{"id":"cb_practice_pdf:t9-math-m1-q24","source_id":"cb_practice_pdf","source_item_id":"t9-math-m1-q24","external_id":null,"ibn":null,"question_id":null,"program":"SAT","module":"math","domain_code":null,"domain":null,"skill_code":null,"skill":null,"difficulty":null,"score_band":null,"answer_type":"mcq","answer_source":"rationale","layout_warning":"Part of this question was typeset outside the text flow — a fraction, subscript or similar — and PDF extraction re-inserted it in the wrong place. The wording is reliable; the symbols may be out of order.","form_id":"cb_practice_pdf:test-9","module_number":1,"position":24,"stem":"f ( x) = 4x^2 +64x+262 = f(x+5) The function g is defined by g(x). For g(x) what value of x does reach its minimum? −13","stimulus":"","stimulus_html":null,"stem_html":"<p>f ( x) = 4x^2 +64x+262 = f(x+5) The function g is defined by g(x). For g(x) what value of x does reach its minimum? −13</p>","rationale_html":"<p>= + =4 2+64 +262 Choice A is correct. It’s given that g^xh f^x 5h. Since f^xh x x , it +5 =4 +5 2+64 +5 +262 +5 2 follows that f^x h ^x h ^x h . Expanding the quantity ^x h +5 =4 2+10 +25 +64 +5 +262 in this equation yields f^x h ^x x h ^x h . Distributing 64 +5 =4 2+40 +100+64 +320+262 the 4 and the yields f^x h x x x . Combining +5 =4 2+104 +682 =4 2+104 +682 like terms yields f^x h x x . Therefore, g^xh x x . = - 2+ For a quadratic function defined by an equation of the form g^xh a^x hh k, where a, h, and k are constants and a is positive, g^xh reaches its minimum, k, 37 SAT PRACTICE TEST #9 ANSWER EXPLANATIONS SAT ANSWER EXPLANATIONS n MATH: MODULE 1 =4 2+104 +682 when the value of x is h. The equation g^xh x x can be rewritten in this form by completing the square. This equation is equivalent to =4 2+26 +682 =4 2+26 +169-169 +682 g^xh ^x xh , or g^xh ^x x h . This =4 +13 2-169 +682 equation can be rewritten as g^xh ^^x h h , or =4 +13 2-4 169 +682 =4 +13 2+6 g^xh ^x h ^ h , which is equivalent to g^xh ^x h . = - 2+ = =-13 = This equation is in the form g^xh a^x hh k, where a 4, h , and k 6. -13 Therefore, g^xh reaches its minimum when the value of x is . Choice B is incorrect. This is the value of x for which f^xh, rather than g^xh, reaches its minimum. Choice C is incorrect and may result from conceptual or - calculation errors. Choice D is incorrect. This is the value of x for which f^x 5h, + rather than f^x 5h, reaches its minimum.</p>","correct_answer":["A"],"has_media":false,"has_mathml":false,"has_table":false,"source_created_at":null,"source_updated_at":null,"options":[{"label":"A","content_html":"<p>−8</p>","ord":0},{"label":"B","content_html":"<p>−5</p>","ord":1},{"label":"C","content_html":"<p>−3</p>","ord":2},{"label":"D","content_html":"","ord":3}],"attribution":{"source_id":"cb_practice_pdf","source_name":"College Board full-length SAT practice tests (PDF)","rights_holder":"College Board","canonical_url":"https://satsuite.collegeboard.org/practice/practice-tests/paper","retrieved_from":"https://satsuite.collegeboard.org/practice/practice-tests/paper","attribution":"Full-length SAT practice tests 4–11 © College Board, published free at satsuite.collegeboard.org.","disclaimer":"Extracted from PDF layout: equations, figures and tables do not survive intact. Use the original PDFs for anything the text renders poorly. SAT® is a trademark registered by the College Board, which is not affiliated with and does not endorse this project.","license":"College Board copyright. Free to download; not licensed for redistribution.","redistributable":false,"item_count":905}}