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Answer:
[tex]f(x)=2\sin\left(\dfrac{1}{3}x-\dfrac{\pi}{2}\right)+4[/tex]
Step-by-step explanation:
To write the equation of the graphed function, we can use the general form of the sine function:
[tex]f(x)=A\sin\left(B\left(x+C\right)\right)+D[/tex]
The amplitude |A| is half the distance between the maximum and minimum values. Given that the maximum value is 6 and the minimum value is 2, then:
[tex]|A|=\dfrac{6-2}{2}=\dfrac{4}{2}=2[/tex]
The vertical shift (D) is the average of the maximum and minimum values. Therefore:
[tex]D=\dfrac{6+2}{2}=\dfrac{8}{2}=4[/tex]
The period (2π/B) is the distance between two consecutive peaks (or troughs). To determine the period given a minimum and the subsequent maximum, subtract the x-coordinate of the minimum from the x-coordinate of the maximum, then multiply the result by 2:
[tex]\dfrac{2\pi}{B}=(3\pi - 0) \cdot 2\\\\\\\dfrac{2\pi}{B}=6\pi\\\\\\B=\dfrac{2\pi}{6\pi}\\\\\\B=\dfrac{1}{3}[/tex]
The phase shift (C) is the horizontal shift from the standard position. For the parent sine function y = sin(x), the midpoint between a minimum and the subsequent maximum occurs at x = 0. Since the minimum of the graphed function occurs at x = 0, there has been a phase shift of a quarter of the period to the right. Therefore, as the period of the function is 6π:
[tex]C=-\dfrac{6\pi}{4}=-\dfrac{3\pi}{2}[/tex]
Substitute the values of A, B, C and D into the general equation:
[tex]f(x)=2\sin\left(\dfrac{1}{3}\left(x-\dfrac{3\pi}{2}\right)\right)+4[/tex]
Simplify:
[tex]f(x)=2\sin\left(\dfrac{1}{3}x-\dfrac{\pi}{2}\right)+4[/tex]
Therefore, the equation of the sinusoidal function is:
[tex]\Large\boxed{f(x)=2\sin\left(\dfrac{1}{3}x-\dfrac{\pi}{2}\right)+4}[/tex]
