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# How to find the sine function from its graph?

Question:

The figure shows a part of a sine graph y = A sin (Bx + C). Find its analytical expression. Solution:

The maximum value is 3 and the minimum value is -3, so the amplitude A = 3. The sine wave from the maximum to the minimum is half period. Let T be period, then T/2 = 7pi/12 - pi/12 = 6pi/12 = pi/2. T = pi. B = 2pi/T = 2pi/pi = 2. The sine function is y = 3 sin (2x + C). Now we determine the initial phase C. The point A has coordinate A (pi/12, 3). Substitute the point A into the function y = 3 sin (2x + C).

3 = 3 sin (2 × pi/12 + C)
1 = sin (2 × pi/12 + C)
1 = sin (pi/6 + C)
pi/6 + C = pi/2
C = - pi/6 + pi/2
= -pi/6 + 3pi/6
= 2pi/6
= pi/3

Therefore, the sine function is y = 3 sin (2x + pi/3). Look the figure above, the blue curve is the graph of y = 3 sin (2x + pi/3). When x = pi/12, the sine graph reaches the maximum point. When x = 7pi/12, the sine graph reaches the minimum point. Watch the video for more details.