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Answer:
[tex]f(x)=-9\sin\left(\dfrac{1}{3}x\right)+2[/tex]
Step-by-step explanation:
To write the equation for 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]
where:
The midline is the horizontal line (y = D) that the function oscillates around. Given that the function intersects its midline at (0, 2), the midline is y = 2, so:
[tex]D = 2[/tex]
The amplitude is the distance from the midline to the maximum or minimum value of the function. Given the minimum point (3π/2, -7) and the midline y = 2, the amplitude is:
[tex]|A |= 2 - (-7) \\\\|A |=2+7\\\\|A |=9[/tex]
For the parent sine function y = sin(x), the midpoint between a trough and the subsequent peak occurs at x = 0. In the given function, the midpoint at x = 0 is between a peak and the following trough, indicating that the graph has been reflected across the x-axis. Therefore:
[tex]A = -9[/tex]
Additionally, as the midpoint between a maximum and minimum (or minimum and maximum) is at x = 0, there has been no horizontal shift from the standard position. Therefore:
[tex]C = 0[/tex]
For a sinusoidal function, the horizontal distance between the intersection of the midline and the minimum point is one-quarter of the period of the function. Therefore, to find the period T of the function, multiply the difference between the x-coordinates of the midline intersection point (x = 0) and the minimum point (x = 3π/2) by 4:
[tex]T= 4 \cdot \left(\dfrac{3\pi}{2} - 0\right) \\\\\\T= 4 \cdot \left(\dfrac{3\pi}{2} \right) \\\\\\T=\dfrac{12\pi}{2}\\\\\\T=6\pi[/tex]
Since the period is equal to 2π/B, then:
[tex]\dfrac{2\pi}{B}=6\pi \\\\\\B=\dfrac{2\pi}{6\pi}\\\\\\B=\dfrac{1}{3}[/tex]
Substitute the values of A, B, C and D into the general equation:
[tex]f(x)=-9\sin\left(\dfrac{1}{3}\left(x+0\right)\right)+2[/tex]
Simplify:
[tex]f(x)=-9\sin\left(\dfrac{1}{3}x\right)+2[/tex]
Therefore, the equation of the sinusoidal function, where x is in radians, is:
[tex]\Large\boxed{\boxed{f(x)=-9\sin\left(\dfrac{1}{3}x\right)+2}}[/tex]
