Respuesta :
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
The trigonometric function equation becomes [tex]g(x)=3 sin(-1.88x-1.88)+6[/tex]
The function has the form:
[tex]g(x)=a sin(bx+c)+d[/tex]
The midline is at (-1, 6)
So, the midline is y = 6 = d
[tex]g(x)=a sin(bx+c)+6[/tex]
We know that the midline is the value of g(x) when the sin(x) part is equal to zero
Given that The function intersects its midline at (-1,6) and has a minimum point at (-3.5,3).
[tex]sin(-1*b + c) = 0,[/tex]
The minimum is the y-value when the sin(x) part is equal to -1
[tex]sin(-3.5*b + c) = -1[/tex]
Since, [tex]g(-3.5) = 3 = a*(-1) + 6[/tex]
[tex]a = -3 + 6 = 3[/tex]
Now equation become,
[tex]g(x) = 3*sin(b*x + c) + 6[/tex]
To find the values of b and c, we need to use the two remaining equations:
[tex]sin(-3.5b + c) = -1=sin( (3/2)* \pi)\\\\ -3.5b + c = (3/2)* \pi[/tex]
[tex]sin(-1b + c) = 0=sin(0)\\\\-b+c=0\\\\b=c[/tex]
Substituting b = c in above equation and [tex]\pi=3.14[/tex]
We get,
[tex]-3.5c + c = (3/2)*\pi\\\\-2.5c=4.71\\\\c=-1.88[/tex]
also [tex]b=-1.88[/tex]
Therefore, equation becomes [tex]g(x)=3 sin(-1.88x-1.88)+6[/tex]
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