To find the formula of the sinusoidal function, we can use the general form of a sine function:

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

where:

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

Step 1: Determine the Amplitude $A$

The amplitude is the vertical distance from the midline to either the maximum or minimum point. The maximum is at $-4$ and the minimum is at $-10$, so the amplitude $A$ is: $A = \frac{\text{Max} - \text{Min}}{2} = \frac{-4 - (-10)}{2} = \frac{6}{2} = 3$

Step 2: Find the Midline $D$

The midline $D$ is the average of the maximum and minimum values: $D = \frac{\text{Max} + \text{Min}}{2} = \frac{-4 + (-10)}{2} = \frac{-14}{2} = -7$

Step 3: Determine the Period and $B$

The period of a sine function is the distance required for the function to complete one full cycle. Here, the minimum point is at $x = 0$ and the maximum point is at $x = 2$. This distance represents a quarter period (since a sine function goes from minimum to maximum in a quarter cycle), so the full period $T$ is: $T = 4 \times 2 = 8$ Since $B = \frac{2\pi}{T}$, we have: $B = \frac{2\pi}{8} = \frac{\pi}{4}$

Step 4: Determine the Phase Shift $C$

Since the minimum occurs at $x = 0$, we don't need a phase shift. Therefore, $C = 0$.

Step 5: Write the Equation

Substitute $A = 3$, $B = \frac{\pi}{4}$, $C = 0$, and $D = -7$ into the formula:

$f(x) = 3 \sin\left(\frac{\pi}{4} x \right) - 7$

Final Answer

The formula for the sinusoidal function is:

$f(x) = 3 \sin\left(\frac{\pi}{4} x \right) - 7$

Would you like more details, or do you have any questions?

Here are 5 related questions to expand your understanding:

1. How would the formula change if the minimum point were at $(0, -12)$?
2. What would happen if there was a phase shift in the function?
3. How can we derive the formula if the points were given in degrees instead of radians?
4. What effect does changing the amplitude $A$ have on the graph of the function?
5. How would the function change if the period were halved?

Tip: Remember that the period of a sinusoidal function is determined by the coefficient $B$, where $T = \frac{2\pi}{B}$.