Answer:
f(x) = 4cos(πx/6) +3 or f(x) = 4sin(πx/6 +π/2) +3
Explanation:
You want a sinusoidal function that has a peak at (0, 7) and crosses the midline at (3, 3).
Amplitude
The amplitude of the function is the difference between the peak value and the midline value: 7 -3 = 4.
Vertical translation
The vertical translation of the function is the midline value.
Period
The x-value difference between the peak and the midline crossing is 1/4 period: (1/4)P = 3 - 0, so P = 12. The multiplier of x in the sinusoidal function argument is 2π/P = 2π/12 = π/6.
Horizontal translation
The peak of a cosine function is at x=0, so we can use a cosine function directly with no horizontal translation:
f(x) = 4cos(πx/6) +3 . . . . . . . . . amplitude 4, midline 3, period 12
If you insist on a sine function, then we can make use of the relation ...
cos(x) = sin(x +π/2)
and the function will be written ...
f(x) = 4sin(πx/6 +π/2) +3
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Additional comment
The generic form of the function can be written as ...
f(x) = Asin(B(x+C))+D
where A = amplitude, B = 2π/period, C = left shift of rising midline crossing, D = vertical shift of midline
For a cosine function, C = left shift of positive peak.
The attached graph shows both the sine and cosine versions of the function.
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