document.write( "Question 1143426: The graph of cos(T) is shifted left 270°, has the period decreased to 180°, and is shifted up 2 units. What is the transformed equation? \n" ); document.write( "

Algebra.Com's Answer #764242 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
The graph of $\"y=cos%28T%29\"$ (with $\"T\"$ in degrees) has a period of $\"360%5Eo\"$ .
\n" ); document.write( "In every period there is a single maximum value of $\"1\"$ , such as at $\"T=0%5Eo\"$ .
\n" ); document.write( " $\"y=cos%28T%29\"$ goes through a maximum at $\"T=0%5Eo\"$ , and as $\"T\"$ increases,
\n" ); document.write( "it has the next maximum at $\"T=360%5Eo\"$ .
\n" ); document.write( "After that, the next maximum is at $\"T=2%28360%5Eo%29=720%5Eo\"$ , and so on.
\n" ); document.write( "Here is a piece of the graph of $\"y=cos%28x%29\"$ with $\"x\"$ in degrees:
\n" ); document.write( " $\"graph%28900%2C200%2C-72%2C478%2C-1.1%2C1.1%2Ccos%28pi%2Ax%2F180%29%29\"$ ,
\n" ); document.write( "
\n" ); document.write( "We want to do 3 changes to get to the \"transformed equation\", and there often is more than one way to get to a destination, but some ways may be \"smoother sailing\" compared to others.
\n" ); document.write( "
\n" ); document.write( "CHANGES IN THE ORDER LISTED:
\n" ); document.write( "If we want shift the graph $\"270%5Eo\"$ to the left, we would end with a maximum at $\"-270%5Eo\"$ .
\n" ); document.write( "A way textbooks suggest to do that shift is replacing the variable with another, such as replacing $\"T=U%2B270%5Eo\"$ .
\n" ); document.write( "hat way the point for $\"T=U%2B270%5Eo=0%5Eo\"$ is at $\"U=-270%5Eo\"$ .
\n" ); document.write( "We would have $\"y=cos%28U%2B270%5Eo%29\"$ .
\n" ); document.write( "The graph of $\"y=cos%28x%2B270%5Eo%29\"$ as a function of $\"x\"$ is shown below.
\n" ); document.write( " $\"graph%28900%2C200%2C-342%2C208%2C-1.1%2C1.1%2Ccos%28pi%2A%28x%2B272%29%2F180%29%29\"$
\n" ); document.write( "
\n" ); document.write( "If after that we want to change the period to $\"180%5Eo=%281%2F2%29%2A%28360%5Eo%29\"$ ,
\n" ); document.write( "we can do that by changing to variable $\"x\"$ , so that it increases twice as fast as $\"U\"$ .
\n" ); document.write( "If we still want the maximum at $\"270%5Eo\"$ , we need to make $\"U%2B270%5Eo=2%28x%2B270%5Eo%29\"$ ,
\n" ); document.write( "and change the function from $\"y=cos%28U%2B270%5Eo%29\"$ to $\"y=cos%282x%2B540%5Eo%29\"$ . The graph of $\"y=cos%282x%2B540%5Eo%29\"$ as a function of $\"x\"$ is shown below.
\n" ); document.write( " $\"graph%28900%2C200%2C-342%2C208%2C-1.1%2C1.1%2Ccos%28pi%2A%282x%2B540%29%2F180%29%29\"$
\n" ); document.write( "
\n" ); document.write( "Finally, to shift an x-y graph up by two units, we just add $\"2\"$ to the expression for $\"y\"$ .
\n" ); document.write( "The graph of $\"y=cos%282x%2B540%5Eo%29%2B2\"$ as a function of $\"x\"$ is shown below.
\n" ); document.write( " $\"graph%28900%2C450%2C-342%2C208%2C-0.2%2C3.1%2Ccos%28pi%2A%282x%2B540%29%2F180%29%2B2%29\"$
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "A DIFFERENT WAY:
\n" ); document.write( "With $\"T=2W\"$ , we change the function from $\"y=cos%28T%29\"$ to $\"y=cos%282W%29\"$ ,
\n" ); document.write( "and that changes the period from $\"360%5Eo\"$ to $\"%28360%5Eo%29%2A%281%2F2%29=180%5Eo\"$ .
\n" ); document.write( "The graph of $\"y=cos%282x%29\"$ as a function of $\"x\"$ is shown below.
\n" ); document.write( " $\"graph%28900%2C200%2C-342%2C208%2C-1.1%2C1.1%2Ccos%28pi%2A%282x%29%2F180%29%29\"$ \r
\n" ); document.write( "\n" ); document.write( "To shift the graph $\"270%5Eo\"$ to the left we change $\"W\"$ to $\"x%2B270%5Eo\"$ to get $\"y=cos%282%28x%2B270%5Eo%29%29=cos%282x%2B540%5Eo%29\"$ \r
\n" ); document.write( "\n" ); document.write( "Finally, we add $\"2\"$ to shift the graph up by $\"2\"$ units, and get $\"y=cos%282x%2B540%5Eo%29%2B2\"$ .
\n" ); document.write( "
\n" ); document.write( "NOTE: There are other expressions with the same graph, such as $\"y=cos%282x-180%5Eo%29%2B2\"$ :
\n" ); document.write( " $\"graph%28900%2C450%2C-342%2C208%2C-0.2%2C3.1%2Ccos%28pi%2A%282x-180%29%2F180%29%2B2%29\"$ . \n" ); document.write( "

\n" );