The sine function is zero when x = 0.
A cosine function is 1 when x = 0.
So we need to choose the acos(bx) here.
a is the amplitude. Amplitude is how much above or how much below the mean line (which is x axis here) the curve swings. Could you figure out the amplitude from the diagram?
Yes. a = 3.
So the equation is y = 3cos(bx). We still need to figure out b.
The period of a cosine or a sine function is how long before the curve repeats itself. Could you look at two successive peaks in the diagram and figure out after how much x the function repeats itself?
No. The curve attains a peak at x = 0. The next peak occurs at x = 6pi.
The period is how much x has passed before the curve starts to repeat itself.
So what is the x distance between peak-to-peak?
No. At x = 0, first peak. At x = 6pi the next peak.
The difference in x is: 6pi - 0 = 6pi.
Every 6pi the curve repeats itself. You can take the distance between two successive low points also and it will still be 6pi.
So the period of this curve is 6pi.
The formula for the period of cos(bx) is: Period = 2pi / b
Looking at the graph we found the period to be 6pi. Plug that into the formula:
6pi = 2pi / b
Find b.