I need help writing an equation for a Sine Wave... I have 3 coordinates, but I can't get the eq. right... (4, 25), (7, 83), (10, 25). If anyone can help, I will be more than grategull. My group bailed on me. \m/

25 is the lowpoint and 83 is the high point. Thanks to anyone who can help!

To write an equation for a sine wave given three coordinates, you need to know the amplitude, period, and vertical translation of the sine wave. Since you mentioned that the low point is 25 and the high point is 83, we can conclude that the amplitude is half the distance between these two points. Therefore, the amplitude is (83 - 25) / 2 = 29.

To determine the period of the sine wave, we need to find the horizontal distance between two consecutive peaks. From the given coordinates, we can see that there are two peaks at x = 4 and x = 10. So the period is 10 - 4 = 6.

Now, let's find the vertical translation (also known as the phase shift or vertical displacement) by examining the y-coordinate of one of the points. In this case, the low point is at y = 25, which means there is no vertical translation.

Putting it all together, the equation for the sine wave is:

y = 29 * sin((2π / 6) * (x - h)) + k

Since there is no vertical translation, the equation simplifies to:

y = 29 * sin((π / 3) * (x - h))

Now, let's find the value of h (the horizontal translation or phase shift) by plugging in one of the x-values and its corresponding y-value:

25 = 29 * sin((π / 3) * (4 - h))

Rearranging this equation, we get:

sin((π / 3) * (4 - h)) = 25 / 29

To solve for h, you will need to find the inverse sine (or arcsine) of (25 / 29) and solve for (π / 3) * (4 - h). After determining the value of (π / 3) * (4 - h), you can isolate h and substitute it back into the equation to get the final equation of the sine wave.

Since this involves some calculations, you may use a scientific calculator or a math software tool to find the inverse sine and perform the necessary calculations.

Remember to check if the other given points lie on the derived sine wave equation to verify its accuracy.