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## Q. 5.3.4

Horizontal translation

Graph two cycles of y = sin(x + π/6), and determine the phase shift of the graph.

## Verified Solution

Since x + π/6 = x – (-π/6) and C < 0, the graph of y = sin(x + π/6) is obtained by moving y = sin x a distance of π/6 to the left. Since the phase shift is – π/6, label the x-axis with multiples of π/6, as shown in Fig. 5.46. Concentrate on moving the fundamental cycle of y = sin x. The three x-intercepts (0, 0), (π, 0), and (2π, 0) move to (-π/6, 0), (5π/6, 0), and (11π/6, 0). The high and low points, (π/2, 1) and (3π/2, -1), move to (π/3, 1) and (4π/3, -1). Draw one cycle through these five points and continue the pattern for another cycle, as shown in Fig. 5.46. The second cycle could be drawn to the right or left of the first cycle.

• Use π/6 as the x-scale on your calculator, as we did in Fig. 5.46. Set the viewing window to show approximately the same two cycles as Fig. 5.46. The graphs of $y_{1} = \sin(x)$ and $y_{2} = \sin(x + \pi/6)$ in Fig. 5.47 support the conclusion that the shift is π/6 to the left.