Write a sine function that has an amplitude of 3, a midline of y, equals, 2y=2 and a period of start fraction, 8, pi, divided by, 7, end fraction 78π​ .

The sine function you're asking for can be written as:

f(x) = 3sin((7x/4π) + 2

Here's the breakdown:

1. Amplitude: The amplitude is the 'height' of the wave. In this case, it's 3. This is represented by the coefficient of the sine function.

2. Midline: The midline is the 'middle' of the wave. In this case, it's y=2. This is represented by the constant added to the sine function.

3. Period: The period is how long it takes for the wave to repeat. In this case, it's 8π/7. This is represented by the coefficient of x inside the sine function. To get this, you divide the normal period of the sine function (2π) by the desired period (8π/7). This gives you 7/4, which is the coefficient of x.

So, putting it all together, you get f(x) = 3sin((7x/4π) + 2.