658 CHAPTER 6 The Circular Functions and Their Graphs Connecting Graphs with Equations EXAMPLE 3 Determining Equations for Graphs Determine an equation for each graph. (a) x y –3 –2 –1 1 0 2 3 4p 3p 2 p p (b) 0 3p 2 1 y x –p 2 p 2 SOLUTION (a) This graph is that of a cosecant function that is stretched horizontally having period 4p. If y = csc bx, where b 70, then we must have b = 1 2 and y = csc 1 2 x. 2p 1 2 = 2p # 2 = 4p Horizontal stretch (b) This is the graph of y = sec x, translated up 1 unit. An equation is y = 1 + sec x. Vertical translation S Now Try Exercises 27 and 29. −2 2 11p 4 11p 4 − Figure 62 −2 2 11p 4 11p 4 − Figure 63 −2 2 11p 4 11p 4 − Figure 64 Addition of Ordinates A function formed by combining two other functions, such as y3 = y1 + y2, has historically been graphed using a method known as addition of ordinates. (The x-value of a point is sometimes called its abscissa, and its y-value is called its ordinate.) EXAMPLE 4 Illustrating Addition of Ordinates Use the functions y1 = cos x and y2 = sin x to illustrate addition of ordinates for y3 = cos x + sin x with the value p 6 for x. SOLUTION In Figures 62 – 64, y1 = cos x is graphed in blue, y2 = sin x is graphed in red, and their sum, y1 + y2 = cos x + sin x, is graphed as y3 in green. If the ordinates (y-values) for x = p 6 (approximately 0.52359878) in Figures 62 and 63 are added, their sum is found in Figure 64. Verify that 0.8660254 + 0.5 = 1.3660254. (This would occur for any value of x.) S Now Try Exercise 45.
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