document.write( "Question 1206910: 1. Determine the period, amplitude, vertical displacement, and phase shift for the following function.\r
\n" ); document.write( "\n" ); document.write( "y = − 1/2 cos (6𝜋x/5 + 3)-7\r
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\n" ); document.write( "\n" ); document.write( "2. A Ferris wheel has a radius of 16 m and its centre is 18 m above the ground. It rotates once every 60 s. Ethan gets on the Ferris wheel at its lowest point and then the wheel starts to rotate. Determine both a sine and cosine function that gives the height, ℎ, of point P above the ground at any time,𝑡𝑡, where ℎ is in metres and 𝑡𝑡 is in seconds. Both functions must have the smallest possible phase shift. \n" ); document.write( "
Algebra.Com's Answer #844666 by ikleyn(52835)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "        In part 1,  the solution by @MathLover for the shift is  INCORRECT.\r
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\n" ); document.write( "\n" ); document.write( "        In part 2,  the final formula for sine function in the post by @MathLover1 is  INCORRECT.\r
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\n" ); document.write( "\n" ); document.write( "                        Below is my solution for Part 1.\r
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\n" ); document.write( "\n" ); document.write( "1. Determine the period, amplitude, vertical displacement, and phase shift for the following function.\r
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\n" ); document.write( "\n" ); document.write( "            y = -(1/2)*cos(6𝜋x/5 + 3)-7\r
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document.write( "In this problem, the key point is to determine correctly the phase shift \r\n" );
document.write( "of the given function relative to the parent function cos(x).\r\n" );
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document.write( "To apply the STANDARD procedure of analyzing the function y = -(1/2)*cos(6𝜋x/5 + 3)-7,\r\n" );
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document.write( "it should be first written with POSITIVE LEADING COEFFICIENT.\r\n" );
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document.write( "So, instead of -(1/2) at the beginning, the POSITIVE (1/2} must go.\r\n" );
document.write( "For it, the shift in the cosine argument must be changed by adding or subtracting \"pi\"\r\n" );
document.write( "to shift cosine for half of the period, because shifting half the period for cosine\r\n" );
document.write( "is equivalent to changing the sign at cosine.\r\n" );
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document.write( "Thus, we should replace the given function by THIS ONE  y = (1/2)(cos(6𝜋x/5 + 3-𝜋).\r\n" );
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document.write( "Now we can apply the standard analyzing procedure, which gives\r\n" );
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document.write( "    - the amplitude is 1/2;\r\n" );
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document.write( "    - the period is \"2pi%2F%28%28%286pi%29%2F5%29%29\" = \"5%2F3\";\r\n" );
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document.write( "    - vertical displacement, or the midline of the plot is -7;\r\n" );
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document.write( "    - phase shift is  \"-%28%283-pi%29%2F%28%28%282pi%29%2F5%29%29%29\" = \"%28pi-3%29%2F%28%28%282pi%29%2F5%29%29\".\r\n" );
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\n" ); document.write( "\n" ); document.write( "It is a correct solution and the correct answer for Part 1.\r
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\n" ); document.write( "\n" ); document.write( "In part 2,  the correct answer for sine function is\r
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\n" ); document.write( "\n" ); document.write( "         \"h%28t%29\" = \"16sin%28%28pi%2F30%29t%2B3pi%2F2%29%2B18\",\r
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\n" ); document.write( "\n" ); document.write( "with the shift of  \"3pi%2F2\".\r
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\n" ); document.write( "\n" ); document.write( "This shift converts  \"16sin%28%28pi%2F30%29t%2B3pi%2F2%29%2B18\"   into   \"-16cos%28%28pi%2F30%29t%29%2B18\".\r
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