Substituting <math alttext=\"left parenthesis x minus 6 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis x plus 6 right parenthesis\"><mfenced><mrow><mi>x</mi><mo>-</mo><mn>6</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></mfenced></math> for <math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> in the equation <math alttext=\"g left parenthesis x right parenthesis equals f left parenthesis x right parenthesis plus 4\"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>+</mo><mn>4</mn></math> yields <math alttext=\"g left parenthesis x right parenthesis equals left parenthesis x minus 6 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis x plus 6 right parenthesis plus 4\"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>6</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></mfenced><mo>+</mo><mn>4</mn></math>. Substituting <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> in this equation yields&nbsp;<math alttext=\"g left parenthesis 0 right parenthesis equals left parenthesis 0 minus 6 right parenthesis left parenthesis 0 minus 2 right parenthesis left parenthesis 0 plus 6 right parenthesis plus 4\"><mi>g</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mfenced><mrow><mn>0</mn><mo>-</mo><mn>6</mn></mrow></mfenced><mfenced><mrow><mn>0</mn><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>0</mn><mo>+</mo><mn>6</mn></mrow></mfenced><mo>+</mo><mn>4</mn></math>, or&nbsp;<math alttext=\"g left parenthesis 0 right parenthesis equals 76\"><mi>g</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>76</mn></math>. Thus, the value of <math alttext=\"g left parenthesis 0 right parenthesis\"><mi>g</mi><mfenced><mn>0</mn></mfenced></math> is <math alttext=\"76\"><mn>76</mn>\n</math>.</p>","correct_answer":["76"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.828000+00:00","source_updated_at":"2023-08-02T20:25:59.828000+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}}