Find the formula for a function of the form y=Asin(Bx)+C with a maximum at (0,50), a minimum at (1.5,−4), and no critical points between these two points.
Sorry the first point is wrong..Find the formula for a function of the form y=Asin(Bx)+C with a maximum at (0.5,0), a minimum at (1.5,−4), and no critical points between these two points.
To find the formula for a function of the form y = Asin(Bx) + C with the given conditions, we can use the information provided to determine the values of A, B, and C.
1. Maximum Point: The maximum point occurs at (0,50), which means that when x = 0, y = 50. This implies that C is equal to 50, since sin(0) = 0 and A * 0 = 0.
2. Minimum Point: The minimum point occurs at (1.5, -4), meaning when x = 1.5, y = -4. By substituting these values into the equation, we have -4 = Asin(B * 1.5) + 50.
Now, let's solve for A and B.
Subtracting 50 from both sides, we get: -4 - 50 = Asin(B * 1.5)
-54 = Asin(1.5B)
Since we know that the sine function has a maximum value of 1, the minimum value of -54 will occur when sin(1.5B) = -1.
Therefore, we have -1 = sin(1.5B).
To find the value of B, we need to find the angle whose sine is -1, which is -π/2 or -90 degrees. Thus,
1.5B = -π/2 or -90 degrees
B = (-π/2) / 1.5 or (-90 degrees) / 1.5
B = -π/3 or -60 degrees
Next, we can substitute B = -π/3 into the equation -54 = Asin(1.5B). This gives us:
-54 = Asin(1.5*(-π/3))
-54 = Asin(-π/2)
-54 = A * (-1)
Simplifying, we find:
-54 = -A
A = 54
Finally, we have determined the values for A, B, and C. Therefore, the formula for the given function is:
y = 54sin(-π/3x) + 50