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Sin and Cos Graphs Amplitude and Period Objective: The student, given one of the six trigonometric functions in standard form [e.g., y = A sin (Bx + C) + D, where A, B, C, and D are real numbers], will: a) state the domain and the range of the function; b) determine the amplitude, period, phase shift, and vertical shift; and c) sketch the graph of the function by using transformations for at least a one-period interval. Sin and Cos Graphs •To graph the equation y = sin x or y = cos x, begin by making a table of values of x and y that satisfy the equation: x 0 y = sin x 0 4 2 2 6 1 2 Decimal Approx. 3 3 2 0.7 0.9 2 1 2 3 5 3 4 6 3 2 1 2 2 2 0 7 5 4 3 5 7 11 2 3 6 6 4 3 4 2 1 2 3 3 2 1 -1 0 2 2 2 2 2 2 -0.7 -0.9 0.9 0.7 -0.9 -0.7 y = sin x 1 x -1 6 4 3 2 2 3 5 3 4 6 7 5 4 6 4 3 3 2 5 7 11 3 4 6 2 Sin and Cos Graphs •To graph the equation y = sin x or y = cos x, begin by making a table of values of x and y that satisfy the equation: x 0 y = cos x 1 Decimal Approx. 4 2 2 6 3 2 2 3 5 7 5 4 3 3 4 3 6 6 4 2 1 2 3 3 2 1 0 0 -1 2 2 2 2 2 2 3 1 2 2 -0.7 -0.9 0.9 0.7 5 3 1 2 -0.9 -0.7 7 11 2 6 4 2 3 1 2 2 0.7 0.9 y = cos x 1 x -1 6 4 3 2 2 3 5 3 4 6 7 5 4 6 4 3 3 2 5 7 11 3 4 6 2 Sin and Cos Graphs • The sine and cosine functions are periodic functions, due to the nature of the values periodically repeating themselves. Amplitude and Period • Period: – The period of a function f(x) is the smallest positive number, p, for which f(x + p) = f(x) for all x. The period is the length of the graph before it begins to repeat itself. – For the sine and cosine functions, the period is 2 since it is the smallest positive number for which the values begin to repeat themselves. • Amplitude: – If the greatest value of y is M and the least value of y is m, then the amplitude of the graph of y is defined to be A = ½ |M – m|. The amplitude determines the height of the graph. 5 Point Method of Graphing: • When graphing the sine and cosine functions, it is not necessary to graph every point along the way. There are FIVE critical points for each function. • The five points occur at the beginning, and end of the period, halfway through the period, and at the ¼ and ¾ points of the period. Critical Points for Sine: • • • • • Beginning: (0, 0) ¼ : (/2, 1) Halfway: (, 0) y = sin x ¾ : (3/2, -1) End: (2, 0) 1 x / -1 2 3/ 2 2 Critical Points for Cosine: • • • • • Beginning: (0, 1) ¼ : (/2, 0) Halfway: (, -1) y = cos x ¾ : (3/2, 0) End: (2, 1) 1 x / -1 2 3/ 2 2 Sine and Cosine Graphs • The graphs of y = sin x and y = cos x both had a period of 2 and an amplitude of 1. • Both the period and amplitude of the graphs of these trig functions can be changed to produce variations of these graphs. • In general the graph of y = a sin bx and y = a cos bx have an amplitude of |a| and a period of 2/b. Examples: • State the amplitude and period for each of the following trig functions. a b 1. y = 2 cos 2x a b 2. y = -3 sin 4x Amp = |2| = 2 Amp = |-3| = 3 Per = Per = 2/ 2 = 2/ 4 = /2 a b 3. y = ½ sin 2/3 x Amp = |½| = ½ Per = 2/ 1 • 3/2 = 3 Sine and Cosine Graphs: • We can graph these functions the same way that we graph the regular functions y = sin x and y = cos x, using the 5 point method. y = sin x baseline maximum baseline minimum baseline y = cos x maximum baseline minimum baseline maximum Sine and Cosine Graphs: • To graph each of these functions, you should: – Find the amplitude and period. – Divide the period into fourths and label the x-axis. – Use the amplitude as the maximum and minimum to label the y-axis. – Plot the 5 critical points and connect them with a smooth curve. Examples: • Determine the period and amplitude of each function. Then graph the function. – y = 3 sin 2x • Period: • Amp: 3 y 3 2 1 Domain: −∞, ∞ Range: [-3,3] / -1 -2 -3 4 / 2 3/ 4 x Examples: • Determine the period and amplitude of each function. Then graph the function. – y = 2 cos 0.5x • Period: 4 • Amp: 2 y 3 2 1 Domain: −∞, ∞ Range: [-2 ,2] -1 -2 -3 2 3 4 x Examples: • Determine the period and amplitude of each function. Then graph the function. – y = -0.5 sin 3x 2/ • Period: • Amp: 0.5 y 3 3 2 1 Domain: −∞, ∞ Range: [-0.5 0.5] / -1 -2 -3 6 / 3 / 2 2/ x 3 Examples: • Determine the period and amplitude of each function. Then graph the function. – y = 1.5 cos 2/3 x • Period: 3 • Amp: 1.5 y 3 2 1 Domain: −∞, ∞ Range: [-1.5, 1.5] 3/ -1 -2 -3 4 3/ 2 9/ 4 3 x