Horizontal and vertical translation
Graph two cycles of y = cos(x – π/4) + 2, and determine the phase shift of the graph.
The graph of y = cos(x – π/4) + 2 is obtained by moving y = cos x a distance of π/4 to the right and two units upward. Since the phase shift is π/4, label the x-axis with multiples of π/4, as shown in Fig. 5.48. Concentrate on moving the fundamental cycle of y = cos x. The points (0, 1), (π, -1), and (2π, 1) move to (π/4, 3), (5π/4, 1), and (9π/4, 3). The x-intercepts (π/2, 0) and (3π/2, 0) move to (3π/4, 2) and (7π/4, 2). Draw one cycle through these five points and continue the pattern for another cycle, as shown in Fig. 5.48.
Use π/4 as the x-scale on your calculator, as we did in Fig. 5.48. Set the viewing window about the same as in Fig. 5.48. The calculator graph in Fig. 5.49 supports the conclusion that the shift is π/4 to the right and two units upward.