{"id":"cb_question_bank:0cae707e-cb54-4c88-bc1c-1fcb0773e335","source_id":"cb_question_bank","source_item_id":"0cae707e-cb54-4c88-bc1c-1fcb0773e335","external_id":"8defe594-1c91-42f1-b36d-c8fc7e515441","ibn":null,"question_id":"8490cc45","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":"The function f is defined by f left parenthesis x right parenthesis equals left parenthesis negative 8 right parenthesis left parenthesis 2 right parenthesis Superscript x Baseline plus 22 . What is the y -intercept of the graph of y equals f left parenthesis x right parenthesis in the xy -plane?","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 negative 8 right parenthesis left parenthesis 2 right parenthesis Superscript x Baseline plus 22\"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>(</mo><mrow><mo>-</mo><mn>8</mn></mrow><mo>)</mo><msup><mrow><mo>(</mo><mrow><mn>2</mn></mrow><mo>)</mo></mrow><mi>x</mi></msup><mo>+</mo><mrow><mn>22</mn></mrow></math>. What is 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><mo>(</mo><mi>x</mi><mo>)</mo></math> in the <em>xy</em>-plane?</p>","rationale_html":"<p style=\"text-align: left;\">Choice A is correct. The <em>y</em>-intercept of 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 occurs at the point on the graph where <math alttext=\"x equals 0\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math>. In other words, 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>, the corresponding value of <math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is the <em>y</em>-coordinate of the <em>y</em>-intercept. Substituting <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> in the given equation yields&nbsp;<math alttext=\"f left parenthesis 0 right parenthesis equals left parenthesis negative 8 right parenthesis left parenthesis 2 right parenthesis Superscript 0 Baseline plus 22\"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mfenced><mrow><mo>-</mo><mn>8</mn></mrow></mfenced><msup><mfenced><mn>2</mn></mfenced><mn>0</mn></msup><mo>+</mo><mn>22</mn></math>, which is equivalent to&nbsp;<math alttext=\"f left parenthesis 0 right parenthesis equals left parenthesis negative 8 right parenthesis left parenthesis 1 right parenthesis plus 22\"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mfenced><mrow><mo>-</mo><mn>8</mn></mrow></mfenced><mfenced><mn>1</mn></mfenced><mo>+</mo><mn>22</mn></math>, or&nbsp;<math alttext=\"f left parenthesis 0 right parenthesis equals 14\"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>14</mn></math>. Thus, 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>, the corresponding value of <math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is <math alttext=\"14\"><mn>14</mn>\n</math>. Therefore, 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> in the <em>xy</em>-plane is <math alttext=\"left parenthesis 0 comma 14 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>14</mn></mrow></mfenced></math>.</p>\n<p style=\"text-align: left;\">Choice B is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice C is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice D is incorrect. This could be the <em>y</em>-intercept for <math alttext=\"f left parenthesis x right parenthesis equals left parenthesis negative 8 right parenthesis left parenthesis 2 right parenthesis Superscript x\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mo>-</mo><mn>8</mn></mrow></mfenced><msup><mfenced><mn>2</mn></mfenced><mi>x</mi></msup></math>, not <math alttext=\"f left parenthesis x right parenthesis equals left parenthesis negative 8 right parenthesis left parenthesis 2 right parenthesis Superscript x Baseline plus 22\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mo>-</mo><mn>8</mn></mrow></mfenced><msup><mfenced><mn>2</mn></mfenced><mi>x</mi></msup><mo>+</mo><mn>22</mn></math>.</p>","correct_answer":["A"],"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":[{"label":"A","content_html":"<p><math alttext=\"left parenthesis 0 comma 14 right parenthesis\"><mo>(</mo><mn>0</mn><mo>,</mo><mrow><mn>14</mn></mrow><mo>)</mo></math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"left parenthesis 0 comma 2 right parenthesis\"><mo>(</mo><mn>0</mn><mo>,</mo><mrow><mn>2</mn></mrow><mo>)</mo></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"left parenthesis 0 comma 22 right parenthesis\"><mo>(</mo><mn>0</mn><mo>,</mo><mrow><mn>22</mn></mrow><mo>)</mo></math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"left parenthesis 0 comma negative 8 right parenthesis\"><mo>(</mo><mn>0</mn><mo>,</mo><mrow><mo>-</mo><mn>8</mn></mrow><mo>)</mo></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}}