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).