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

To write a positive sine equation for a function with a maximum at \((\frac{\pi}{4}, 5)\) and a minimum at \((\frac{\pi}{2}, 1)\), we need to use the general form of a sine function:

\[
y = A \sin(Bx - C) + D
\]

Where:
- \(A\) is the amplitude (half the distance between the maximum and minimum),
- \(B\) affects the period of the function,
- \(C\) is the horizontal shift,
- \(D\) is the vertical shift (midline).

### Step 1: Determine amplitude, \(A\)
The amplitude is half the distance between the maximum and minimum values:
\[
A = \frac{5 - 1}{2} = 2
\]

### Step 2: Find the vertical shift, \(D\)
The midline is the average of the maximum and minimum values:
\[
D = \frac{5 + 1}{2} = 3
\]

### Step 3: Determine the period and \(B\)
The sine function completes one cycle between the maximum and minimum. The distance between \(\frac{\pi}{4}\) and \(\frac{\pi}{2}\) is:
\[
\frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4}
\]
This represents one-quarter of the full sine cycle, so the period \(T\) is \(4 \times \frac{\pi}{4} = \pi\).

The period \(T\) is related to \(B\) by:
\[
T = \frac{2\pi}{B}
\]
Solving for \(B\):
\[
\pi = \frac{2\pi}{B} \quad \Rightarrow \quad B = 2
\]

### Step 4: Determine the phase shift, \(C\)
The sine function typically reaches its maximum at \(x = 0\). In this case, the maximum occurs at \(x = \frac{\pi}{4}\), which means the sine function is shifted to the right by \(\frac{\pi}{4}\). Thus, \(C = \frac{\pi}{4}\).

### Step 5: Write the equation
Now, we can write the sine equation using the values of \(A\), \(B\), \(C\), and \(D\):
\[
y = 2 \sin\left(2x - \frac{\pi}{2}\right) + 3
\]

This is the sine function with a maximum at \(\left(\frac{\pi}{4}, 5\right)\) and a minimum at \(\left(\frac{\pi}{2}, 1\right)\).

Would you like more details on any step? 

Here are some related questions:
1. What is the general form of a trigonometric function?
2. How do you calculate the phase shift of a sine function?
3. Can the amplitude of a sine function be negative, and what does it mean if it is?
4. How do you calculate the period of a sine function from its equation?
5. What are the effects of horizontal and vertical shifts in trigonometric functions?

**Tip:** When shifting sine functions, always consider whether the shift is horizontal (phase shift) or vertical (midline adjustment).

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

Learn how to write a positive sine equation with a maximum at (π/4, 5) and a minimum at (π/2, 1). This problem covers key trigonometric concepts, including amplitude, phase shift, and vertical shift. Suitable for high school students in Grades 10-12.

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