SOLUTION: A sinusoidal function has a maximum value of -5, a minimum value of -29, and consecutive minimum values when {{{x=-pi/5}}} and {{{x= 4pi/15}}}. Determine the phase shift

Algebra ->  Trigonometry-basics -> SOLUTION: A sinusoidal function has a maximum value of -5, a minimum value of -29, and consecutive minimum values when {{{x=-pi/5}}} and {{{x= 4pi/15}}}. Determine the phase shift      Log On


   

Question 1103234: A sinusoidal function has a maximum value of -5, a minimum value of -29, and
consecutive minimum values when x=-pi%2F5 and x=+4pi%2F15.
Determine the phase shift
Answer by greenestamps(13215) About Me  (Show Source):
You can put this solution on YOUR website!


Note that "sinusoidal" can mean either a sine function or a cosine function. I will choose to use a sine function.

The general form for the equation of a sine function is

y+=+A%2Asin%28B%28x-C%29%29%2BD

We are asked to find the phase shift, which is C.

The function has consecutive minimum values at x=-pi%2F5 and x=+4pi%2F15. The difference between those two values is 7pi%2F15.
So the period is 7pi%2F15; that means B is 2pi%2F%287pi%2F15%29=30%2F7.

Since the given points are minimum values of the sine function, we want the phase shift to be 3/4 of the period (because sin(x) has its minimum value 3/4 of the way through a period), minus -pi%2F5:
C=%283%2F4%29%287pi%2F15%29%2B%28pi%2F5%29+=+7pi%2F20%2B4pi%2F20+=+11pi%2F20

The phase shift is 11pi/20.

The maximum and minimum values of the function are -5 and -29; in the formula, that makes A=12 and D=-17.

So the complete function is
12%2Asin%28%2830%2F7%29%28x%2B11pi%2F20%29%29-17

Here is a graph....

graph%28300%2C200%2C-pi%2Cpi%2C-35%2C5%2C12%2Asin%28%2830%2F7%29%28x%2B11pi%2F20%29%29-17%29

The two minimum points closest to x=0 and on either of x=0 are at x=-pi%2F5 and x=+4pi%2F15.