https://file.aiquickdraw.com/mathbot/file/1726086236193q4v9qp9l.jpg

Find the equation for the graph shown. Type it in y=A sin (wx).

To determine the equation of the sine function in the form \( y = A \sin(wx) \) based on the graph, let's analyze the key features:

### 1. **Amplitude \( A \):**
   The amplitude is the maximum vertical displacement from the centerline (y = 0). Based on the graph, the maximum and minimum values on the vertical axis are \( 4 \) and \( -4 \), respectively. Hence, the amplitude \( A \) is:

   \[
   A = 4
   \]

### 2. **Period and Angular Frequency \( w \):**
   The period \( T \) is the horizontal length of one complete cycle. From the graph, it takes a horizontal distance of \( 2\pi/9 \) units to complete one cycle. The relationship between the period and the angular frequency is given by:

   \[
   T = \frac{2\pi}{w}
   \]

   Since the period \( T \) is \( \frac{2\pi}{9} \), we can solve for \( w \):

   \[
   \frac{2\pi}{w} = \frac{2\pi}{9} \implies w = 9
   \]

### 3. **Final Equation:**
   With the amplitude \( A = 4 \) and the angular frequency \( w = 9 \), the equation of the sine wave is:

   \[
   y = 4 \sin(9x)
   \]

If you need further clarification or details on this solution, let me know!

### Follow-up Questions:
1. What would be the equation if the amplitude were doubled?
2. How would the equation change if the frequency were halved?
3. What is the significance of the angular frequency \( w \) in a sine function?
4. Can this graph be represented as a cosine function instead of a sine function?
5. What happens to the graph if the equation is shifted horizontally?

### Tip:
Always check the amplitude and period carefully to correctly identify key features of a trigonometric function from its graph!

Find the equation for the graph shown. Type it in y = A sin(wx).

Learn how to find the equation of a sine wave given its graph. This problem involves identifying the amplitude and period from the graph and using them to write the equation in the form y = A sin(wx). Suitable for students in Grades 10-12 studying trigonometry.

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