Write a cosine function that has an amplitude of 4, a midline of y, equals, 5y=5 and a period of one quarter 41​ .

The general form of a cosine function is y = A cos(B(x - C)) + D, where:

• A is the amplitude
• B is the frequency, which is related to the period (P) by the formula B = 2π/P
• C is the phase shift
• D is the vertical shift, which is the midline of the function

Given that the amplitude is 4, the midline is y = 5 (so D = 5), and the period is 1/4 (so B = 2π * 4 = 8π), and assuming there is no phase shift (so C = 0), the cosine function that meets these criteria is:

y = 4 cos(8πx) + 5

Please note that the midline y = 5 is usually written in the form y = D, where D is a constant. The form y = 5y = 5 seems to be a typo or misunderstanding. If the midline is supposed to be y = 5, then D = 5. If the midline is supposed to be y = 5y, then D = 5y, but this would mean that the midline is not a constant, which is unusual for a cosine function.