1. 1 Chapter 4 Graphing and Inverse Functions

2. 2 The line from the center sweeps out a central angle  in an amount time t, then the angular velocity, (omega) Angular Speed: the amount of rotation per unit of time, where  is the angle of rotation and t is the time. 3.5 Angular and Linear Speed

3. 3 Periodic Function (page 165) A periodic function is a function f such that f(x) = f(x + np), for every real number x in the domain of f, every integer n, and some positive real number p. The smallest possible positive value of p is the period of the function.

4. 4 Sine Function f(x) = sin x

5. 5 Cosine Function f(x) = cos x

6. 6 Example: Graph y = 3 sin x compare to y = sin x. Make a table of values. 0 33 0303sin x 0 11 010sin x  3  /2  /2 0x

7. 7 Amplitude The amplitude of a periodic function is half the difference between the maximum and minimum values. The graph of y = a sin x or y = a cos x, with a  0, will have the same shape as the graph of y = sin x or y = cos x, respectively, except the range will be [  |a|, |a|]. The amplitude is |a|.

8. 8 Example: Graph y = sin 2x The period is 2  /2 = . The graph will complete one period over the interval [0,  ]. The endpoints are 0 and , the three middle points are: Plot points and join in a smooth curve.

9. 9 Example: Graph y = sin 2x continued

10. 10 Period For b > 0, the graph of y = sin bx will resemble that of y = sin x, but with period 2  /b. The graph of y = cos bx will resemble that of y = cos x, with period 2  /b.

11. 11 Graph y = cos 2x/3 over one period The period is 3 . Divide the interval into four equal parts. – Obtain key points for one period. 10 11 01 cos 2x/3 22 3  /2  /2 02x/3 33 9  /43  /23  /4 0x

12. 12 Graph y = cos 2x/3 over one period continued The amplitude is 1. Join the points and connect with a smooth curve.

13. 13 Guidelines for Sketching Graphs of Sine and Cosine Functions To graph y = a sin bx or y = a cos bx, with b > 0, follow these steps. Step 1Find the period, 2  /b. Start with 0 on the x-axis, and lay off a distance of 2  /b. Step 2Divide the interval into four equal parts. Step 3Evaluate the function for each of the five x-values resulting from Step 2. The points will be maximum points, minimum points, and x-intercepts.

14. 14 Guidelines for Sketching Graphs of Sine and Cosine Functions continued Step 4Plot the points found in Step 3, and join them with a sinusoidal curve having amplitude |a|. Step 5Draw the graph over additional periods, to the right and to the left, as needed.

15. 15 Graph y =  2 sin 4x Step 1Period = 2  /4 =  /2. The function will be graphed over the interval [0,  /2]. Step 2Divide the interval into four equal parts. Step 3Make a table of values 020 22 0  2 sin 4x 0 11 010sin 4x 22 3  /2  /2 04x4x 3  /8  /4  /8 0x

16. 16 Graph y =  2 sin 4x continued Plot the points and join them with a sinusoidal curve with amplitude 2.

17. 17 Tangent Functions

18. 18 Cosecant Function

19. 19 Cotangent Functions

20. 20 Secant Function

21. 21 Guidelines for Sketching Graphs of Cosecant and Secant Functions To graph y = csc bx or y = sec bx, with b > 0, follow these steps. Step 1Graph the corresponding reciprocal function as a guide, using a dashed curve. y = cos bxy = a sec bx y = a sin bxy = a csc bx Use as a GuideTo Graph

22. 22 Guidelines for Sketching Graphs of Cosecant and Secant Functions continued Step 2Sketch the vertical asymptotes. They will have equations of the form x = k, where k is an x-intercept of the graph of the guide function. Step 3Sketch the graph of the desired function by drawing the typical U-shapes branches between the adjacent asymptotes. The branches will be above the graph of the guide function when the guide function values are positive and below the graph of the guide function when the guide function values are negative.

23. 23 Graph y = 2 sec x/2 Step 1Graph the corresponding reciprocal function y = 2 cos x/2. The function has amplitude 2 and one period of the graph lies along the interval that satisfies the inequality Divide the interval into four equal parts.

24. 24 Graph y = 2 sec x/2 continued

25. 25 Graph y = 2 sec x/2 continued Sketch the vertical asymptotes. These occur at x-values for which the guide function equals 0, such as x =  3 , x = 3 , x = , x = 3 . Sketch the graph of y = 2 sec x/2 by drawing the typical U-shaped branches, approaching the asymptotes.

26. 26 Guidelines for Sketching Graphs of Tangent and Cotangent Functions To graph y = tan bx or y = cot bx, with b > 0, follow these steps. Step 1Determine the period,  /b. To locate two adjacent vertical asymptotes solve the following equations for x:

27. 27 Guidelines for Sketching Graphs of Tangent and Cotangent Functions continued Step 2 Sketch the two vertical asymptotes found in Step 1. Step 3Divide the interval formed by the vertical asymptotes into four equal parts. Step 4Evaluate the function for the first-quarter point, midpoint, and third-quarter point, using the x-values found in Step 3. Step 5Join the points with a smooth curve, approaching the vertical asymptotes. Indicate additional asymptotes and periods of the graph as necessary.

28. 28 Graph y = tan 2x Step 1The period of the function is  /2. The two asymptotes have equations x =  /4 and x =  /4. Step 2Sketch two vertical asymptotes x =   /4. Step 3Divide the interval into four equal parts. This gives the following key x-values. –First quarter:  /8 –Middle value: 0Third quarter:  /8

29. 29 Graph y = tan 2x continued Table of values 10 11 tan 2x  /4 0  /4 2x2x  /8 0  /8 x

30. 30 Graph y = 2 + tan x Every y value for this function will be 2 units more than the corresponding y in y = tan x, causing the graph to be translated 2 units up compared to y = tan x.