Given the information that the graph of a sinusoidal function has a minimum point at $(0, -3)$ and intersects its midline at $(1, 1)$, we can deduce the key parameters of the sinusoidal function. Let's break this down step by step:

Step 1: General form of the sinusoidal function

The general form of a sinusoidal function is:

$y = A \sin(B(x - C)) + D$

Where:

• $A$ is the amplitude,
• $B$ is the frequency (affects the period),
• $C$ is the phase shift (horizontal translation),
• $D$ is the vertical shift (midline).

Step 2: Vertical shift ($D$)

Since the sinusoidal function intersects its midline at $(1, 1)$, the midline is at $y = 1$. Therefore, the vertical shift is:

$D = 1$

So the equation becomes:

$y = A \sin(B(x - C)) + 1$

Step 3: Minimum point at $(0, -3)$

The minimum point at $(0, -3)$ tells us that at $x = 0$, $y = -3$. The minimum value of the sinusoidal function occurs when $\sin(B(x - C)) = -1$, because the sine function oscillates between -1 and 1.

Thus, at $x = 0$:

$-3 = A \cdot (-1) + 1$

Solving for $A$:

$-3 - 1 = -A$ $A = 4$

Now the equation is:

$y = 4 \sin(B(x - C)) + 1$

Step 4: Finding the frequency $B$ and phase shift $C$

We know the function intersects its midline at $(1, 1)$, meaning that at $x = 1$, the function crosses the midline, where $y = 1$. This happens at the point where $\sin(B(x - C)) = 0$.

So, at $x = 1$:

$1 = 4 \sin(B(1 - C)) + 1$

Simplifying:

$0 = 4 \sin(B(1 - C))$ $\sin(B(1 - C)) = 0$

This means that $B(1 - C) = n\pi$, where $n$ is an integer. Let's start with the simplest case, where $n = 0$, so:

$B(1 - C) = 0$ $1 - C = 0 \quad \Rightarrow \quad C = 1$

Now, the equation is:

$y = 4 \sin(B(x - 1)) + 1$

Step 5: Conclusion

Thus, the equation of the sinusoidal function is:

$y = 4 \sin(B(x - 1)) + 1$

To determine $B$, we need more information, such as the period or another specific point, but from the current details, we can express the function in this form.

Would you like further clarification or help in finding the value of $B$? Here are some related questions that might expand your understanding:

1. How does the value of $B$ affect the period of the sinusoidal function?
2. What would happen if the amplitude ($A$) were different, say $A = 2$?
3. How can we derive the phase shift if the sinusoidal function has a peak instead of a minimum at the origin?
4. What is the general method for finding the equation of a sinusoidal function from a graph?
5. How do vertical shifts ($D$) influence the graph of a sinusoidal function?

Tip: To identify the amplitude, look at the distance between the midline and the maximum or minimum points of the graph.