Question 1134825
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The function has a maximum value of 13 at x=4 and again at x=7.6, so the period is 7.6-4 = 3.6.<br>
The function has a minimum value of -4 halfway through the period, at x=5.8.  So the oscillation is between -4 and 13; that makes the amplitude 8.5 and the centerline 4.5.<br>
The basic cosine graph has its maximum at 0; since this function has its "first" maximum at x=4, the phase shift is 4.<br>
In the equation<br>
{{{y = a*cos(b(x-c))+d}}}<br>
a is the amplitude, c is the phase shift, and d is the vertical shift.<br>
At this point we know a=8.5, c=4, and d=4.5; we need to determine b, which determines the period.<br>
The value of b, to get the length of the period equal to 3.6, is<br>
b = (2pi)/3.6<br>
So the complete function is<br>
{{{y = 8.5*cos((2pi/3.6)(x-4))+4.5}}}<br>
Here is a graph, with horizontal lines showing the maximum and minimum values and the centerline of the function.<br>
{{{graph(400,400,-2,10,-6,16,8.5*cos((2pi/3.6)(x-4))+4.5,13,-4,4.5)}}}