Question 1161305
<br>
{{{y = a*sin(b(x-c))+d}}}<br>
a is the amplitude
b determines, or is determined by, the period
c is the horizontal (phase) shift
d is the vertical shift (determines the center line of the oscillation)<br>
The range is from -7 to +2, a difference of 9.  The amplitude a is half of that, 4.5.<br>
The center line is halfway between the minimum and maximum; d = -2.5.<br>
The function has a minimum at -30 degrees and a next maximum at 60 degrees; that difference of 90 degrees is half the period, so the period is 180 degrees.  That is half the period of the basic sine function, so b = 360/180 = 2.<br>
At this point our function is<br>
{{{y = 4.5*sin(2(x-c))-2.5}}}<br>
When the angle is 0 degrees, the value of sine is 0 and increasing.  In this example, the angle is 0 when x=c.<br>
With a minimum at -30 degrees and a maximum at 60 degrees, this function (before the vertical shift) is 0 and increasing halfway between -30 degrees and +60 degrees -- at +15 degrees.<br>
So c = 15 (degrees), and the function is<br>
{{{y = 4.5*sin(2(x-15))-2.5}}}<br>
A graph, showing the sine function and the constants representing the minimum and maximum values and the center line.<br>
Note the minimum at -30 degrees, the maximum at +60 degrees, and the function value at the center line and increasing at +15 degrees.<br>
{{{graph(900,400,-90,90,-10,6,4.5*sin(2(pi/180)(x-15))-2.5,-7,-2.5,2)}}}<br>