{"id":"cb_question_bank:6f304113-85cd-4930-ac36-c777b4c1a583","source_id":"cb_question_bank","source_item_id":"6f304113-85cd-4930-ac36-c777b4c1a583","external_id":"664fb796-9a6b-4891-a961-52cdf5f5b989","ibn":null,"question_id":"44076c7d","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":"x f left parenthesis x right parenthesis negative 1 10 0 14 1 20 For the quadratic function f , the table shows three values of x and their corresponding values of f left parenthesis x right parenthesis . Which equation defines f ?","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
x\nfx
-1\n10\n
0\n14\n
1\n20\n
\n

For the quadratic function f\n, the table shows three values of x\n and their corresponding values of fx. Which equation defines f\n?

","rationale_html":"

Choice D is correct. The equation of a quadratic function can be written in the form fx=ax-h2+k, where a\n, h\n, and k\n are constants. It’s given in the table that when \n\tx\n\t=\n\t-1\n\n, the corresponding value of fx is 10\n. Substituting -1\n for x\n and 10\n for fx in the equation fx=ax-h2+k gives 10=a-1-h2+k, which is equivalent to 10=a1+2h+h2+k, or 10=a+2ah+ah2+k. It’s given in the table that when \n\tx\n\t=\n\t0\n\n, the corresponding value of fx is 14\n. Substituting 0\n for x\n and 14\n for fx in the equation fx=ax-h2+k gives 14=a0-h2+k, or 14=ah2+k. It’s given in the table that when \n\tx\n\t=\n\t1\n\n, the corresponding value of fx is 20\n. Substituting 1\n for x\n and 20\n for fx in the equation fx=ax-h2+k gives 20=a1-h2+k, which is equivalent to 20=a1-2h+h2+k, or 20=a-2ah+ah2+k. Adding 20=a-2ah+ah2+k to the equation 10=a+2ah+ah2+k gives 30=2a+2ah2+2k. Dividing both sides of this equation by 2\n gives 15=a+ah2+k. Since 14=ah2+k, substituting 14\n for ah2+k  into the equation 15=a+ah2+k gives 15=a+14. Subtracting 14\n from both sides of this equation gives \n\ta\n\t=\n\t1\n\n. Substituting 1\n for a\n in the equations \n\t14\n\t=\n\t\n\t\t\n\t\t\ta\n\t\t\t\n\t\t\t\th\n\t\t\t\t2\n\t\t\t\n\t\t\n\t\t+\n\t\tk\n\t\n\n and \n\t20\n\t=\n\t\n\t\t\n\t\t\ta\n\t\t\t\n\t\t\t\th\n\t\t\t\t2\n\t\t\t\n\t\t\n\t\t-\n\t\t\n\t\t\t2\n\t\t\ta\n\t\t\th\n\t\t\n\t\t+\n\t\ta\n\t\t+\n\t\tk\n\t\n\n gives 14=h2+k and 20=1-2h+h2+k, respectively. Since \n\t14\n\t=\n\t\n\t\t\n\t\t\th\n\t\t\t2\n\t\t\n\t\t+\n\t\tk\n\t\n\n, substituting 14\n for \n\t\n\t\th\n\t\t2\n\t\n\t+\n\tk\n\n in the equation 20=1-2h+h2+k gives 20=1-2h+14, or 20=15-2h. Subtracting 15\n from both sides of this equation gives 5=-2h. Dividing both sides of this equation by -2\n gives -52=h. Substituting \n\t-\n\t\n\t\t5\n\t\t2\n\t\n\n for h\n into the equation 14=h2+k gives 14=-522+k, or 14=254+k. Subtracting 254 from both sides of this equation gives 314=k. Substituting 1\n for a\n, -52 for h\n, and 314 for k\n in the equation fx=ax-h2+k gives fx=x+522+314, which is equivalent to fx=x2+5x+254+314, or fx=x2+5x+14. Therefore, fx=x2+5x+14 defines f\n.

\n

Choice A is incorrect. If fx=3x2+3x+14, then when \n\tx\n\t=\n\t-1\n\n, the corresponding value of fx is 14\n, not 10\n.

\n

Choice B is incorrect. If fx=5x2+x+14, then when \n\tx\n\t=\n\t-1\n\n, the corresponding value of fx is 18\n, not 10\n.

\n

Choice C is incorrect. If fx=9x2-x+14, then when \n\tx\n\t=\n\t-1\n\n, the corresponding value of fx is 24\n, not 10\n, and when \n\tx\n\t=\n\t1\n\n, the corresponding value of fx is 22\n, not 20\n.

","correct_answer":["D"],"has_media":false,"has_mathml":true,"has_table":true,"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":"

fx=3x2+3x+14

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

fx=5x2+x+14

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

fx=9x2-x+14

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

fx=x2+5x+14

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