Digital Lesson Graphs of Trigonometric Functions HWQ 1. A weight attached to the end of a spring is pulled down 3 cm below its equilibrium point and released. It takes 2 seconds for it to complete one cycle of moving from 3 cm. below the equilibrium point to 3 cm. above and then returning to its low point. Find the sinusoidal function that best represents the position of the moving weight and the approximate position of the weight 5 seconds after it is released. Warm-Up • • • • Graph y = tanx on your calculator. Where does the function seem to be undefined? Why do you think it is undefined there? Discuss with your partner. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 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 x k 2 2. range: (–, +) 3. period: 4. vertical asymptotes: x k 2 2 3 2 3 2 x 2 period: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Sketching the Graph of y=tanx • The standard form is y=atan(bx-c)+d • For one cycle, determine the endpoints of the interval by solving: – Left endpoint: bx-c = - /2 – Right endpoint: bx-c = /2 This interval is bounded by asymptotes. • Graph the asymptotes, plot the x-intercept, and sketch the characteristic curve. Example: Find the period and asymptotes and sketch the graph y 1 x x of y tan 2 x 4 4 3 1. Find consecutive vertical asymptotes by solving for x: 2x , 2x 2 2 3 8 1 , 8 3 2 1 , 8 3 Vertical asymptotes: x , x 4 4 2. Plot several points between asymptotes (midpoint b/w asymptotes is x-int) 3. Sketch one branch and repeat. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. x 8 1 1 y tan 2 x 3 3 0 0 x 3 1 , 3 8 3 8 8 1 1 3 3 6 Example • Graph x y 2 tan 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Sketching the Graph of cotx • The standard form is y=acot(bx-c)+d • For one cycle, determine the endpoints of the interval by solving: – Left endpoint: bx-c = 0 – Right endpoint: bx-c = This interval is bounded by asymptotes. • Graph the asymptotes, plot the x-intercept, and sketch the characteristic curve. Graph of the Cotangent Function cos x To graph y = cot x, use the identity cot x . sin x At values of x for which sin x = 0, the cotangent function is undefined and its graph has vertical asymptotes. y Properties of y = cot x y cot x 1. domain : all real x x k k 2. range: (–, +) 3. period: 4. vertical asymptotes: x k k vertical asymptotes Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 2 2 x x0 3 2 2 x x 2 x 2 9 Example • Graph y 3 cot 2 x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Sketching secx and cscx • These are the reciprocal functions cscx=1/sinx secx=1/cosx You do not graph these functions directly: Graph the reciprocal function as a “support” Add asymptotes where the support function equals zero (WHY?) Add cscx or secx :they are parabolic curves rising up/down from the support function Remove the support function Graph of the Secant Function The graph y = sec x, use the identity sec x 1 . cos x At values of x for which cos x = 0, the secant function is undefined and its graph has vertical asymptotes. y y sec x Properties of y = sec x 1. domain : all real x x k ( k ) 2 2. range: (–,–1] [1, +) 3. period: 2 4. vertical asymptotes: x k k 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 y cos x x 2 2 3 2 2 5 3 2 4 12 Example Graph 1 y sec x 4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Graph of the Cosecant Function 1 To graph y = csc x, use the identity csc x . sin x At values of x for which sin x = 0, the cosecant function is undefined and its graph has vertical asymptotes. y Properties of y = csc x 4 y csc x 1. domain : all real x x k k 2. range: (–,–1] [1, +) 3. period: 2 4. vertical asymptotes where sine = 0: x k k Copyright © by Houghton Mifflin Company, Inc. All rights reserved. x 2 2 3 2 2 5 2 y sin x 4 14 Example Graph y csc2 x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Homework 4.6 pg 305 7, 9, 31-47 odd, 53,55,67 Quiz Thursday on Graphing Trig. Functions and Inverse Trig Functions. (4.5/7) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16