{"id":"cb_question_bank:fa960192-79ad-461b-b791-4511cac8417a","source_id":"cb_question_bank","source_item_id":"fa960192-79ad-461b-b791-4511cac8417a","external_id":null,"ibn":"08453-DC","question_id":"f5247e52","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":"M","score_band":5,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"In the equation above, a and c are positive constants. How many times does the graph of the equation above intersect the graph of the equation y equals, a plus c in the xy -plane?","stimulus":"y equals, a times x squared minus c","stimulus_html":"<div class=\"stimulus_reference \">\n        <p class=\"standalone_statement style:1 \">\n          <span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e7e5650900aa3da341aeb6a8e2bd64eb989f5443f10c2f3635f6c9a7818d225c\" alt=\"y equals, a times x squared minus c\"></span></span>\n        </p>\n      </div>\n","stem_html":"<p class=\"stem_paragraph \">In the equation above, <span class=\"italic\">a</span> and <span class=\"italic\">c</span> are positive constants. How many times does the graph of the equation above intersect the graph of the equation <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b81fefafd648c35b0fb333685b01347c7428bf66708af86d7b92a17b3624bfae\" alt=\"y equals, a plus c\"></span></span> in the <span class=\"italic\"><span class=\"formatted_text font_style:italic \">xy</span></span>-plane?</p>\n","rationale_html":"<p>Choice C is correct. It is given that the constants <span class=\"italic\">a</span> and <span class=\"italic\">c</span> are both positive; therefore, the graph of the given quadratic equation is a parabola that opens up with a vertex on the <span class=\"italic\">y</span>-axis at a point below the <span class=\"italic\">x</span>-axis. The graph of the second equation provided is a horizontal line that lies above the <span class=\"italic\">x</span>-axis. A horizontal line above the <span class=\"italic\">x</span>-axis will intersect a parabola that opens up and has a vertex below the <span class=\"italic\">x</span>-axis in exactly two points.</p><p>Choices A, B, and D are incorrect and are the result of not understanding the relationships of the graphs of the two equations given. Choice A is incorrect because the two graphs intersect. Choice B is incorrect because in order for there to be only one intersection point, the horizontal line would have to intersect the parabola at the vertex, but the vertex is below the <span class=\"italic\">x</span>-axis and the line is above the <span class=\"italic\">x</span>-axis. Choice D is incorrect because a line cannot intersect a parabola in more than two points.</p><p></p>\n","correct_answer":["C"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.636000+00:00","source_updated_at":"2023-08-02T20:25:59.636000+00:00","options":[{"label":"A","content_html":"<p>Zero</p>\n","ord":0},{"label":"B","content_html":"<p>One</p>\n","ord":1},{"label":"C","content_html":"<p>Two</p>\n","ord":2},{"label":"D","content_html":"<p>More than two</p>\n","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}}