The given information describes the characteristics of a sinusoidal function. Let the function be represented as:

$f(x) = A \sin(Bx + C) + D$

where:

• $A$ is the amplitude,
• $B$ affects the period,
• $C$ is the phase shift, and
• $D$ is the vertical shift or the midline.

Step 1: Midline and Vertical Shift ($D$)

The point $(0, 2)$ is where the function intersects its midline. Therefore, the midline is at $y = 2$, which means:

$D = 2$

Thus, the function becomes:

$f(x) = A \sin(Bx + C) + 2$

Step 2: Amplitude ($A$)

The minimum point of the function is at $(3, -6)$, and since the midline is $y = 2$, the amplitude $A$ is the distance from the midline to the minimum point:

$A = 2 - (-6) = 8$

Therefore:

$f(x) = 8 \sin(Bx + C) + 2$

Step 3: Period and $B$

A sinusoidal function completes one full cycle in a specific period, which is given by:

$\text{Period} = \frac{2\pi}{B}$

The minimum point at $(3, -6)$ suggests that the quarter-period (from the midline to the minimum) is 3 units. Since the distance from the midline crossing to the minimum corresponds to a quarter of the period, we have:

$\frac{\text{Period}}{4} = 3 \quad \Rightarrow \quad \text{Period} = 12$

Thus,

$\frac{2\pi}{B} = 12 \quad \Rightarrow \quad B = \frac{\pi}{6}$

Step 4: Phase Shift ($C$)

Since the function crosses the midline at $x = 0$, there is no horizontal shift. Therefore, $C = 0$.

Final Function

The function is now:

$f(x) = 8 \sin\left(\frac{\pi}{6}x\right) + 2$

Conclusion

The formula of the sinusoidal function is:

$f(x) = 8 \sin\left(\frac{\pi}{6}x\right) + 2$

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Would you like further details on any of the steps? Here are 5 related questions for practice:

1. How do you determine the phase shift of a sinusoidal function?
2. What is the period of $f(x) = 3\sin\left(\frac{\pi}{4}x\right) + 1$?
3. How do you find the amplitude of a sinusoidal function from its graph?
4. What effect does the coefficient $B$ have on the graph of a sine function?
5. How would the function change if the minimum point was at $(3, -4)$ instead?

Tip: To quickly find the amplitude of a sinusoidal function, calculate the vertical distance between the midline and the maximum or minimum point.