Final answer:
To determine the equation of the trigonometric function with the given properties, one must calculate the amplitude, apply the given midline, period, and y-intercept to construct the cosine function that best fits the specifications.
Step-by-step explanation:
The student's question appears to relate to finding the equation of a trigonometric function based on certain properties: the midline, minimum value, period, and the y-intercept. Given the information that the equation of the midline is y=1, the minimum value is y=-4, the period is 8, and the y-intercept is 1, we can construct a sine or cosine function that fits these specifications.
Firstly, the amplitude can be found by calculating the difference between the midline and the minimum value: Amplitude = (Midline - Minimum Value) / 2 = (1 - (-4)) / 2 = 2.5. Next, since the midline is y=1, this would shift the standard sine or cosine curve vertically by +1. The period being 8 indicates that the function completes a full cycle over an interval of 8 units on the x-axis, which means the coefficient of the x variable inside the sine or cosine function, representing the frequency, is (2π / Period) = (2π / 8) = π/4. Finally, with a y-intercept of 1, we can start with a cosine function, which naturally intersects the y-axis at its maximum value when there is no horizontal shift.
Consequently, the function would be of the form y = A cos(B(x - C)) + D, with the amplitude A = 2.5, B as the frequency which is π/4 (which implies no horizontal shift, so C=0), and D as the vertical shift, which is 1. Therefore, the equation is y = 2.5 cos(π/4 x) + 1.