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90 80 70 60 50 40 30 20 10 0 0 5/24/2017 10 20 30 2 Sine 5/24/2017 The most fundamental sine wave, y=sin(x), has the graph shown. It fluctuates from 0 to a high of 1, down to –1, and back to 0, in a space of 2. 3 The graph of y a sin b( x h) k is determined by four numbers, a, b, h, and k. The amplitude, a, tells the height of each peak and the depth of each trough. The frequency, b, tells the number of full wave patterns that are completed in a space of 2. The period of the function is The two remaining numbers, h and k, tell the 2 translation of the wave from the origin. b 5/24/2017 4 5 4 3 2 1 2 1 1 1 2 3 4 5 5/24/2017 2 Which of the following equations best describes the graph shown? (A) y = 3sin(2x) - 1 (B) y = 2sin(4x) (C) y = 2sin(2x) - 1 (D) y = 4sin(2x) - 1 (E) y = 3sin(4x) 5 5 4 3 2 1 2 1 1 1 2 2 4 5 y = 3sin(2x) - 1 Graph is translated -1 vertically. Find height of each peak. 3 5/24/2017 Find the baseline between the high and low points. Amplitude is 3 Count number of waves in 2 Frequency is 2 6 Cosine 5/24/2017 The graph of y=cos(x) resembles the graph of y=sin(x) but is shifted, or translated, units to the left. 2 It fluctuates from 1 to 0, down to –1, back to 0 and up to 1, in a space of 2. 7 The values of a, b, h, and k change the shape and location of the wave as for the sine. y a cos b( x h) k Amplitude Frequency Period Translation 5/24/2017 a b 2/b h, k Height of each peak Number of full wave patterns Space required to complete wave Horizontal and vertical shift 8 Which of the following equations best describes the graph? 5/24/2017 (A) y = 3cos(5x) + 4 (B) y = 3cos(4x) + 5 (C) y = 4cos(3x) + 5 (D) y = 5cos(3x) +4 (E) y = 5sin(4x) +3 8 6 4 2 2 1 1 2 9 Find the baseline 6 4 2 Amplitude = 5 Number of waves in 2 5/24/2017 Vertical translation + 4 Find the height of peak 8 Frequency =3 2 1 1 2 y = 5cos(3x) + 4 10 Tangent The tangent function has a discontinuous graph, repeating in a period of . Cotangent Like the tangent, cotangent is discontinuous. Discontinuities of the cotangent are units left of those for tangent. 2 5/24/2017 11 Secant and Cosecant 5/24/2017 The secant and cosecant functions are the reciprocals of the cosine and sine functions respectively. Imagine each graph is balancing on the peaks and troughs of its reciprocal function. 12