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- 1. Digital LessonGraphs of Trigonometric Functions
- 2. Properties of Sine and Cosine Functions The graphs of y = sin x and y = cos x have similar properties: 1. The domain is the set of real numbers. 2. The range is the set of y values such that − 1 ≤ y ≤ 1 . 3. The maximum value is 1 and the minimum value is –1. 4. The graph is a smooth curve. 5. Each function cycles through all the values of the range over an x-interval of 2π . 6. The cycle repeats itself indefinitely in both directions.Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2
- 3. Graph of the Sine Function To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. π 3π x 0 π 2π 2 2 sin x 0 1 0 -1 0 Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y y = sin x 3π π 1 π 3π 5π − − 2 −π 2 2 π 2 2π 2 x −1Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3
- 4. Graph of the Cosine Function To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts. π 3π x 0 π 2π 2 2 cos x 1 0 -1 0 1 Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y y = cos x 3π π 1 π 3π 5π − − 2 −π 2 2 π 2 2π 2 x −1Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4
- 5. Example: Sketch the graph of y = 3 cos x on the interval [–π, 4π]. Find the key points; graph one cycle; then extend the graph in both directions for the required interval. π 3π x 0 2 π 2 2π y = 3 cos x 3 0 -3 0 3 max x-int min x-int max y (0, 3) (2π, 3) 2 −π 1 π 2π 3π 4π x −1 π ( 3π , 0) − 2 ( 2 , 0) 2 −3 ( π, –3)Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5
- 6. The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function. amplitude = |a| If |a| > 1, the amplitude stretches the graph vertically. If 0 < |a| > 1, the amplitude shrinks the graph vertically. If there is a negative in front (a < 0), the graph is reflected in the x- axis.When I ask for yamplitude I will not 4ask what kind ofstretch it is. Instead, y = sin x π 3πI will ask for the 2 π 2 2π xvalue of the 1amplitude. y = 2 sin x y = – 4 sin x y = 2 sin x reflection of y = 4 sin x y = 4 sin x −4Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6
- 7. The period of a function is the x interval needed for the function to complete one cycle. For b > 0, the period of y = a sin bx is 2π . b For b > 0, the period of y = a cos bx is also 2π . b If 0 < b < 1, the graph of the function is stretched horizontally. y y = sin 2π period: 2π period: π y = sin x x −π π 2π If b > 1, the graph of the function is shrunk horizontally. y y = cos x 1 y = cos x period: 2π 2 −π π 2π 3π 4π x period: 4πCopyright © by Houghton Mifflin Company, Inc. All rights reserved. 7
- 8. Use basic trigonometric identities to graph y = f (–x) Example 1: Sketch the graph of y = sin (–x). The graph of y = sin (–x) is the graph of y = sin x reflected in the x-axis. y y = sin (–x) Use the identity x sin (–x) = – sin x π 2π y = sin x Example 2: Sketch the graph of y = cos (–x). The graph of y = cos (–x) is identical to the graph of y = cos x. y Use the identity cos (–x) = – cos x x π 2π y = cos (–x)Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8
- 9. Example: Sketch the graph of y = 2 sin (–3x). Rewrite the function in the form y = a sin bx with b > 0 Use the identity sin (– x) = – sin x: y = 2 sin (–3x) = –2 sin 3x period: 2 π 2π amplitude: |a| = |–2| = 2 = b 3 Calculate the five key points. x π π π 2π 0 6 3 2 3 y = –2 sin 3x 0 –2 0 2 0 y ( π , 2) 2 2 π π π π 2π 5π 6 6 3 2 3 6 π x (0, 0) ( π , 0) 2π −2 3 ( , 0) ( π , -2) 3 6Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9
- 10. Graph of the Tangent Function sin x To graph y = tan x, use the identity tan x = . cos x At values of x for which cos x = 0, the tangent function is undefined and its graph has vertical asymptotes. y Properties of y = tan x 1. domain : all real x π x ≠ kπ + ( k ∈ Ζ ) π 3π 2 2. range: (–∞, +∞) 2 2 x 3. period: π − 3π −π 2 2 4. vertical asymptotes: π 3π x = , ( repeatseveryπ ) 2 2 period: πCopyright © by Houghton Mifflin Company, Inc. All rights reserved. 10
- 11. Example: Find the period and asymptotes and sketch the graph 1 π π of y = tan 2 x x=− y x= 3 4 4 1. Period of y = tan x is . π π → Period of y = tan 2 x is . 2 − 3π π 1 ,− π 8 8 3 2 2. Find consecutive vertical x asymptotes by solving for x: π 1 , 3π 1 π π 8 3 ,− 2x = − , 2x = 8 3 2 2 π π Vertical asymptotes: x = − , x = 4 4 π π π 3π 3. Plot several points in (0, ) x − 0 2 8 8 8 1 1 1 1 y = tan 2 x − 0 − 4. Sketch one branch and repeat. 3 3 3 3Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11

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