{"id":"cb_question_bank:2b76569b-ce88-4efe-97ac-8140d8ed9884","source_id":"cb_question_bank","source_item_id":"2b76569b-ce88-4efe-97ac-8140d8ed9884","external_id":"30c5bb08-3642-4126-a066-371cb194f2d1","ibn":null,"question_id":"252a3b3a","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"M","score_band":5,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"In quadrant 3: The curve rises sharply to touch the x axis at point (negative 5 comma 0). The curve falls gradually to a relative minimum at point (negative 2 comma negative 11). The curve rises gradually to cross both axes at the origin. In quadrant 1: The curve rises gradually to a relative maximum at point (2.5 comma 21). The curve falls sharply to cross the x axis at point (4 comma 0). In quadrant 4 the curve falls sharply. Which of the following could be the equation of the graph shown in the xy -plane?","stimulus":"","stimulus_html":null,"stem_html":"

\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n

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Which of the following could be the equation of the graph shown in the xy-plane?

","rationale_html":"

Choice B is correct. Each of the given choices is an equation of the form y=-110xx-amx+bn, where a\n, b\n, m\n, and n\n are positive constants. In the xy-plane, the graph of an equation of this form has x-intercepts at \n\tx\n\t=\n\t0\n\n, \n\tx\n\t=\n\ta\n\n, and \n\tx\n\t=\n\t\n\t\t-\n\t\tb\n\t\n\n. The graph shown has x-intercepts at \n\tx\n\t=\n\t0\n\n, \n\tx\n\t=\n\t4\n\n, and \n\tx\n\t=\n\t-5\n\n. Therefore, \n\ta\n\t=\n\t4\n\n and \n\tb\n\t=\n\t5\n\n. Of the given choices, only choices A and B have \n\ta\n\t=\n\t4\n\n and \n\tb\n\t=\n\t5\n\n. For an equation in the form y=-110xx-amx+bn, if all values of x\n that are less than \n\t-\n\tb\n\n or greater than a\n correspond to negative y-values, then the sum of all the exponents of the factors on the right-hand side of the equation is even. In the graph shown, all values of x\n less than -5\n or greater than 4\n correspond to negative y-values. Therefore, the sum of all the exponents of the factors on the right-hand side of the equation y=-110xx-4mx+5n must be even. For choice A, the sum of these exponents is 1+1+1, or 3\n, which is odd. For choice B, the sum of these exponents is 1+1+2, or 4\n, which is even. Therefore, y=-110xx-4x+52 could be the equation of the graph shown.

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Choice A is incorrect. For the graph of this equation, all values of x\n less than -5\n correspond to positive, not negative, y-values.

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Choice C is incorrect. The graph of this equation has x-intercepts at \n\tx\n\t=\n\t-4\n\n, \n\tx\n\t=\n\t0\n\n, and \n\tx\n\t=\n\t5\n\n, rather than x-intercepts at \n\tx\n\t=\n\t-5\n\n, \n\tx\n\t=\n\t0\n\n, and \n\tx\n\t=\n\t4\n\n.

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Choice D is incorrect. The graph of this equation has x-intercepts at \n\tx\n\t=\n\t-4\n\n, \n\tx\n\t=\n\t0\n\n, and \n\tx\n\t=\n\t5\n\n, rather than x-intercepts at \n\tx\n\t=\n\t-5\n\n, \n\tx\n\t=\n\t0\n\n, and \n\tx\n\t=\n\t4\n\n.

","correct_answer":["B"],"has_media":true,"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":"

y=-110xx-4x+5

","ord":0},{"label":"B","content_html":"

y=-110xx-4x+52

","ord":1},{"label":"C","content_html":"

y=-110xx-5x+4

","ord":2},{"label":"D","content_html":"

y=-110xx-52x+4

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