Let's find the equation of the sinusoidal function with the given properties.

Step 1: General Form of the Sinusoidal Function

The general form of a sinusoidal function is:

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

Where:

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

Step 2: Identify the Given Parameters

1. Amplitude ($A$): Given as 1, so $A = 1$.
2. Period: Given as $2\pi$. The period is given by $\frac{2\pi}{|B|}$. Since the period is $2\pi$, we have: $\frac{2\pi}{|B|} = 2\pi$ Simplifying gives $|B| = 1$. So, $B = 1$.
3. Point: The function passes through the point $\left(\frac{\pi}{2}, -3\right)$.

Step 3: Set Up the Equation

Now we substitute the known values into the equation:

$y = \sin(x + C) + D$

Step 4: Use the Point $\left(\frac{\pi}{2}, -3\right)$ to Find $C$ and $D$

Substitute $x = \frac{\pi}{2}$ and $y = -3$ into the equation:

$-3 = \sin\left(\frac{\pi}{2} + C\right) + D$

This simplifies to:

-3 = 1 + D \quad \text{(since \sin\left(\frac{\pi}{2} + C\right)$reaches its maximum value of 1 when$C = 0)}

$D = -4$

Now the equation is:

$y = \sin(x + C) - 4$

Step 5: Determine $C$ Using the Known Point

Given that $\sin\left(\frac{\pi}{2} + C\right) = 1$, and the sine function equals 1 at $\frac{\pi}{2}$, $C$ should be 0.

Therefore, the equation becomes:

$y = \sin(x) - 4$

Final Answer:

The equation of the sinusoidal function that meets all the given conditions is:

$\boxed{y = \sin(x) - 4}$

Would you like more details or have any questions?

Relative Questions:

1. How does the vertical shift $D$ affect the graph of a sinusoidal function?
2. What would the equation look like if the function had a different phase shift $C$?
3. How can you determine the maximum and minimum values of a sinusoidal function?
4. What happens to the function $y = \sin(x)$ if the amplitude is changed to 2?
5. How does the sine function compare to the cosine function in terms of phase shifts?

Tip:

Remember that the vertical shift $D$ moves the entire graph up or down, depending on its sign. It represents the midline of the sinusoidal wave.