Write a positive sin equation of a function what has a maxium at (pi/4,5) and minimum point at (pi/2,1)

To create a positive sine equation with the given characteristics, we need to determine the amplitude, period, phase shift, and vertical shift. Here's how we break it down:

### Key information:
- **Maximum at (π/4, 5)**: This tells us the maximum value of the function is 5.
- **Minimum at (π/2, 1)**: This tells us the minimum value is 1.

### Step-by-step process:
1. **Amplitude (A)**:
   - The amplitude is half the distance between the maximum and minimum values:
     \[
     A = \frac{5 - 1}{2} = 2
     \]

2. **Vertical Shift (D)**:
   - The vertical shift is the average of the maximum and minimum values:
     \[
     D = \frac{5 + 1}{2} = 3
     \]
   This means the sine function oscillates around 3.

3. **Period (T)**:
   - The sine wave completes a full cycle between maximum and minimum points. The distance between the maximum at \( \pi/4 \) and minimum at \( \pi/2 \) is \( \pi/2 - \pi/4 = \pi/4 \).
   - Since this distance represents half a period (from maximum to minimum), the full period \( T \) is:
     \[
     T = 2 \times \frac{\pi}{4} = \frac{\pi}{2}
     \]
   - The period of a sine function is given by \( T = \frac{2\pi}{B} \), so:
     \[
     \frac{2\pi}{B} = \frac{\pi}{2} \implies B = 4
     \]

4. **Phase Shift (C)**:
   - Since the maximum occurs at \( x = \pi/4 \), the sine function has been shifted horizontally by \( \pi/4 \). For a sine function of the form \( \sin(B(x - C)) \), \( C = \pi/4 \).

### Putting it all together:
The general form of a sine function is:
\[
y = A \sin(B(x - C)) + D
\]
Substituting the values we found:
\[
y = 2 \sin(4(x - \frac{\pi}{4})) + 3
\]

This is the equation of the sine function that has a maximum at \( (\pi/4, 5) \) and a minimum at \( (\pi/2, 1) \).

Would you like further details on any part of this process?

### Related Questions:
1. What is the phase shift of the sine function in more complex transformations?
2. How would the function change if the maximum were at \( x = \pi/2 \) instead?
3. How does changing the amplitude affect the maximum and minimum values?
4. What would happen if we used a cosine function instead of a sine function?
5. How can you derive the equation if you are given a different period and phase shift?

### Tip:
Always remember that the amplitude and vertical shift affect the function's range, while the period and phase shift influence its horizontal behavior.

Write a positive sin equation of a function that has a maximum at (π/4, 5) and a minimum point at (π/2, 1).

Learn how to derive the equation of a sine function with a maximum at (π/4, 5) and a minimum at (π/2, 1). This step-by-step guide covers amplitude, vertical shift, period, and phase shift, with practical applications for Grades 10-12.

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