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.