Algebra.Com's Answer #841623 by greenestamps(13200)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "You write in your post:
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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\".
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document.write( "It looks as if your confusion is about the \"+ C\" and the \"- c\".
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document.write( "In that case, you probably didn't write the two forms of the formula correctly.
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document.write( "(I am also concerned about your use of capital letters in one form and lower case letters in the other form. I hope you aren't using a resource that uses capital letters if the form has \"+ C\" and lower case letters if the form has \"- c\". That would be VERY confusing and therefore most unfortunate....)
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document.write( "ASSUMING there is no difference between the forms with capital and lower case letters, it is certainly NOT TRUE that
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document.write( "y = A sin B(x+C) + D
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document.write( "and
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document.write( "y = A(sin(Bx+C) + D
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document.write( "are the same.
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document.write( "The \"B\" outside the parentheses in one form and inside the parentheses in the other make the two forms very different.
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document.write( "I suspect you meant to write both forms with the \"B\" outside the parentheses, as the form with it inside the parentheses is much less useful.
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document.write( "So in order to try to help you with this, I am going to assume that the two forms you are looking at are
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document.write( "y = A sin B(x+C) + D
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document.write( "and
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document.write( "y = A sin B(x-C) + D
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document.write( "so that the only difference is the \"+C\" and \"-C\", which is what appears to be confusing you.
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document.write( "Unfortunately, the example in this problem is a very bad one for trying to clear that confusion, because a phase shift of EITHER pi/3 or -pi/3 produces the SAME graph, so using the \"+C\" or \"-C\" form both give correct answers.
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document.write( "In my experience, the form that is virtually always used is the one with \"-C\". That makes it consistent with the discussion of other types of (non-cyclic) graphs, such as a parabola, where  is always used to represent a shift h units to the right.
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document.write( "So I myself would object to being asked to write the equation of the function shown in the graph in the form
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document.write( "y = A sin B(x + C) + D
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document.write( "But, as I pointed out a bit earlier, with this particular graph it doesn't matter which form you use, because, with either form, C can be either pi/3 or -pi/3.
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document.write( "So given all that, I would use the form
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document.write( "y = A sin B(x-C) + D
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document.write( "and analyze the given graph as follows.
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document.write( "The maximum and minimum values are 1 and -3, so the midline is -1 and the amplitude is 2. That gives us D=-1 and A=2.
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document.write( "The period is 2pi/3. B is (2pi) divided by the period, so B=3.
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document.write( "For the parent sine graph, the function is at the midline and is increasing at 0. In this example, that happens at both pi/3 and -pi/3. So for this graph I can use either C=pi/3 or C=-pi/3.
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document.write( "That gives me answers that are those shown in your answer key:
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document.write( "y = 2 sin 3(x-pi/3) + (-1)
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document.write( "Finally, it should be pointed out that, although \"amplitude\" is always positive, in the equation A can be negative.
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document.write( "For the basic sine graph with A negative, the function is at the midline and is decreasing at 0, which is the case with this graph. So a different correct equation for the given graph would have A=-2 with 0 phase shift, so C would be 0.
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document.write( "That would give us different correct answer to the problem:
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document.write( "A=-2; B=3; C=0; D=-1
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document.write( "y = -2 sin 3(x) + (-1)
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