Answer:
y = -3cos(πx/4) +3
Explanation:
You want a possible trig function that describes the values in the given table.
Observation
Looking at the table, we find the minimum y-value is 0, and it corresponds to x=0 and x=8. This means the period (P) of the function is 8.
The maximum y-value is 6. This means the midline (M) of the function is 3, the average of the minimum and maximum: (0 +6)/2 = 3.
The amplitude (A) of the function is the difference between the maximum value and the midline value: 6 -3 = 3.
Trig functions
The sine and cosine functions both oscillate back and forth between a value of -1 and a value of +1. The cosine function has its maximum at x=0, but the function needed here will have a minimum at x=0. We can use the cosine function, but it needs to be reflected over the x-axis (or midline).
In general the trig function we're looking for will have the form ...
y = Acos(2πx/P) +M
As discussed above, the value of A will need to be negative to reflect the function over its midline. For A=-3, M=3, P=8, the function is ...
y = -3cos(2πx/8) +3
y = -3cos(πx/4) +3
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Additional comment
In the case where the extreme values (cosine) or the midline values (sine) are not at x=0, there is a "phase shift" involved. That is, the trig function will need to be translated horizontally to match values in the table. That translation is accomplished by adding or subtracting a value from x. Translation of the cosine function is not needed in this case.