Question 1205013
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You write in your post:<br>
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".<br>
It looks as if your confusion is about the "+ C" and the "- c".<br>
In that case, you probably didn't write the two forms of the formula correctly.<br>
(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....)<br>
ASSUMING there is no difference between the forms with capital and lower case letters, it is certainly NOT TRUE that<br>
y = A sin B(x+C) + D<br>
and<br>
y = A(sin(Bx+C) + D<br>
are the same.<br>
The "B" outside the parentheses in one form and inside the parentheses in the other make the two forms very different.<br>
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.<br>
So in order to try to help you with this, I am going to assume that the two forms you are looking at are<br>
y = A sin B(x+C) + D<br>
and<br>
y = A sin B(x-C) + D<br>
so that the only difference is the "+C" and "-C", which is what appears to be confusing you.<br>
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.<br>
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 {{{y=(x-h)^2+k}}} is always used to represent a shift h units to the right.<br>
So I myself would object to being asked to write the equation of the function shown in the graph in the form<br>
y = A sin B(x + C) + D<br>
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.<br>
So given all that, I would use the form<br>
y = A sin B(x-C) + D<br>
and analyze the given graph as follows.<br>
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.<br>
The period is 2pi/3.  B is (2pi) divided by the period, so B=3.<br>
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.<br>
That gives me answers that are those shown in your answer key:<br>
y = 2 sin 3(x-pi/3) + (-1)<br>
Finally, it should be pointed out that, although "amplitude" is always positive, in the equation A can be negative.<br>
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.<br>
That would give us different correct answer to the problem:<br>
A=-2; B=3; C=0; D=-1<br>
y = -2 sin 3(x) + (-1)<br>