{"id":"cb_practice_pdf:t8-math-m1-q25","source_id":"cb_practice_pdf","source_item_id":"t8-math-m1-q25","external_id":null,"ibn":null,"question_id":null,"program":"SAT","module":"math","domain_code":null,"domain":null,"skill_code":null,"skill":null,"difficulty":null,"score_band":null,"answer_type":"mcq","answer_source":"rationale","layout_warning":null,"form_id":"cb_practice_pdf:test-8","module_number":1,"position":25,"stem":"- - -_the_graph_of_the - -_equation - ~ In the xy-plane, y = −x^2 +9x−100 y = c intersects the line at exactly one point. What is the value of c? −","stimulus":"","stimulus_html":null,"stem_html":"<p>- - -_the_graph_of_the - -_equation - ~ In the xy-plane, y = −x^2 +9x−100 y = c intersects the line at exactly one point. What is the value of c? −</p>","rationale_html":"<p>= Choice C is correct. In the xy-plane, the graph of the line y c is a horizontal line = that crosses the y-axis at y c and the graph of the quadratic equation =- 2+9 -100 y x x is a parabola. A parabola can intersect a horizontal line at exactly one point only at its vertex. Therefore, the value of c should be equal to the y-coordinate of the vertex of the graph of the given equation. For a quadratic = - 2+ equation in vertex form, y a^x hh k, the vertex of its graph in the xy-plane is =- 2+9 -100 ^h,kh. The given quadratic equation, y x x , can be rewritten as =- 2-2 9 + 9 2 + 9 2 -100 =- - 9 2 + - 319 y dx b 2 lx b 2 l n b 2 l , or y bx 2 l b 4 l. Thus, the value of c is 319 - equal to 4 . Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["C"],"has_media":false,"has_mathml":false,"has_table":false,"source_created_at":null,"source_updated_at":null,"options":[{"label":"A","content_html":"<p>481/4 −100</p>","ord":0},{"label":"B","content_html":"<p>−</p>","ord":1},{"label":"C","content_html":"<p>319/4 −</p>","ord":2},{"label":"D","content_html":"<p>9/2</p>","ord":3}],"attribution":{"source_id":"cb_practice_pdf","source_name":"College Board full-length SAT practice tests (PDF)","rights_holder":"College Board","canonical_url":"https://satsuite.collegeboard.org/practice/practice-tests/paper","retrieved_from":"https://satsuite.collegeboard.org/practice/practice-tests/paper","attribution":"Full-length SAT practice tests 4–11 © College Board, published free at satsuite.collegeboard.org.","disclaimer":"Extracted from PDF layout: equations, figures and tables do not survive intact. Use the original PDFs for anything the text renders poorly. SAT® is a trademark registered by the College Board, which is not affiliated with and does not endorse this project.","license":"College Board copyright. Free to download; not licensed for redistribution.","redistributable":false,"item_count":905}}