SOLUTION: Determine the equation of a sine function that has a range of {y ∈ R|−7 ≤ y ≤ 2}, the first minimum value at −30° and the first maximum value at 60°.

Question 1161305: Determine the equation of a sine function that has a range of {y ∈ R|−7 ≤ y ≤ 2}, the first minimum value at −30° and the first maximum value at 60°.
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

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)

The range is from -7 to +2, a difference of 9. The amplitude a is half of that, 4.5.

The center line is halfway between the minimum and maximum; d = -2.5.

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.

At this point our function is

When the angle is 0 degrees, the value of sine is 0 and increasing. In this example, the angle is 0 when x=c.

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.

So c = 15 (degrees), and the function is

A graph, showing the sine function and the constants representing the minimum and maximum values and the center line.

Note the minimum at -30 degrees, the maximum at +60 degrees, and the function value at the center line and increasing at +15 degrees.

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