https://file.aiquickdraw.com/mathbot/file/1725818244561n64tgn01.jpg

The graph you provided is a sinusoidal wave, which resembles a standard trigonometric function. Specifically, this appears to be a graph of a cosine function due to its symmetry about the vertical axis and the wave starting at a peak.

### Key features of this graph:
- **Period:** The distance between two consecutive peaks or troughs is \( 2\pi \). The graph repeats itself every \( 4\pi \), suggesting the period is \( 4\pi \).
- **Amplitude:** The maximum and minimum values of the wave (from peak to trough) suggest an amplitude of 1.
- **Phase shift:** The graph starts at the peak, which is typical for a cosine function, so there's no horizontal phase shift.
- **Equation:** Given that the graph repeats every \( 4\pi \), the equation could be written as \( y = \cos\left(\frac{x}{2}\right) \), where the horizontal compression factor \( \frac{1}{2} \) adjusts the period to \( 4\pi \).

### General form of the function:
The function can be modeled as:
\[ y = A \cos(Bx) + C \]
Where:
- \( A \) is the amplitude,
- \( B \) relates to the period by \( \text{Period} = \frac{2\pi}{B} \),
- \( C \) is the vertical shift (here it seems to be zero).

The likely equation for this graph is:
\[ y = \cos\left(\frac{x}{2}\right) \]

Do you want more detailed steps for deriving the equation, or have any questions about the graph?

### Relative Questions:
1. What is the difference between a cosine and sine wave in terms of phase shift?
2. How would the equation change if the amplitude were doubled?
3. What happens to the period if the coefficient of \( x \) is increased?
4. How would you graph a sine wave with a period of \( 2\pi \)?
5. What real-world phenomena can be modeled by cosine functions?

### Tip:
To determine the period of a trigonometric function, use the formula \( \frac{2\pi}{B} \), where \( B \) is the coefficient of \( x \) in \( \cos(Bx) \).

What is the equation of the given sinusoidal graph?

Explore the graph of a cosine function with a period of 4π. Learn how to analyze the period, amplitude, and equation of the function, and how to graph cosine waves. This lesson is suitable for students in Grades 9-12 studying trigonometry and graphing sinusoidal functions.

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