{"id":"cb_question_bank:f8fb91a9-aa48-4f39-a1ae-d14214dd94cc","source_id":"cb_question_bank","source_item_id":"f8fb91a9-aa48-4f39-a1ae-d14214dd94cc","external_id":null,"ibn":"020462-DC","question_id":"a7711fe8","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":"What is the minimum value of the function f defined by f of x equals, open parenthesis, x minus 2, close parenthesis, squared, minus 4 ?","stimulus":"","stimulus_html":null,"stem_html":"<p class=\"stem_paragraph \">What is the minimum value of the function&nbsp;<span class=\"italic\">f</span> defined by <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/462168bfb76134275ec1196f974501c473e595bbce4d3b5684785a2ff03129e6\" alt=\"f of x equals, open parenthesis, x minus 2, close parenthesis, squared, minus 4 \"></span></span>&nbsp;?</p>\n","rationale_html":"<p>Choice A is correct. The given quadratic function <span class=\"italic\">f</span> is in vertex form, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/ee0a9e846c5d9f43c51e09bb387b04447007bf970c24df4cdd55ad29f137ce3e\" alt=\"f of x equals, open parenthesis, x minus h, close parenthesis, squared, plus k \"></span>, where <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0fbcdf3067da8003b9a2185c18d77c5f31212a47481a805cefb139af286f21d4\" alt=\"the point with coordinates h comma k \"></span> is the vertex of the graph of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/ff2211e29d87dc264b50005a18fc2444a78e14708a9844009b3ed81ade385a63\" alt=\"y equals f of x \"></span> in the <span class=\"italic\">xy</span>-plane. Therefore, the vertex of the graph of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/ff2211e29d87dc264b50005a18fc2444a78e14708a9844009b3ed81ade385a63\" alt=\"y equals f of x \"></span> is <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f1aea5f2ed44df857a2812a904eb572e2e8932fb2a2ca4eb28b8e544def08924\" alt=\"the point with coordinates 2 comma negative 4\"></span>. In addition, the <span class=\"italic\">y</span>-coordinate of the vertex represents either the minimum or maximum value of a quadratic function, depending on whether the graph of the function opens upward or downward. Since the leading coefficient of <span class=\"italic\">f</span> (the coefficient of the term <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6e550fcee6c9c5498a0d40e184a2d504737dc34e0b126a0dabe86c5ae18f29a7\" alt=\"open parenthesis, x minus 2, close parenthesis, squared\"></span>) is 1, which is positive, the graph of <span class=\"italic\"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/ff2211e29d87dc264b50005a18fc2444a78e14708a9844009b3ed81ade385a63\" alt=\"y equals f of x \"></span></span> opens upward. It follows that at <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/401169281090f4f1d9aafeeb928524e672d737cdce814b4e9256c5e06a42d325\" alt=\"x equals 2 \"></span>, the minimum value of the function <span class=\"italic\">f</span> is <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/bc0609d3d5523eaf8e35097c2a9c872ba31206db61297d0a21597123b1c843d7\" alt=\"negative 4 \"></span>.</p><p>Choice B is incorrect and may result from making a sign error and from using the <span class=\"italic\">x</span>-coordinate of the vertex. Choice C is incorrect and may result from using the <span class=\"italic\">x</span>-coordinate of the vertex. Choice D is incorrect and may result from making a sign error.</p><p>&nbsp;</p><p></p>\n","correct_answer":["A"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.616000+00:00","source_updated_at":"2023-08-02T20:25:59.616000+00:00","options":[{"label":"A","content_html":"<p><span class=\"choice_cell align:right \"><span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/bc0609d3d5523eaf8e35097c2a9c872ba31206db61297d0a21597123b1c843d7\" alt=\"negative 4\"></span></span> </span></p>\n","ord":0},{"label":"B","content_html":"<p><span class=\"choice_cell align:right \"><span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/ce74cbabbaf45617e6561aad72315be422e849fe95e17b32d71443c7ee5c57e7\" alt=\"negative 2\"></span></span> </span></p>\n","ord":1},{"label":"C","content_html":"<p><span class=\"choice_cell align:right \"><span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/d870b3a6c89096490c39cc291dea5fd94193a953487d36d45d7f8548ac0b8f4e\" alt=\"2\"></span></span> </span></p>\n","ord":2},{"label":"D","content_html":"<p><span class=\"choice_cell align:right \"><span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/8bf3fb601176e63eeff4df74d894a4d3ef6f5b84afee5e9254a35c1ddf0f625c\" alt=\"4\"></span></span> </span></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}}