https://file.aiquickdraw.com/mathbot/file/17261897974057ztkjfjb.jpg

To determine the formula for the function \( f(x) \) based on the graph, we will use the following characteristics of the sinusoidal graph:

1. **Amplitude**: The maximum and minimum values give the amplitude.
2. **Period**: The distance between two repeating points (such as peaks or troughs) helps us find the period.
3. **Vertical shift**: A shift up or down from the x-axis.
4. **Phase shift**: Any horizontal shift from the origin of the typical sine or cosine wave.

### Step-by-step analysis:

1. **Amplitude**: 
   - The maximum value of the function is 1, and the minimum is -1, so the amplitude is \( A = 1 \).

2. **Period**: 
   - From the graph, it seems that one full cycle occurs between \( -\frac{2}{9} \) and \( \frac{2}{9} \), meaning the period is \( \frac{4}{9} \).
   - The period of a sine function is given by \( \frac{2\pi}{B} \), so we can equate \( \frac{2\pi}{B} = \frac{4}{9} \), and solve for \( B \):
     \[
     B = \frac{2\pi}{\frac{4}{9}} = \frac{9\pi}{2}
     \]

3. **Vertical shift**: 
   - There is no vertical shift because the graph oscillates between -1 and 1, symmetric around the x-axis.

4. **Phase shift**: 
   - The graph appears to start at 0 when \( x = -\frac{2}{9} \), so the phase shift is \( -\frac{2}{9} \).

Thus, the equation of the function is:
\[
f(x) = \sin\left( \frac{9\pi}{2}x + \frac{\pi}{2} \right)
\]
since the phase shift corresponds to adding \( \frac{\pi}{2} \) inside the sine function.

Would you like more details on any of the steps?

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Here are 5 related questions to think about:
1. How would the formula change if the amplitude were doubled?
2. What if the function were vertically shifted up by 3 units?
3. How would the period change if the coefficient inside the sine function were halved?
4. Can this function be expressed as a cosine function instead?
5. What is the significance of the phase shift in terms of the function's behavior?

**Tip**: The phase shift inside a sine or cosine function determines the horizontal movement of the graph from its standard position.

Find the formula for the graph of the function f(x) given in the graph above.

This page explains how to determine the formula for a sinusoidal function based on its graph, focusing on amplitude, period, and phase shift. Suitable for high school students, this problem covers key trigonometric concepts.

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