The graph of a sinusoidal function intersects its midline at 
\[(0,2)\] and then has a minimum point at 
\[(3,-6)\].
Write the formula of the function, where 
\[x\] is entered in radians.

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:

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

---

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.

The graph of a sinusoidal function intersects its midline at (0,2) and then has a minimum point at (3,-6). Write the formula of the function, where x is entered in radians.

Learn how to derive the formula of a sinusoidal function given its midline at (0, 2) and minimum point at (3, -6). This trigonometry problem covers key concepts such as amplitude, period, and phase shift. Suitable for students in Grades 10-12.

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