{"id":"cb_question_bank:b5f4c481-a808-4e51-a12a-09e08f52c338","source_id":"cb_question_bank","source_item_id":"b5f4c481-a808-4e51-a12a-09e08f52c338","external_id":"b47b1926-6e95-4552-8fc8-d6ec1af4c692","ibn":null,"question_id":"1178f2df","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"H","score_band":7,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"x y 21 negative 8 23 8 25 negative 8 The table shows three values of x and their corresponding values of y , where y equals f left parenthesis x right parenthesis plus 4 and f is a quadratic function. What is the y -coordinate of the y -intercept of the graph of y equals f left parenthesis x right parenthesis 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
x\ny\n
21\n-8\n
23\n8\n
25\n-8\n
\n

The table shows three values of x\n and their corresponding values of y\n, where y=fx+4 and f\n is a quadratic function. What is the y-coordinate of the y-intercept of the graph of y=fx in the xy-plane?

","rationale_html":"

The correct answer is -2,112. It's given that f\n is a quadratic function. It follows that f\n can be defined by an equation of the form fx=ax-h2+k, where a\n, h\n, and k\n are constants. It's also given that the table shows three values of x\n and their corresponding values of y\n, where y=fx+4. Substituting ax-h2+k for fx in this equation yields y=ax-h2+k+4. This equation represents a quadratic relationship between x\n and y\n, where \n\tk\n\t+\n\t4\n\n is either the maximum or the minimum value of y\n, which occurs when \n\tx\n\t=\n\th\n\n. For quadratic relationships between x\n and y\n, the maximum or minimum value of y\n occurs at the value of x\n halfway between any two values of x\n that have the same corresponding value of y\n. The table shows that x-values of 21\n and 25\n correspond to the same y-value, -8. Since 23\n is halfway between 21\n and 25\n, the maximum or minimum value of y\n occurs at an x-value of 23\n. The table shows that when \n\tx\n\t=\n\t23\n\n, \n\ty\n\t=\n\t8\n\n. It follows that \n\th\n\t=\n\t23\n\n and \n\t\n\t\tk\n\t\t+\n\t\t4\n\t\n\t=\n\t8\n\n. Subtracting 4\n from both sides of the equation \n\t\n\t\tk\n\t\t+\n\t\t4\n\t\n\t=\n\t8\n\n yields \n\tk\n\t=\n\t4\n\n. Substituting 23\n for h\n and 4\n for k\n in the equation y=ax-h2+k+4 yields y=ax-232+4+4, or y=ax-232+8. The value of a\n can be found by substituting any x-value and its corresponding y-value for x\n and y\n, respectively, in this equation. Substituting 25\n for x\n and -8 for y\n in this equation yields -8=a25-232+8, or -8=a22+8. Subtracting 8\n from both sides of this equation yields -16=a22, or -16=4a. Dividing both sides of this equation by 4\n yields -4=a. Substituting -4 for a\n, 23\n for h\n, and 4\n for k\n in the equation fx=ax-h2+k yields fx=-4x-232+4. The y-intercept of the graph of y=fx in the xy-plane is the point on the graph where \n\tx\n\t=\n\t0\n\n. Substituting 0\n for x\n in the equation fx=-4x-232+4 yields f0=-40-232+4, or f0=-4-232+4. This is equivalent to f0=-2,112, so the y-intercept of the graph of y=fx in the xy-plane is 0,-2,112. Thus, the y-coordinate of the y-intercept of the graph of y=fx in the xy-plane is -2,112.

","correct_answer":["-2112"],"has_media":false,"has_mathml":true,"has_table":true,"source_created_at":"2023-08-02T20:25:59.828000+00:00","source_updated_at":"2023-08-02T20:25:59.828000+00:00","options":[],"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}}