1. Write a sinusoidal function for the function with a period of 6. The function has a max of 3 at x=2 and a low point of -1.

2. Write a sinusoidal function for the function with a period of 5. The function has a max of 7 at x=1.

explain it in full detail please

To write a sinusoidal function, we need to consider the given information about the period, maximum or minimum points, and any horizontal shifts.

1. Sinusoidal function with a period of 6, maximum of 3 at x=2, and low point of -1:

Step 1: Determine the amplitude:
The amplitude is the distance from the midline (the average value of the function) to the maximum or minimum point. In this case, the maximum is 3 and the low point is -1. The amplitude can be calculated by finding the half the difference between the maximum and minimum values:
Amplitude = (3 - (-1)) / 2 = 4 / 2 = 2.

Step 2: Determine the midline:
The midline is the average value of the function. In this case, it is the average of the maximum and minimum points:
Midline = (3 + (-1)) / 2 = 2 / 2 = 1.

Step 3: Determine the vertical shift (if any):
Since the midline is already at y = 1, there is no vertical shift.

Step 4: Determine the phase shift (if any):
The phase shift represents a horizontal shift of the function. There is no information given about a phase shift, so we assume it is at its default position, centered at x = 0.

Step 5: Write the equation:
The general form of a sinusoidal function is:

f(x) = A * sin(B(x - C)) + D,

where A is the amplitude, B determines the period, C represents the phase shift, and D is the vertical shift.

Using the given information, the equation becomes:

f(x) = 2 * sin[(2π/6)(x - 0)] + 1

Simplifying,

f(x) = 2 * sin[(π/3)(x)] + 1.

Therefore, the sinusoidal function is f(x) = 2 * sin[(π/3)(x)] + 1, with a period of 6, a maximum of 3 at x = 2, and a low point of -1.

2. Sinusoidal function with a period of 5 and a maximum of 7 at x=1:

Step 1: Determine the amplitude:
The amplitude is not given in this case. We assume it to be the maximum value, which is 7.

Step 2: Determine the midline:
The midline is the average value of the function. Since no information is given, we assume it to be at y = 0.

Step 3: Determine the vertical shift (if any):
Since the midline is already at y = 0, there is no vertical shift.

Step 4: Determine the phase shift (if any):
The phase shift represents a horizontal shift of the function. There is no information given about a phase shift, so we assume it is at its default position, centered at x = 0.

Step 5: Write the equation:
The general form of a sinusoidal function is:

f(x) = A * sin(B(x - C)) + D,

where A is the amplitude, B determines the period, C represents the phase shift, and D is the vertical shift.

Using the given information, the equation becomes:

f(x) = 7 * sin[(2π/5)(x - 0)] + 0

Simplifying,

f(x) = 7 * sin[(2π/5)(x)].

Therefore, the sinusoidal function is f(x) = 7 * sin[(2π/5)(x)], with a period of 5 and a maximum of 7 at x = 1.

To write a sinusoidal function, we need to know the following information: period, amplitude, phase shift, and vertical shift.

1. Sinusoidal function with a period of 6, a maximum of 3 at x=2, and a low point of -1:
The general form for a sinusoidal function is y = A * sin(B * (x - 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, the period is 6. Since the period represents the width of one complete cycle, we can calculate the value of B by using the formula B = 2π / period. Therefore, B = 2π / 6 = π / 3.

The maximum value is 3 at x=2, which means the midline of the graph is at y = (3 - (-1)) / 2 = 2. The amplitude is the distance between the maximum and the midline, which is 3 - 2 = 1. Therefore, A = 1.

Since the maximum occurs at x=2, the phase shift is the horizontal displacement from the starting point of the sine wave. For this equation, the phase shift is -2 because x = 2 - (-2) = 4, which corresponds to one-quarter of the period. Therefore, C = -2.

The vertical shift is the movement of the entire graph up or down. In this case, the graph is shifted down 2 units because the midline is at y = 2. Therefore, D = -2.

Putting all the values together, the sinusoidal function representing the given conditions is y = 1 * sin((π/3) * (x - (-2))) - 2.

2. Sinusoidal function with a period of 5, a maximum of 7 at x=1:
Using the same general form, we can determine the values of A, B, C, and D.

The period is 5, so B = 2π / 5.

The maximum value is 7 at x=1, and the midline is at y = (7 + 0) / 2 = 3.5. Therefore, the amplitude, A, is 7 - 3.5 = 3.5.

The phase shift is 1 because the maximum occurs at x=1, which corresponds to one-fifth of the period. Hence, C = 1.

There is no vertical shift mentioned, so D = 0.

Combining all the values, the sinusoidal function representing the given conditions is y = 3.5 * sin((2π/5) * (x - 1)).

did you not read my replies to those same questions just a few posts ago ???