Determine the general sine function, 𝑓=𝐴+𝐵sin(𝐶(𝑥+𝐷))
 that has a minimum value of −47
 at 𝑥=6
, a maximum value of 153
 and a period of 76
.

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 \( C \).

Determine the general sine function, f = A + B sin(C(x + D)), that has a minimum value of -47 at x = 6, a maximum value of 153, and a period of 76.

Learn how to determine the general sine function f(x) = A + B sin(C(x + D)) that satisfies the conditions of a maximum value of 153, a minimum value of -47 at x = 6, and a period of 76. This problem covers key trigonometric concepts such as amplitude, phase shift, period, and vertical shift. Suitable for Grades 11-12.

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