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 $\cos(Bx)$.