Hopefully you already know what a basic sine curve looks like - the y value starts at 0, rises to 1, drops through 0 to -1 and returns to zero in the course of the x value increasing from 0 to 2pi.
The regular sine curve, y = sin(x) has a "midline" of 0 (goes up and down "from" a "height",or y value, of 0). and has a "period" (repeats itself) in a distance of 2pi on the x axis.
Your info says the curve has a midline of y = 5, so once we find the basic equation of the sine curve we will shift it UP by 5 by adding 5 to the end of the equation. That was the easy part.
You are told the maximums occur at 3.8 and 11.8. So the maximums. thus the whole curve, repeat in an interval of 11.8 - 3.8, or 8 units, on the x axis. This is the period. The period is the distance on the x axis that the curve repeats itself.
Since a sine curve naturally takes 2 pi units to repeat itself, you have to "scale" the curve on the s axis so that it repeats in 8 units. You have to match the period to the given information.
8 times what is 2pi? ---> pi/4. This is the number you need to SCALE the sine curve to give you a period of 8 units. Thinking of SCALING the curve to match your given data can help you make sense of terms like period and frequency, and helps you understand why the equation works the way it does. Your x SCALE FACTOR is pi/4.
We will assume the amplitude, the height of the sine wave, is still up 1 and down 1 from the midline, since you are given no information on this. So the curve is not "stretched" or "compressed", by a factor which would be a leading coefficient. In the case here, the leading coefficient is 1, so you can leave that out.
Now you have the information you need to out together the equation, which will take the form
y = sin ( ("x SCALE FACTOR") x ) + VERTICAL SHIFT.
