document.write( "Question 940505: Determine the amplitude period and phase shift of y = -2 cos(pi x -3) \n" ); document.write( "

Algebra.Com's Answer #854638 by ikleyn(53937) $\"\"$ $\"About$

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\n" ); document.write( "Determine the amplitude, period and phase shift of y = -2 cos(pi x -3)
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\n" ); document.write( "\n" ); document.write( "        In the Internet, where are many similar problems.\r
\n" ); document.write( "\n" ); document.write( "        But this one is SPECIAL: it is a TRAP, and its solution in the post by @lwsshar3 is incorrect.\r
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\n" ); document.write( "\n" ); document.write( "        Below is my correct solution with complete explanations.\r
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document.write( "Regarding the amplitude, the question is trivial: I hope, 9 of 10 tutors will answer correctly  \" amplitude is 2 units \".\r\n" );
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document.write( "With the period, the question is trivial, too: I hope that many tutors will give a correct answer  \" the period is 2 \".\r\n" );
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document.write( "But regarding the phase shift, 9 of 10 tutors and 99 of 100 students will answer incorrectly, \r\n" );
document.write( "saying that the phase shift is  .\r\n" );
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document.write( "To answer correctly, we should remember that the parent function to compare with is cos(x),  BY DEFAULT.\r\n" );
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document.write( "When we consider y = , we should first write it equivalently with the positive coefficient \r\n" );
document.write( "before cosine\r\n" );
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document.write( "    y = ,\r\n" );
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document.write( "shifting right by half-period .        we can rewrite the function, extracting the shift explicitly\r\n" );
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document.write( "    y =  = .\r\n" );
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document.write( "Now it becomes obvious that the shift is    units right, comparing with the parent function cos(x).\r\n" );
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document.write( "ANSWER.  The shift is    units right.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved correctly with complete explanations.\r
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\n" ); document.write( "\n" ); document.write( "REMEMBER: The sign \" - \" before the trigonometric functions \"sine\" and \"cosine\"
\n" ); document.write( "is not a harmless symbol that can be ignored in such analysis.\r
\n" ); document.write( "\n" ); document.write( "Actually, this sign means the same and works the same as the half-period shift of the argument. \r
\n" ); document.write( "\n" ); document.write( "Therefore, it must be taken into account, and it changes the game completely, turning everything upside down.\r
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\n" ); document.write( "\n" ); document.write( "Driven by my curiosity, I posted this problem today (May 22, 2026, about 12:05 pm) to two Artificial Intelligence
\n" ); document.write( "web-sites. One website was Google Overview; the other one was www.math-gpt.org/\r
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\n" ); document.write( "\n" ); document.write( "Both answered incorrectly, similar to the \"solution\" by @lwsshar3, giving the shift $\"3%2Fpi\"$ units right.\r
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\n" ); document.write( "\n" ); document.write( "Actually, this standard trap is well known to those who learned from good teachers, studied Math following
\n" ); document.write( "good educational programs and read relevant popular Math literature.\r
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