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LESSON 6: GRAPHING SINE AND COSINE FUNCTIONS Learning Outcome: To sketch the graphs of y = sin x and y = cos x To determine the characteristics of the graphs of y = sin x and y = cos x To demonstrate an understanding of the effects of vertical and horizontal stretches on the graphs of sinusoidal functions To solve a problems by analyzing the graph of a trigonometric function Periodic Functions: A periodic function is a function whose graph repeats regularly over some interval of the domain. The length of this interval is called the period of the function. The amplitude of a periodic function is defined as half the distance between the maximum and minimum values of the function. max min amplitude 2 Graph y sin and y cos for 0 2 Sin x Cos x Must be in radians mode: 0, 4 , 2, 2,1 2 Graph y 2sin for 0 2 amplitude 2 1 -1 2 3 2 -2 Period 2 For y a sin As a increases or decreases, the period stays the same and the amplitude increases or decreases. a is a vertical stretch. Ex. Graph: y=sin x, y=3sin x, and y=0.5sin x, for 0 2 . State the amplitude of each. Ex. Sketch y=cos x and y=cos 2x, for 0 360 Ex. Now graph y=cos 3x and y=cos0.5x over the same interval. What did you notice? In general: To find the period of a function in the form y=a sin bx or y=a cos bx: period = 2 360 (for degrees) period = (for radians) b b Ex. State the amplitude and period of each: a. y 2 cos 3 : b. y 1 1 cos : 2 3 Now we will consider the graphs of the functions whose equations are; y a sin b x c d and y a cos b x c d What transformations are occurring in the following examples: a. b. c. d. y= 2 sin x : y=sin 2x: y=-3sin x: y=sin (-3x): Notice: vertical stretches affect amplitude, and horizontal stretches affect period. Ex. a. List the steps involved in graphing the function y = 3 sin 4x. b. Determine: Amplitude Period Maximum and minimum value x-intercept, y-intercept Domain and range One cycle of sin x has the following characteristics: Max value of 1, min value of -1, amplitude of 1, period of 2π, y-intercept of 0, x-intercepts of 0, π and 2π, domain of 𝜃𝜖𝑅, and a range of −1 ≤ 𝑦 ≤ 1 How many of the parameters above will change if we compare to the graph of y = cos x Max/min value: Amplitude/period: Domain/range: y-intercept x-intercepts Assignment: pg.233-237 #1-12, 14, 15, 19, 20, 21, 23