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To find the formula of the sinusoidal function, we can use the general form of a sine function:

\[
f(x) = A \sin(B(x - C)) + D
\]

where:
- \( A \) is the amplitude,
- \( B \) affects the period of the function,
- \( C \) is the phase shift,
- \( D \) is the vertical shift.

### Step 1: Determine the Amplitude \( A \)
The amplitude is the vertical distance from the midline to either the maximum or minimum point. The maximum is at \(-4\) and the minimum is at \(-10\), so the amplitude \( A \) is:
\[
A = \frac{\text{Max} - \text{Min}}{2} = \frac{-4 - (-10)}{2} = \frac{6}{2} = 3
\]

### Step 2: Find the Midline \( D \)
The midline \( D \) is the average of the maximum and minimum values:
\[
D = \frac{\text{Max} + \text{Min}}{2} = \frac{-4 + (-10)}{2} = \frac{-14}{2} = -7
\]

### Step 3: Determine the Period and \( B \)
The period of a sine function is the distance required for the function to complete one full cycle. Here, the minimum point is at \(x = 0\) and the maximum point is at \(x = 2\). This distance represents a quarter period (since a sine function goes from minimum to maximum in a quarter cycle), so the full period \( T \) is:
\[
T = 4 \times 2 = 8
\]
Since \( B = \frac{2\pi}{T} \), we have:
\[
B = \frac{2\pi}{8} = \frac{\pi}{4}
\]

### Step 4: Determine the Phase Shift \( C \)
Since the minimum occurs at \( x = 0 \), we don't need a phase shift. Therefore, \( C = 0 \).

### Step 5: Write the Equation
Substitute \( A = 3 \), \( B = \frac{\pi}{4} \), \( C = 0 \), and \( D = -7 \) into the formula:

\[
f(x) = 3 \sin\left(\frac{\pi}{4} x \right) - 7
\]

### Final Answer
The formula for the sinusoidal function is:

\[
f(x) = 3 \sin\left(\frac{\pi}{4} x \right) - 7
\]

Would you like more details, or do you have any questions?

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Here are 5 related questions to expand your understanding:

1. How would the formula change if the minimum point were at \( (0, -12) \)?
2. What would happen if there was a phase shift in the function?
3. How can we derive the formula if the points were given in degrees instead of radians?
4. What effect does changing the amplitude \( A \) have on the graph of the function?
5. How would the function change if the period were halved?

**Tip:** Remember that the period of a sinusoidal function is determined by the coefficient \( B \), where \( T = \frac{2\pi}{B} \).

The graph of a sinusoidal function has a minimum point at (0, -10) and then has a maximum point at (2, -4). Write the formula of the function, where x is entered in radians.

Find the formula of a sinusoidal function with a minimum at (0, -10) and maximum at (2, -4). This problem explores key trigonometric concepts, including amplitude, period, phase shift, and vertical shift, suitable for high school students.

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