Question 940505
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Determine the amplitude, period and phase shift of y = -2 cos(pi x -3)
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        In the  Internet,  where are many similar problems.

        But this one is  SPECIAL:  it is a  TRAP,  and its solution in the post by @lwsshar3 is incorrect.


        Below is my correct solution with complete explanations.



<pre>
Regarding the amplitude, the question is trivial: I hope, 9 of 10 tutors will answer correctly  " amplitude is 2 units ".


With the period, the question is trivial, too: I hope that many tutors will give a correct answer  " the period is 2 ".


But regarding the phase shift, 9 of 10 tutors and 99 of 100 students will answer incorrectly, 
saying that the phase shift is  {{{3/pi}}}.


To answer correctly, we should remember that the parent function to compare with is cos(x),  BY DEFAULT.


When we consider y = {{{-2*cos(pi*x-3)}}}, we should first write it equivalently with the positive coefficient 
before cosine

    y = {{{2*cos(pi*x-3-pi)}}},


shifting right by half-period {{{pi}}}.  {{{highlight(highlight(ONLY))}}}  {{{highlight(highlight(AFTER))}}}  {{{highlight(highlight(THAT))}}}  we can rewrite the function, extracting the shift explicitly

    y = {{{2*cos(pi*(x - (3+pi)/pi))}}} = {{{2*cos(pi*(x-(3/pi+1)))}}}.


Now it becomes obvious that the shift is  {{{3/pi+1)}}}  units right, comparing with the parent function cos(x).


<U>ANSWER</U>.  The shift is  {{{3/pi+1)}}}  units right.
</pre>

Solved correctly with complete explanations.



REMEMBER: &nbsp;&nbsp;The sign &nbsp;" - " &nbsp;before the trigonometric functions "sine" and "cosine" 
is not a harmless symbol that can be ignored in such analysis.

Actually, &nbsp;this sign means the same and works the same as the half-period shift of the argument. 

Therefore, &nbsp;it must be taken into account, &nbsp;and it changes the game completely, &nbsp;turning everything upside down.



//////////// I N T E R E S T I N G ////////////



Driven by my curiosity, &nbsp;I posted this problem today &nbsp;(May 22, 2026, about 12:05 pm) &nbsp;to two &nbsp;Artificial &nbsp;Intelligence 
web-sites. &nbsp;One website was &nbsp;Google &nbsp;Overview; &nbsp;the other one was www.math-gpt.org/


Both answered incorrectly, &nbsp;similar to the &nbsp;"solution" &nbsp;by @lwsshar3, &nbsp;giving the shift &nbsp;{{{3/pi}}} &nbsp;units right.


Actually, &nbsp;this standard trap is well known to those who learned from good teachers, &nbsp;studied &nbsp;Math following 
good educational programs and read relevant popular &nbsp;Math literature.