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Find an equation for the graph shown to the right.
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Part 1
Type the equation in the form y equals Upper A sine left parenthesis omega x right parenthesisy=Asin(ωx) or y equals Upper A cosine left parenthesis omega x right parenthesisy=Acos(ωx).

To determine the equation of the sinusoidal graph shown, we need to observe a few key properties:

1. **Amplitude (A):** The amplitude is the distance from the midline of the graph to the peak. In this case, the maximum value on the y-axis is 2 and the minimum value is -2, so the amplitude \( A \) is \( 2 \).

2. **Period (T):** The period is the length of one complete cycle of the graph. From the graph, one complete cycle occurs from \( 0 \) to \( 4\pi \). Thus, the period is \( 4\pi \).

   The period \( T \) is related to the angular frequency \( \omega \) by the formula \( T = \frac{2\pi}{\omega} \). Using the period of \( 4\pi \):
   \[
   T = \frac{2\pi}{\omega} = 4\pi \quad \Rightarrow \quad \omega = \frac{2\pi}{4\pi} = \frac{1}{2}
   \]

3. **Phase shift:** There is no horizontal shift visible in the graph, so the phase shift is zero.

4. **Midline:** The midline of the graph appears to be at \( y = 0 \), so there is no vertical shift.

Since the graph starts at \( y = 0 \) and increases, it matches the form of a sine function. Therefore, the equation is:

\[
y = 2 \sin\left(\frac{1}{2}x\right)
\]

This is the equation of the graph.

Would you like more details or have further questions on this? Here are 5 questions to further explore this concept:

1. What is the relationship between amplitude and the maximum/minimum values of the graph?
2. How do you calculate the period from a sinusoidal graph?
3. What is the difference between a sine and cosine function graph?
4. How does phase shift affect the graph of a trigonometric function?
5. Can you find the equation if the graph represents a cosine function?

**Tip:** When determining the period of a sine or cosine function, always remember the formula \( T = \frac{2\pi}{\omega} \), where \( T \) is the period and \( \omega \) is the angular frequency.

Find an equation for the graph shown. Type the equation in the form y = A sin(ωx) or y = A cos(ωx).

This problem involves finding the equation of a sinusoidal graph with an amplitude of 2 and a period of 4π. Learn how to use the relationship between amplitude, angular frequency, and period to solve for the equation of the graph in the form y = A sin(ωx).

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