Answer:
g(x) = 3*sin((-15/4)*pi*x + (-15/4)*pi) + 6
Step-by-step explanation:
The function has the form:
g(x) = a*sin(b*x + c) + d
We know that:
The midline is at (-1, 6)
The midline is the value of g(x) when the sin(x) part is equal to zero
Then the midline is y = 6 = d
g(x) = a*sin(b*x + c) + 6
And from this we also know that:
sin(b*-1 + c) = 0,
We also know that the minimum is at (-3.5, 3)
The minimum is the y-value when the sin(x) part is equal to -1
Then
sin(b*-3.5 + c) = -1
And:
g(-3.5) = 3 = a*(-1) + 6
3 = -a + 6
a = -3 + 6 = 3
The equation is something like:
g(x) = 3*sin(b*x + c) + 6
To find the values of b and c, we need to use the two remaining equations:
sin(b*-3.5 + c) = -1
sin(b*-1 + c) = 0
We also know that:
Sin(0 ) = 0
sin( (3/2)*pi) = -1
where pi = 3.14
Then we can just write:
b*-1 + c = 0
b*-3.5 + c = (3/2)*pi
From the first one, we get:
-b + c = 0
c = b
Replacing that on the other equation we get:
c*-3.5 + c = (3/2)*pi
c*(-3.5 + 1) = (3/2)*pi
c*(-2.5) = (3/2)*pi
c = (3/2)*pi/(-2.5)
and:
-2.5 = -5/2
c = (3/2)*(-5/2)*pi = (-15/4)*pi
Then the equation becomes:
g(x) = 3*sin((-15/4)*pi*x + (-15/4)*pi) + 6
