{"id":"cb_question_bank:f411ee4a-ef4d-430c-b3ea-8d1bb99afc65","source_id":"cb_question_bank","source_item_id":"f411ee4a-ef4d-430c-b3ea-8d1bb99afc65","external_id":"2dabc1f7-ef1b-4948-88ab-4a73835ddf68","ibn":null,"question_id":"a54753ca","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.B.","skill":"Nonlinear equations in one variable and systems of equations in two variables","difficulty":"H","score_band":7,"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 equation y equals minus x squared plus 9 x minus 100 intersects the line y equals c at exactly one point. What is the value of c ?","stimulus":"","stimulus_html":null,"stem_html":"<p>In the <em>xy</em>-plane, the graph of the equation <math alttext=\"y equals minus x squared plus 9 x minus 100\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mo>-</mo>\n\t\t\t<msup>\n\t\t\t\t<mi>x</mi>\n\t\t\t\t<mn>2</mn>\n\t\t\t</msup>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mrow>\n\t\t\t<mn>9</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>-</mo>\n\t\t<mn>100</mn>\n\t</mrow>\n</mrow>\n</math> intersects the line <math alttext=\"y equals c\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mi>c</mi>\n</mrow>\n</math> at exactly one point. What is the value of <math alttext=\"c\"><mi>c</mi>\n</math>?</p>","rationale_html":"<p>Choice C is correct. In the <em>xy</em>-plane, the graph of the line <math alttext=\"y equals c\"><mi>y</mi><mo>=</mo><mi>c</mi></math> is a horizontal line that crosses the <em>y</em>-axis at <math alttext=\"y equals c\"><mi>y</mi><mo>=</mo><mi>c</mi></math> and the graph of the quadratic equation <math alttext=\"y equals minus x squared plus 9 x minus 100\"><mi>y</mi><mo>=</mo><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><mi>x</mi><mo>-</mo><mn>100</mn></math> is a parabola. A parabola can intersect a horizontal line at exactly one point only at its vertex. Therefore, the value of <math alttext=\"c\"><mi>c</mi>\n</math> should be equal to the <em>y</em>-coordinate of the vertex of the graph of the given equation. For a quadratic equation in vertex form, <math alttext=\"y equals a left parenthesis x minus h right parenthesis squared plus k\"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>h</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mi>k</mi></math>, the vertex of its graph in the <em>xy</em>-plane is <math alttext=\"left parenthesis h comma k right parenthesis\"><mfenced><mrow><mi>h</mi><mo>,</mo><mi>k</mi></mrow></mfenced></math>. The given quadratic equation, <math alttext=\"y equals minus x squared plus 9 x minus 100\"><mi>y</mi><mo>=</mo><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><mi>x</mi><mo>-</mo><mn>100</mn></math>, can be rewritten as <math alttext=\"y equals minus left parenthesis x squared minus 2 left parenthesis nine halves right parenthesis x plus left parenthesis nine halves right parenthesis squared right parenthesis plus left parenthesis nine halves right parenthesis squared minus 100\"><mi>y</mi><mo>=</mo><mo>-</mo><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mfenced><mfrac><mn>9</mn><mn>2</mn></mfrac></mfenced><mi>x</mi><mo>+</mo><msup><mfenced><mfrac><mn>9</mn><mn>2</mn></mfrac></mfenced><mn>2</mn></msup></mrow></mfenced><mo>+</mo><msup><mfenced><mfrac><mn>9</mn><mn>2</mn></mfrac></mfenced><mn>2</mn></msup><mo>-</mo><mn>100</mn></math>, or&nbsp;<math alttext=\"y equals minus left parenthesis x minus nine halves right parenthesis squared plus left parenthesis negative StartFraction 319 Over 4 EndFraction right parenthesis\"><mi>y</mi><mo>=</mo><mo>-</mo><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mn>9</mn><mn>2</mn></mfrac></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mfenced><mrow><mo>-</mo><mfrac><mn>319</mn><mn>4</mn></mfrac></mrow></mfenced></math>. Thus, the value of <math alttext=\"c\"><mi>c</mi>\n</math> is equal to <math alttext=\"negative StartFraction 319 Over 4 EndFraction\"><mo>-</mo><mfrac><mn>319</mn><mn>4</mn></mfrac></math>.</p>\n<p>Choice A is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice B 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>\n<p>&nbsp;</p>","correct_answer":["C"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.824000+00:00","source_updated_at":"2023-08-02T20:25:59.824000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"negative StartFraction 481 Over 4 EndFraction\"><mrow>\n\t<mo>-</mo>\n\t<mfrac>\n\t\t<mn>481</mn>\n\t\t<mn>4</mn>\n\t</mfrac>\n</mrow>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"negative 100\"><mo>-</mo><mn>100</mn>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"negative StartFraction 319 Over 4 EndFraction\"><mrow>\n\t<mo>-</mo>\n\t<mfrac>\n\t\t<mn>319</mn>\n\t\t<mn>4</mn>\n\t</mfrac>\n</mrow>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"negative nine halves\"><mrow>\n\t<mo>-</mo>\n\t<mfrac>\n\t\t<mn>9</mn>\n\t\t<mn>2</mn>\n\t</mfrac>\n</mrow>\n</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}}