To find the equation of a sinusoidal function with the given properties, let's start by considering the general form of a sinusoidal function:

$y = A \sin(Bx + C) + D$

where:

• $A$ is the amplitude.
• $B$ affects the period.
• $C$ is the phase shift.
• $D$ is the vertical shift.

Given:

• Amplitude $A = 1$.
• Period $T = 2\pi$, so $B = 1$ (since $T = \frac{2\pi}{B}$).
• Vertical shift $D = 0$ (no mention of any vertical shift).

So, the equation simplifies to:

$y = \sin(x + C)$

Next, we'll use the given point $(- \pi, 0)$ to determine the phase shift $C$.

$0 = \sin(-\pi + C)$

The sine function equals 0 at multiples of $\pi$, so:

$-\pi + C = k\pi \quad \text{for some integer } k$

For simplicity, let's choose $k = -1$:

$-\pi + C = -\pi$

This implies:

$C = 0$

Thus, the equation is:

$y = \sin(x)$

However, this equation does not satisfy the condition at $(- \pi, 0)$, so let's reconsider another possibility: using the cosine function.

Given the same amplitude and period:

$y = \cos(Bx + C)$

Given $A = 1$, $B = 1$, and using the point $(- \pi, 0)$:

$0 = \cos(-\pi + C)$

The cosine function equals 0 at odd multiples of $\frac{\pi}{2}$, so:

$-\pi + C = \frac{\pi}{2} + k\pi$

Let's solve for $C$ using $k = 1$:

$-\pi + C = \frac{3\pi}{2}$

$C = \frac{3\pi}{2} + \pi = \frac{5\pi}{2}$

Thus, the cosine-based sinusoidal function is:

$y = \cos\left(x - \frac{\pi}{2}\right)$

Alternatively, you could write it as:

$y = \sin\left(x + \frac{\pi}{2}\right)$

This equation satisfies the conditions.

Do you have any further questions or need clarification on any step?

---

Here are some related questions to consider:

1. How do you determine whether to use sine or cosine for a sinusoidal function?
2. What are the phase shifts in sinusoidal functions?
3. How do you graph sinusoidal functions with phase shifts?
4. What is the significance of different values of $C$ in sinusoidal functions?
5. How can you check if a given point lies on a sinusoidal curve?

Tip: When given a specific point, consider both sine and cosine functions, as either might be a better fit depending on the phase shift!