{"id":"cb_question_bank:7de01969-e34b-42cd-9ecf-25735658ecfd","source_id":"cb_question_bank","source_item_id":"7de01969-e34b-42cd-9ecf-25735658ecfd","external_id":"a5817e09-28f5-4ad8-9f99-d3d5b3651286","ibn":null,"question_id":"bba18ecb","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":"When the quadratic function f is graphed in the xy -plane, where y equals f left parenthesis x right parenthesis , its vertex is left parenthesis negative 3 comma 6 right parenthesis . One of the x -intercepts of this graph is left parenthesis negative StartFraction 17 Over 4 EndFraction comma 0 right parenthesis . What is the other x -intercept of the graph?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">When the quadratic function <math alttext=\"f\"><mi>f</mi>\n</math> is graphed in the <em>xy</em>-plane, 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>, its vertex is <math alttext=\"left parenthesis negative 3 comma 6 right parenthesis\"><mfenced><mrow><mo>-</mo><mrow><mn>3</mn></mrow><mo>,</mo><mrow><mn>6</mn></mrow></mrow></mfenced></math>. One of the <em>x</em>-intercepts of this graph is <math alttext=\"left parenthesis negative StartFraction 17 Over 4 EndFraction comma 0 right parenthesis\"><mfenced><mrow><mrow><mrow><mo>-</mo><mfrac><mn>17</mn><mn>4</mn></mfrac></mrow></mrow><mo>,</mo><mn>0</mn></mrow></mfenced></math>. What is the other <em>x</em>-intercept of the graph?</p>","rationale_html":"<p>Choice B is correct. Since the line of symmetry for the graph of a quadratic function contains the vertex of the graph, the <em>x</em>-coordinate of the vertex 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>&nbsp;is the <em>x</em>-coordinate of the midpoint of its two <em>x</em>-intercepts. The midpoint of two points with <em>x</em>-coordinates&nbsp;<math alttext=\"x 1\"><msub><mi>x</mi><mn>1</mn></msub></math> and&nbsp;<math alttext=\"x 2\"><msub><mi>x</mi><mn>2</mn></msub></math> has <em>x</em>-coordinate&nbsp;<math alttext=\"x Subscript m\"><msub><mi>x</mi><mi>m</mi></msub></math>, where&nbsp;<math alttext=\"x Subscript m Baseline equals StartFraction x 1 plus x 2 Over 2 EndFraction\"><msub><mi>x</mi><mi>m</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub></mrow><mn>2</mn></mfrac></math>. It′s given that the vertex is <math alttext=\"left parenthesis negative 3 comma 6 right parenthesis\"><mfenced><mrow><mo>-</mo><mn>3</mn><mo>,</mo><mn>6</mn></mrow></mfenced></math>&nbsp;and one of the <em>x</em>-intercepts is <math alttext=\"left parenthesis negative StartFraction 17 Over 4 EndFraction comma 0 right parenthesis\"><mfenced><mrow><mo>-</mo><mfrac><mn>17</mn><mn>4</mn></mfrac><mo>,</mo><mn>0</mn></mrow></mfenced></math>. Substituting <math alttext=\"negative 3\"><mo>-</mo><mn>3</mn></math> for <math alttext=\"x Subscript m\"><msub><mi>x</mi><mi>m</mi></msub></math> and&nbsp;<math alttext=\"negative StartFraction 17 Over 4 EndFraction\"><mo>-</mo><mfrac><mn>17</mn><mn>4</mn></mfrac></math> for&nbsp;<math alttext=\"x 1\"><msub><mi>x</mi><mn>1</mn></msub></math> in the equation&nbsp;<math alttext=\"x Subscript m Baseline equals StartFraction x 1 plus x 2 Over 2 EndFraction\"><msub><mi>x</mi><mi>m</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub></mrow><mn>2</mn></mfrac></math> yields <math alttext=\"negative 3 equals StartStartFraction negative StartFraction 17 Over 4 EndFraction plus x 2 OverOver 2 EndEndFraction\"><mo>-</mo><mn>3</mn><mo>=</mo><mfrac><mrow><mo>-</mo><mstyle displaystyle=\"true\"><mfrac><mn>17</mn><mn>4</mn></mfrac></mstyle><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub></mrow><mn>2</mn></mfrac></math>. Multiplying each side of this equation by <math alttext=\"2\"><mn>2</mn>\n</math> yields <math alttext=\"negative 6 equals negative StartFraction 17 Over 4 EndFraction plus x 2\"><mo>-</mo><mn>6</mn><mo>=</mo><mo>-</mo><mfrac><mn>17</mn><mn>4</mn></mfrac><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub></math>. Adding <math alttext=\"StartFraction 17 Over 4 EndFraction\"><mfrac><mn>17</mn><mn>4</mn></mfrac></math> to each side of this equation yields <math alttext=\"negative seven fourths equals x 2\"><mo>-</mo><mfrac><mn>7</mn><mn>4</mn></mfrac><mo>=</mo><msub><mi>x</mi><mn>2</mn></msub></math>. Therefore, the other <em>x</em>-intercept is <math alttext=\"left parenthesis negative seven fourths comma 0 right parenthesis\"><mfenced><mrow><mo>-</mo><mfrac><mn>7</mn><mn>4</mn></mfrac><mo>,</mo><mn>0</mn></mrow></mfenced></math>. &nbsp;</p>\n<p>Choice A is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice C is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice D is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["B"],"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 negative StartFraction 29 Over 4 EndFraction comma 0 right parenthesis\"><mfenced><mrow><mrow><mrow><mo>-</mo><mfrac><mn>29</mn><mn>4</mn></mfrac></mrow></mrow><mo>,</mo><mn>0</mn></mrow></mfenced></math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"left parenthesis negative seven fourths comma 0 right parenthesis\"><mfenced><mrow><mrow><mrow><mo>-</mo><mfrac><mn>7</mn><mn>4</mn></mfrac></mrow></mrow><mo>,</mo><mn>0</mn></mrow></mfenced></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"left parenthesis five fourths comma 0 right parenthesis\"><mfenced><mrow><mrow><mfrac><mn>5</mn><mn>4</mn></mfrac></mrow><mo>,</mo><mn>0</mn></mrow></mfenced></math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"left parenthesis StartFraction 17 Over 4 EndFraction comma 0 right parenthesis\"><mfenced><mrow><mrow><mfrac><mn>17</mn><mn>4</mn></mfrac></mrow><mo>,</mo><mn>0</mn></mrow></mfenced></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}}