Based on your uploaded graph, the sinusoidal function can be analyzed as follows:

Observations:

1. Amplitude ($A$):

   • The graph reaches a maximum of $y = 3$ and a minimum of $y = -3$.
   • Thus, the amplitude is: $A = \frac{\text{max} - \text{min}}{2} = \frac{3 - (-3)}{2} = 3.$

2. Vertical Shift ($C$):

   • The midline of the graph is at $y = 0$, so $C = 0$.

3. Period ($T$):

   • One complete cycle is visible between $x = -15$ and $x = -5$, so the period is: $T = |(-5) - (-15)| = 10.$
   • The period is related to $k$ by the formula: $T = \frac{2\pi}{k} \implies k = \frac{2\pi}{T} = \frac{2\pi}{10} = \frac{\pi}{5}.$

4. Cosine or Sine Function:

   • The graph starts at its maximum at $x = -15$. This behavior corresponds to a cosine function ($\cos$) with no phase shift.

Function:

The function in cosine form is: $y = A\cos(kx) + C$ Substitute $A = 3$, $k = \frac{\pi}{5}$, and $C = 0$: $y = 3\cos\left(\frac{\pi}{5}x\right).$

Final Answer:

$y = 3\cos\left(\frac{\pi}{5}x\right)$

Would you like me to verify or explain any specific step in greater detail? Let me know if you have further questions.

Questions to Explore:

1. What would the function look like if the graph started at the origin ($x = 0, y = 0$)?
2. How does the choice of sine vs. cosine affect the function's appearance?
3. How is the equation modified if the amplitude changes to $4$?
4. What happens to the function if the midline shifts to $y = 2$?
5. Can we rewrite the cosine function using a sine function with a phase shift?

Tip:

For sinusoidal graphs, always identify the amplitude, period, and phase shift before choosing between sine and cosine functions.