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 equation of a function with the given conditions, we need to determine the values of A, B, and C in the general formula y = Asin(Bx) + C.
First, let's consider the maximum at (0.5, 0). This means that when x = 0.5, y = 0. By substituting these values into the equation, we can solve for A and C.
0 = Asin(B * 0.5) + C
Since sin(0) = 0, the equation becomes:
0 = A * 0 + C
0 = C
We now know that C = 0.
Next, let's consider the minimum at (1.5, -4). This means that when x = 1.5, y = -4. By substituting these values into the equation, we can solve for A and B.
-4 = Asin(B * 1.5) + 0
Dividing both sides by A gives us:
-4/A = sin(1.5B)
To determine B, we need to find the inverse of sin. Taking the inverse sine (sin^(-1)) gives us:
1.5B = sin^(-1)(-4/A)
B = (1/sin^(-1)(-4/A)) * sin^(-1)(-4/A)
Now we have the values for A, B, and C. We can plug them back into the general formula y = Asin(Bx) + C:
y = Asin(Bx) + C
y = A*sin(Bx) + 0
y = Asin(Bx)
Thus, the formula for the function is y = Asin(Bx), where A = sin^(-1)(-4/A) and B = (1/sin^(-1)(-4/A)) * sin^(-1)(-4/A).
To find the formula for a function in the form y = Asin(Bx) + C with a maximum at (0.5, 0) and a minimum at (1.5, -4), we need to determine the values of A, B, and C.
The general form of the equation for a sinusoidal function is y = Asin(Bx + C) + D, where A is the amplitude, B is the frequency, C is the phase shift, and D is the vertical shift.
In this case, we are given that the maximum occurs at (0.5, 0) and the minimum occurs at (1.5, -4). Let's break down the information provided:
1. Maximum point: (0.5, 0)
At the maximum point, the value of A * sin(B * x) is maximum, which means sin(B * x) is 1.
Therefore, A * 1 + C = 0 (since sin(0) = 0).
This gives us the equation A + C = 0. -- equation 1
2. Minimum point: (1.5, -4)
At the minimum point, the value of A * sin(B * x) is minimum, which means sin(B * x) is -1.
Therefore, A * -1 + C = -4 (since sin(π) = 0).
This gives us the equation -A + C = -4. -- equation 2
Now, we need to solve the system of equations formed by equations 1 and 2 to find the values of A and C.
Adding equations 1 and 2, we get:
(A + C) + (-A + C) = 0 + (-4)
2C = -4
C = -2
Substituting the value of C into equation 1, we get:
A + (-2) = 0
A = 2
So, we have A = 2 and C = -2.
Now that we know the values of A and C, we can write the equation of the function as:
y = 2sin(Bx) - 2.
We still need to find the value of B.
Since we are given that there are no critical points between (0.5, 0) and (1.5, -4), that means there is one complete period within this interval. The length of one complete period for a function of the form sin(B * x) is 2π/B.
In this case, the length of the interval (1.5 - 0.5) = 1, which should be equal to one complete period. Hence, we have:
1 = 2π/B.
Simplifying the equation, we get:
2π = B.
Therefore, B = 2π.
Now we have A = 2, B = 2π, and C = -2. So, the formula for the function is:
y = 2sin(2πx) - 2.