633 6.4 Translations of the Graphs of the Sine and Cosine Functions Relating Concepts For individual or collaborative investigation (Exercises 63 – 66) Connecting the Unit Circle and Sine Graph Using a TI-84 Plus calculator, adjust the settings to correspond to the following screens. Graph the two equations (which are in parametric form), and watch as the unit circle and the sine function are graphed simultaneously. Press the TRACE key once to obtain the screen shown on the left below. Then press the up-arrow key to obtain the screen shown on the right below. The screen on the left gives a unit circle interpretation of cos 0 = 1 and sin 0 = 0. The screen on the right gives a rectangular coordinate graph interpretation of sin 0 = 0. −2.5 −1.38 2.5 6.67 −2.5 −1.38 2.5 6.67 63. On the unit circle graph, let T = 2. Find X and Y, and interpret their values. 64. On the sine graph, let T = 2. What values of X and Y are displayed? Interpret these values with an equation in X and Y. 65. Now go back and redefine Y2T as cos1T2. Graph both equations. On the cosine graph, let T = 2. What values of X and Y are displayed? Interpret these values with an equation in X and Y. 66. Explain the relationship between the coordinates of the unit circle and the coordinates of the sine and cosine graphs. MODE FORMAT Y= EDITOR Horizontal Translations The graph of the function y =ƒ1x −d2 is translated horizontally compared to the graph of y = ƒ1x2. The translation is to the right d units if d 70 and to the left d units if d 60. See Figure 31. With circular functions, a horizontal translation is a phase shift. In the function y = ƒ1x - d2, the expression x - d is the argument. 6.4 Translations of the Graphs of the Sine and Cosine Functions ■ Horizontal Translations ■ Vertical Translations ■ Combinations of Translations ■ ATrigonometric Model y y = f(x) –3 0 x 4 y = f(x – 4) y = f(x + 3) Horizontal translations of y = f(x) Figure 31
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