1. Graphs of Trigonometric Functions Digital Lesson

2. 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.

3. 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

4. 4 y x Tangent Function Graph of the Tangent Function 2. range: (– , +  ) 3. period:  4. vertical asymptotes: 1. domain : all real x Properties of y = tan x period: To graph y = tan x, use the identity. At values of x for which cos x = 0, the tangent function is undefined and its graph has vertical asymptotes.

5. 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.

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Example: Tangent Function 1. Find consecutive vertical asymptotes by solving for x: 3. Sketch one branch and repeat. Example: Find the period and asymptotes and sketch the graph of Vertical asymptotes: 2. Plot several points between asymptotes (midpoint b/w asymptotes is x-int) y x

7. Example Graph Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7

8. 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.

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Cotangent Function Graph of the Cotangent Function 2. range: (– , +  ) 3. period:  4. vertical asymptotes: 1. domain : all real x Properties of y = cot x y x vertical asymptotes To graph y = cot x, use the identity. At values of x for which sin x = 0, the cotangent function is undefined and its graph has vertical asymptotes.

10. Example Graph Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10

11. 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

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 y x Secant Function Graph of the Secant Function 2. range: (– ,–1]  [1, +  ) 3. period: 2  4. vertical asymptotes: 1. domain : all real x The graph y = sec x, use the identity. Properties of y = sec x At values of x for which cos x = 0, the secant function is undefined and its graph has vertical asymptotes.

13. Example Graph Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13

14. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 x y Cosecant Function Graph of the Cosecant Function 2. range: (– ,–1]  [1, +  ) 3. period: 2  4. vertical asymptotes where sine = 0: 1. domain : all real x To graph y = csc x, use the identity. Properties of y = csc x At values of x for which sin x = 0, the cosecant function is undefined and its graph has vertical asymptotes.

15. Example Graph Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15

16. 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