Copyright © 2007 Pearson Education, Inc. Slide Graphs of the Sine and Cosine Functions Many things in daily life repeat with a predictable pattern. Because sine and cosine repeat their values over and over in a regular pattern, they are examples of periodic functions. Periodic Function A periodic function is a function f such that f (x) = f (x + np), for every real number x in the domain of f, every integer n, and some positive real number p. The smallest possible positive value of p is the period of the function.

Copyright © 2007 Pearson Education, Inc. Slide Graph of the Sine Function From the graph we can see that as t increases, sin t oscillates between  1. Using x rather than t, we can plot points to obtain the graph y = sin x. The graph is continuous on (– ,  ). Its x-intercepts are of the form n , n an integer. Its period is 2 . Its graph is symmetric with respect to the origin, and it is an odd function.

Copyright © 2007 Pearson Education, Inc. Slide Graph of the Cosine Function The graph of y = cos x can be found much the same way as y = sin x. Note that the graph of y = cos x is the graph y = sin x translated units to the left. The graph is continuous on (– ,  ). Its x-intercepts are of the form (2n + 1), n an integer. Its period is 2 . Its graph is symmetric with respect to the y-axis, and it is an even function.

Copyright © 2007 Pearson Education, Inc. Slide Graphing Techniques, Amplitude, and Period ExampleGraph y = 2 sin x, and compare to the graph of y = sin x. SolutionFrom the table, the only change in the graph is the range, which becomes [–2,2].

Copyright © 2007 Pearson Education, Inc. Slide Amplitude The amplitude of a periodic function is half the difference between the maximum and minimum values. For sine and cosine, the amplitude is Amplitude The graph of y = a sin x or y = a cos x, with a  0, will have the same shape as y = sin x or y = cos x, respectively, except with range [–|a|, |a|]. The amplitude is |a|.

Copyright © 2007 Pearson Education, Inc. Slide Period To find the period of y = sin bx or y = cos bx, solve the inequality for b > 0 Thus, the period is Divide the interval into four equal parts to get the values for which y = sin bx or y = cos bx is –1, 0, or 1. These values will give the minimum points, x- intercepts, and maximum points on the graph.

Copyright © 2007 Pearson Education, Inc. Slide Graphing y = cos bx ExampleGraph over one period. Analytic Solution

Copyright © 2007 Pearson Education, Inc. Slide Graphing y = cos bx Graphing Calculator Solution Use window [0,3  ] by [–2,2], with Xscl = 3  /4. Choose Xscl = 3  /4 so that x-intercepts, maximums, and minimums coincide with tick marks on the axis.

Copyright © 2007 Pearson Education, Inc. Slide Sketching Traditional Graphs of the Sine and Cosine Functions To graph y = a sin bx or y = a cos bx, with b > 0, 1.Find the period, Start at 0 on the x-axis, and lay off a distance of 2.Divide the interval into four equal parts. 3.Evaluate the function for each of the five x-values resulting from step 2. The points will be the maximum points, minimum points, and x-intercepts. 4.Plot the points found in step 3, and join them with a sinusoidal curve with amplitude |a|. 5.Draw additional cycles as needed.

Copyright © 2007 Pearson Education, Inc. Slide Graphing y = a sin bx Example Graph y = –2 sin 3x. Solution 1.Period: 2.Divide the interval into four equal parts to get the x-values 3.

Copyright © 2007 Pearson Education, Inc. Slide Graphing y = a sin bx 4.Plot the points (x, –2 sin 3x) from the table. Notice that when a is negative, the graph of y = –2 sin 3x is a reflection across the x-axis of the graph of y = |a| sin bx.

Copyright © 2007 Pearson Education, Inc. Slide Translations Horizontal –The graph of y = f (x – d) translates the graph of y = f (x) d units to the right if d > 0 and |d| units to the left if d < 0. –A horizontal translation is called a phase shift and the expression x – d is called the argument. Vertical –The graph of y = c + f (x) translates the graph of y = f (x) c units upward if c > 0 and |c| units downward if c < 0.

Copyright © 2007 Pearson Education, Inc. Slide Graphing y = c + a sin b(x – d) ExampleGraph y = –1 + 2 sin(4x +  ). SolutionExpress y in the form c + a sin [b(x – d)]. Amplitude = 2 Period Translate |-1| = 1 unit downward and units to the left. Start the first period at x-value and end the first period at