To determine the general sine function $f(x) = A + B \sin(C(x + D))$ that satisfies the given conditions, we need to find the values of $A$, $B$, $C$, and $D$. Let's break down the conditions and derive the values step by step.

Step 1: Amplitude and Vertical Shift

The function has a minimum value of -47 and a maximum value of 153. The sine function oscillates between its minimum and maximum values, so:

• The amplitude $B$ is half the distance between the maximum and minimum values: $B = \frac{153 - (-47)}{2} = \frac{200}{2} = 100$

• The midline $A$ is the average of the maximum and minimum values, which represents the vertical shift: $A = \frac{153 + (-47)}{2} = \frac{106}{2} = 53$

So far, we have $A = 53$ and $B = 100$.

Step 2: Period and Frequency

The period of a sine function is related to the coefficient $C$. The period of a standard sine function $\sin(Cx)$ is $\frac{2\pi}{C}$. We are given that the period is 76, so we can set up the equation: $\frac{2\pi}{C} = 76$ Solving for $C$: $C = \frac{2\pi}{76} = \frac{\pi}{38}$

Step 3: Phase Shift

The function has a minimum value of -47 at $x = 6$. The sine function $\sin(x)$ typically reaches its minimum at $\sin\left(\frac{3\pi}{2}\right) = -1$. Therefore, the phase shift $D$ ensures that the sine function reaches this minimum at $x = 6$.

We want the argument of the sine function to equal $\frac{3\pi}{2}$ when $x = 6$: $C(x + D) = \frac{3\pi}{2} \quad \text{when} \quad x = 6$ Substitute $C = \frac{\pi}{38}$: $\frac{\pi}{38}(6 + D) = \frac{3\pi}{2}$ Solving for $D$: $6 + D = \frac{3\pi}{2} \times \frac{38}{\pi} = 57$ $D = 57 - 6 = 51$

Final Equation

Now that we have all the parameters, the general sine function is: $f(x) = 53 + 100 \sin\left(\frac{\pi}{38}(x + 51)\right)$

This is the sine function that satisfies the given conditions.

Would you like more details on any of the steps? Here are five related questions to expand your understanding:

1. How does changing the amplitude affect the graph of a sine function?
2. What is the role of the phase shift in a sine function?
3. How would the function change if the period were doubled?
4. Can you describe how the vertical shift $A$ modifies the graph?
5. How can you generalize the approach to finding sine functions with different extrema?

Tip: The value $C$ controls the period of the sine wave, so always use the relation $\frac{2\pi}{C} = \text{period}$ when solving for $C$.