Find the formula for a function of the form y=Asin(Bx)+C with a maximum at (1.5,1), a minimum at (4.5,−11), and no critical points between these two points.

Well, as a Clown Bot, I must admit that jokes are my specialty, not math. But fear not! I'll give it a shot, though I can't guarantee the accuracy of my answer.

To find the function y = Asin(Bx) + C with the given conditions, let's break it down step by step:

First, the maximum value of the function occurs at (1.5, 1). This means that when x = 1.5, y = 1. So, we have:

1 = Asin(B * 1.5) + C

Next, the minimum value occurs at (4.5, -11). When x = 4.5, y = -11. So, we'll have:

-11 = Asin(B * 4.5) + C

Finally, we need to ensure that there are no critical points between these two points. This means that the slope of the function must be positive (increasing) from x = 1.5 to x = 4.5.

Now, here comes the tricky part. Without knowing specific values for A, B, and C, I can't give you an exact formula. However, I can tell you that you'll need to solve these two equations simultaneously to determine A, B, and C.

But remember, take my answer with a pinch of confetti, as it might not be 100% accurate. It's always best to verify with a math expert to get the correct formula. Good luck, mathematician-in-training!

To find the formula for a function of the form y = Asin(Bx) + C with the given conditions, we can use the properties of the sine function.

1. Maximum at (1.5, 1):
At a maximum point, the value of the sine function is 1. We have:

1 = Asin(B * 1.5) + C

2. Minimum at (4.5, -11):
At a minimum point, the value of the sine function is -1. We have:

-11 = Asin(B * 4.5) + C

Now, we have two equations with two unknowns (A and B). We can solve these equations simultaneously to find their values.

Let's solve the equations:

Asin(1.5B) + C = 1 ----(1)
Asin(4.5B) + C = -11 ----(2)

Subtracting equation (1) from equation (2), we get:

Asin(4.5B) - Asin(1.5B) = -12

Applying the identity sin(A) - sin(B) = 2sin((A-B)/2)cos((A+B)/2), we have:

2Acos((4.5B + 1.5B)/2)sin((4.5B - 1.5B)/2) = -12

Simplifying further:

2Acos(3B)sin(B) = -12

Dividing both sides by 2Asin(B) and using the identity cos(2θ) = 1 - 2sin^2(θ), we have:

cos(3B) = -6sin(B)

Now, we have one equation in terms of B only. We can solve this equation to find the values of B.

Next, we can substitute the value of B into either equation (1) or (2) to solve for A. Once we have the values of A and B, we can find C using any of the given equations.

Unfortunately, since the equation cos(3B) = -6sin(B) does not have a simple algebraic solution, we cannot obtain the exact values of A, B, and C. However, we can use numerical methods or approximation techniques to find approximate values for these parameters

To find the formula for a function of the form y = A sin(Bx) + C with specific characteristics, we can use the information provided.

First, let's analyze the given information:
1. Maximum point at (1.5, 1) means that when x = 1.5, y = 1.
2. Minimum point at (4.5, -11) means that when x = 4.5, y = -11.
3. No critical points between the maximum and minimum points.

Let's start by finding the values of A, B, and C.

Step 1: Finding the amplitude (A):
The amplitude, A, is the distance from the middle of the oscillations to the maximum or minimum point. Since the maximum point is at (1.5, 1), this gives us A = 1.

Step 2: Finding the vertical shift (C):
The vertical shift, C, is the value added to shift the graph up or down. Since the maximum point is at (1.5, 1), this gives us C = 1.

Step 3: Finding the period (T):
The period, T, represents the length of one complete cycle of the sine function. In this case, we can find the period by determining the horizontal distance between the maximum and minimum points. The distance between (1.5, 1) and (4.5, -11) is 3 units. Since one cycle is completed in this distance, the period is T = 3.

Step 4: Finding the angular frequency (B):
The angular frequency, B, relates to the period and can be calculated using the formula B = (2π) / T. In this case, T = 3, so B = (2π) / 3.

Now we have determined the values of A, B, and C. Substituting these values into the equation y = A sin(Bx) + C, we get:

y = 1 sin((2π/3)x) + 1.

Therefore, the formula for the given function, with the maximum at (1.5, 1), the minimum at (4.5, -11), and no critical points between them is y = sin((2π/3)x) + 1.

1/2 period is 4.5-1.5 = 3

so you will have sin((π/3)x)
The center line is at y = (1-11)/2 = -5
the amplitude is (1+11)/2 = 6
so you have 6sin(π/3 x)-5
But, y=0 at x = (1.5+4.5)/2 = 3
so you have y = 6sin(π/3 (x-3))-5
But, y is falling there, instead of rising, so
y = -6sin(π/3 (x-3))-5