Write a cosine function that has an amplitude of 3, a midline of y, equals, 4y=4 and a period of start fraction, 5, divided by, 4, end fraction 45​ .

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

f(x) = 3cos((4π/5)x) + 4

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 cosine function.

2. Midline: The midline is the 'middle' of the wave. In this case, it's y=4. This is represented by the constant added to the cosine function.

3. Period: The period is how long it takes for the wave to repeat. In this case, it's 5/4. The period of the cosine function is 2π divided by the coefficient of x, so to get a period of 5/4, the coefficient of x should be (4π/5).

So, putting it all together, the function is f(x) = 3cos((4π/5)x) + 4.