document.write( "Question 1205013: The graph shows part of a sine function of the form y = A sin B(x + C) + D. Determine the values of A, B, C, and D.
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\n" ); document.write( "\n" ); document.write( "The answer key is A = 2, B = 3, C = -π/3, D = -1\r
\n" ); document.write( "\n" ); document.write( "But I am in need of a worked-out solution\r
\n" ); document.write( "\n" ); document.write( "Is y = A sin B(x + C) + D the same as y = asin(bx - c) + d? I'm confused. I don't understand why the first formula has \"+ C\" instead of \"- c\". I thought the formula of the parent graph is always written \"- c\" regardless if the value of c is negative or positive. I understood that only after replacing the variables with values and if \"c\" has a negative value and you are subtracting a negative (double negatives) that it becomes addition, but why is the first formula with variables \"+ C\" instead of \"- c\"? And why is \"B\" outside the parenthesis instead of inside? I thought \"b\" was supposed to be a quotient of \"x\". \n" ); document.write( "

Algebra.Com's Answer #841596 by MathLover1(20849) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"y+=+A%2Asin+%28B%28x+%2B+C%29%29+%2B+D+\"$ \r
\n" ); document.write( "\n" ); document.write( "where\r
\n" ); document.write( "\n" ); document.write( " $\"A\"$ is amplitude (The amplitude of the graph is the maximum height the graph reaches from the x-axis.)
\n" ); document.write( " $\"%282pi%29%2FB\"$ is period (The period is the distance along the x-axis that is required for the function to make one full oscillation.)
\n" ); document.write( " $\"C\"$ is phase shift (The phase shift is the measure of how far the graph has shifted horizontally.)
\n" ); document.write( " $\"D+\"$ is vertical shift (The vertical shift is the measure of how far the graph has shifted vertically, either up or down, from its initial position.)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "here is the graph of parent function $\"y=sin%28x%29\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"+graph%28+600%2C+600%2C+-5%2C+5%2C+-5%2C+5%2C+sin%28x%29%29+\"$ \r
\n" ); document.write( "\n" ); document.write( "The graph of $\"+y=sin%28x%29\"$ has:\r
\n" ); document.write( "\n" ); document.write( "- an amplitude of $\"1\"$ \r
\n" ); document.write( "\n" ); document.write( "- a period of $\"2pi\"$ \r
\n" ); document.write( "\n" ); document.write( "- a phase shift of $\"0\"$ \r
\n" ); document.write( "\n" ); document.write( "- a vertical shift of $\"+0\"$ \r
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\n" ); document.write( "\n" ); document.write( "you need to compare it to the given graph to determine the values of $\"A\"$ , $\"B\"$ , $\"C\"$ , and $\"+D\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "as you can see, midline goes through $\"y=-1\"$ ( line parallel to $\"x\"$ -axis, means the graph is $\"shifted\"$ $\"+down\"$ $\"+one\"$ unit)=> $\"D=-1\"$ \r
\n" ); document.write( "\n" ); document.write( "the $\"maximum\"$ height the graph reaches from the x-axis is from $\"-3+\"$ to $\"1\"$ , $\"4\"$ units,
\n" ); document.write( "the distance from $\"-1\"$ to $\"1+\"$ and from $\"-1\"$ to $\"+-3\"$ is $\"2+\"$ units, so the amplitude is\r
\n" ); document.write( "\n" ); document.write( " $\"A=2\"$ \r
\n" ); document.write( "\n" ); document.write( "The period of the graph is $\"%282pi%29%2FB\"$ .\r
\n" ); document.write( "\n" ); document.write( "given function makes one full oscillation at $\"%282pi%29%2F3\"$ (when you start at point ( $\"0\"$ , $\"-1\"$ ) and go to point ( $\"2pi%2F3\"$ , $\"-1\"$ ))\r
\n" ); document.write( "\n" ); document.write( "so, the period is: $\"%282pi%29%2FB=%282pi%29%2F3\"$ => $\"B=3\"$ \r
\n" ); document.write( "\n" ); document.write( "the graph has shifted horizontally $\"pi%2F3+\"$ to the right, so\r
\n" ); document.write( "\n" ); document.write( " $\"C+=+-pi%2F3\"$ \r
\n" ); document.write( "\n" ); document.write( "the graph has shifted vertically one unit down, so\r
\n" ); document.write( "\n" ); document.write( " $\"D=-1\"$ \r
\n" ); document.write( "\n" ); document.write( "your equation is:\r
\n" ); document.write( "\n" ); document.write( " $\"y+=+2sin%283%28x+-pi%2F3%29%29+-1\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"+%0D%0Agraph%28+600%2C+600%2C+-5%2C+5%2C+-5%2C+5%2C+2sin%283%28x+-pi%2F3%29%29+-1%29%29+\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Is $\"y+=+A+%2Asin+%28B%28x+%2B+C%29%29+%2B+D\"$ the same as $\"y+=+a%2Asin%28bx+-+c%29+%2B+d\"$ ? => $\"no\"$ , it’s not same\r
\n" ); document.write( "\n" ); document.write( "reason:\r
\n" ); document.write( "\n" ); document.write( " $\"%28B%28x+%2B+C%29%29%3C%3E%28bx+-+c%29\"$
\n" ); document.write( " when $\"C+\"$ is positive the graph shifts to the left
\n" ); document.write( " when $\"C\"$ is negative the graph shifts to the right \r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

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