Answer: y = (1/2)*sin(x*pi/30) + 3/2
Explanation:
We have the equation:
y = p*sin(q*x) + r.
in this case, p is the amplitude, which is calculated as half of the difference between the value at the peak (the maximum) and the value at the troughs (the minimums).
q is related to the period
r is the mid value of the function, will be equal to the minimum plus the amplitude (or the maximum minus the amplitude).
We know that the maximum is (15, 2) (x = 15, y = 2)
We know that the minimum is (45, 1) (x = 45, y = 1)
Then the value of y at the peak is 2, and the value of y at the trough is 1.
This means that the amplitude is:
p = (2 - 1)/2 = (1/2)
And we know that r is equal to the minimum plus one time the amplitude, then:
r = 1 + 1/2 = 3/2.
Then, for now, our equation is:
y = (1/2)*sin(q*x) + 3/2.
Now we can use the information that (15, 2) is a maximum.
We know that the maximum value of the function sin(x) is 1.
and it is when:
x = (pi/2) + n*2*pi (where n is a whole number)
Then we must have that:
15*q = (pi/2) + n*2*pi
as this is periodic, we can define n = 0 and it will be the same.
15*q = pi/2
q = (pi/2)/15 = pi/30.
Now, let's test this with the minimum.
The minimum of the sin(x) function is when:
x = (3/2)*pi + k*2*pi (where k can be any whole number)
and we have a minimum at x = 45, then:
q*45 = (3/2)*pi + k*2*pi
(pi/30)*45 = (3/2)*pi + k*2*pi
(45/30) = (3/2) + k*2
3/2 = 3/2 + k*2
If we take k = 0, then the equality is true
Then q = pi/30 is consistent.
So we can conclude that the equation is:
y = (1/2)*sin(x*pi/30) + 3/2