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 that satisfies the given conditions, we can start by analyzing the information provided.
1. Maximum point at (0.5, 0):
This means that the function reaches its maximum value when x = 0.5. Plugging these values into the formula, we get:
0 = Asin(B * 0.5) + C
2. Minimum point at (1.5, -4):
This means that the function reaches its minimum value when x = 1.5. Plugging these values into the formula, we get:
-4 = Asin(B * 1.5) + C
We now have two equations with two unknowns (A and B) and one constant term (C). To solve for A and B, we need to eliminate C. We can subtract the first equation from the second equation:
-4 - 0 = A(sin(B * 1.5) + C) - A(sin(B * 0.5) + C)
-4 = A(sin(B * 1.5) - sin(B * 0.5))
Next, we use the trigonometric identity for the difference of sines:
sin(A) - sin(B) = 2cos((A + B)/2)sin((A - B)/2)
Applying this identity to our equation, we have:
-4 = 2Acos((B * 1.5 + B * 0.5)/2)sin((B * 1.5 - B * 0.5)/2)
Simplifying further:
-4 = 2Acos(B)sin(B/2)
Divide both sides by 2A:
-2 = cos(B)sin(B/2)
To find the possible values of B, we notice that when cos(B) = 0, sin(B/2) must be nonzero. This means that B is an odd multiple of π/2. Therefore, B can be represented as B = (2n + 1)(π/2), where n is an integer.
For example, B can be π/2, 3π/2, 5π/2, etc.
Once we determine B, we can substitute it back into the original equations to solve for A and C.
Let's take B = π/2 as an example:
Substituting B = π/2 into the first equation (maximum point at (0.5, 0)):
0 = Asin((π/2) * 0.5) + C
0 = A + C
Substituting B = π/2 into the second equation (minimum point at (1.5, -4)):
-4 = Asin((π/2) * 1.5) + C
-4 = -A + C
From these two equations, we can solve for A and C simultaneously:
0 = A + C
-4 = -A + C
Adding the two equations together:
-4 = 2C
C = -2
Substituting C = -2 into the first equation:
0 = A - 2
A = 2
Therefore, a possible formula for the function that satisfies the given conditions is:
y = 2sin((π/2)x) - 2
Note that there can be multiple valid solutions with different values of A, B, and C, depending on the choice of B.