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Find the initial phase of a sine graph

Question:

If the graph of the function y = 2sin (3x + C) is symmetry to x = pi/4, what is the value of C?

Solution:

show the graph of y = sin x in two period.

Figure above shows the graph of y = sin x, when x = pi/2, y has the maximum value 1 and the graph is symmetry to x = pi/2. When x = 3pi/2, y has minimum value -1 and the graph is symmetry to x = 3pi/2. The graph of y = sin x is symmetry to the lines in which y has either maximum or minimum value. For every pi interval, y change from maximum to minimum or from minimum to maximum. So, the symmetry axis of y = sin x is: x = pi/2 + k × pi, in which k is an integer.

when k = 0, x = pi/2
when k = 1, x = pi/2 + pi = 3pi/2
when k = 2, x = pi/2 + 2pi = 5pi/2

So, when k is an even number, y has a maximum value. When k is an odd number, y has a minimum value.

In our case, the graph of y = 2sin (3x + C) is symmetry to x = pi/4. That is, when x = pi/4, y is ether maximum or minimum value.

that is, sin (3 × pi/4 + C) = +- 1
3pi/4 + C = pi/2 + k × pi
C = pi/2 - 3pi/4 + k × pi
= 2pi/4 - 3pi/4 + k × pi
= -pi/4 + k × pi
when k = 0, C = -pi/4
when k = 1, C = -pi/4 + pi = 3pi/4
when k = 2, C = -pi/4 + 2pi = 7pi/4

In the second slice of the video, shows the case when k = 0, C = -pi/4, y = 2 sin (3x - pi/4), the graph is symmetry to x = pi/4.

show the graph y1 = 2sin (3x - pi/4), y2 = 2sin (3x + 3pi/4) and y3 = 2sin (3x + 7pi/4)

In the third slice of the video or the figure above, there are three curves y1, y2 and y3. When k = 0, C = -pi/4, y1 = 2 sin (3x - pi/4), which is the blue curve. When k = 1, C = 3pi/4, y2 = 2 sin (3x + 3pi/4), which is the orange curve. When k = 2, C = 7pi/4, y3 = 2 sin (3x + 7pi/4), which is grey curve. The blue curve is covered by the grey curve. That is, the graph of y3 = 2sin (3x + 7pi/4) is the same graph as y1 = 2sin (3x - pi/4). So, when k is an even number, the graph of the function is the blue curve, when k is the odd number, the graph of the function is the orange curve. The blue curve and the orange curve have a phase difference pi. When the blue curve reaches the maximum value the orange curve reaches the minimum value. When the orange curve reaches the maximum value, the blue curve reaches the minimum value. Watch the video for more details.