A given sinusoidal function has an amplitude of 8, an axis at 𝑦 = 12, a period of 9𝜋, and
a maximum at 𝑥 = 3�

Given the details of a sinusoidal function, we can determine its equation. A sinusoidal function typically has the form:

\[
y = A \sin(B(x - h)) + k
\]

Where:
- \( A \) is the amplitude.
- \( B \) is related to the period by the formula \( \text{Period} = \frac{2\pi}{B} \).
- \( h \) is the horizontal shift (phase shift).
- \( k \) is the vertical shift (axis of the wave).

From the information provided:
- **Amplitude** = 8, so \( A = 8 \).
- **Axis** = 12, which means the vertical shift \( k = 12 \).
- **Period** = \( 9\pi \), so \( B \) can be found using the relationship \( \text{Period} = \frac{2\pi}{B} \). Thus, \( 9\pi = \frac{2\pi}{B} \), leading to \( B = \frac{2}{9} \).
- **Maximum at \( x = 3 \)**, meaning the sine function reaches its peak at \( x = 3 \). Since the sine function reaches its maximum at \( x = h \), we can say \( h = 3 \).

Now, let's plug these values into the equation:

\[
y = 8 \sin\left(\frac{2}{9}(x - 3)\right) + 12
\]

This is the equation of the sinusoidal function.

### Do you have any questions, or would you like further clarification on any part of the solution?

Here are five related questions you might find helpful:
1. How would the equation change if the period was different, say \( 6\pi \)?
2. What if the maximum occurred at \( x = 5 \) instead of \( x = 3 \)?
3. How do you determine the phase shift of a cosine function given similar parameters?
4. How does changing the amplitude affect the shape of the graph?
5. Can you describe how to find the minimum value of this sinusoidal function?

**Tip**: The amplitude always represents the distance from the axis of the wave to either the maximum or minimum point of the graph.

A given sinusoidal function has an amplitude of 8, an axis at 𝑦 = 12, a period of 9𝜋, and a maximum at 𝑥 = 3.

This detailed guide helps you find the equation of a sinusoidal function with amplitude 8, axis at y = 12, period 9π, and maximum at x = 3. Learn key trigonometry concepts like amplitude, period, and phase shift, and apply the formula y = A sin(B(x - h)) + k. Suitable for high school students in Grades 10-12.

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