Rev.S08 MAC 1114 Module 4 Graphs of the Circular Functions

2 Rev.S08 Learning Objectives Upon completing this module, you should be able to: 1. Recognize periodic functions. 2. Determine the amplitude and period, when given the equation of a periodic function. 3. Find the phase shift and vertical shift, when given the equation of a periodic function. 4. Graph sine and cosine functions. 5. Graph cosecant and secant functions. 6. Graph tangent and cotangent functions. 7. Interpret a trigonometric model. Click link to download other modules.

3 Rev.S08 Graphs of the Circular Functions Click link to download other modules. - Graphs of the Sine and Cosine Functions - Translations of the Graphs of the Sine and Cosine Functions - Graphs of the Other Circular Functions There are three major topics in this module:

4 Rev.S08 Introduction to Periodic Function Click link to download other modules. 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.

5 Rev.S08 Example of a Periodic Function Click link to download other modules.

6 Rev.S08 Example of Another Periodic Function Click link to download other modules.

7 Rev.S08 What is the Amplitude of a Periodic Function? Click link to download other modules. 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 Rev.S08 How to Graph y = 3 sin(x) ? Click link to download other modules. 0−3−30303sin x 0−1−1010sin x  3  /2  /2 0x Note the difference between sin x and 3sin x. What is the difference?

9 Rev.S08 How to Graph y = sin(2x)? Click link to download other modules. 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.

10 Rev.S08 How to Graph y = sin(2x)? (Cont.) Click link to download other modules. Note the difference between sin x and sin 2x. What is the difference?

11 Rev.S08 Period of a Periodic Function Click link to download other modules. Based on the previous example, we can generalize the following: 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.

12 Rev.S08 How to Graph y = cos (2x/3) over one period? Click link to download other modules. The period is 3 . Divide the interval into four equal parts.  Obtain key points for one period. 10−1−101cos 2x/3 22 3  /2  /2 02x/3 33 9  /43  /23  /4 0x

13 Rev.S08 How to Graph y = cos(2x/3) over one period? (Cont.) Click link to download other modules. The amplitude is 1. Join the points and connect with a smooth curve.

14 Rev.S08 Guidelines for Sketching Graphs of Sine and Cosine Functions Click link to download other modules. 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.

15 Rev.S08 Guidelines for Sketching Graphs of Sine and Cosine Functions Continued Click link to download other modules. 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.

16 Rev.S08 How to Graph y = −2 sin(4x)? Click link to download other modules. 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 020−2−20−2 sin 4x 0−1−1010sin 4x 22 3  /2  /2 04x4x 3  /8  /4  /8 0x

17 Rev.S08 How to Graph y = −2 sin(4x)? (Cont.) Click link to download other modules. Plot the points and join them with a sinusoidal curve with amplitude 2.

18 Rev.S08 What is a Phase Shift? Click link to download other modules. In trigonometric functions, a horizontal translation is called a phase shift. In the equation the graph is shifted  /2 units to the right.

19 Rev.S08 How to Graph y = sin (x −  /3) by Using Horizontal Translation or Phase Shift? Click link to download other modules. Find the interval for one period. Divide the interval into four equal parts.

20 Rev.S08 How to Graph y = sin (x −  /3) by Using Horizontal Translation or Phase Shift? (Cont.) Click link to download other modules. 0−1−1010 sin (x −  /3) 22 3  /2  /2 0 x −  /3 7  /311  /64  /35  /6  /3 x

21 Rev.S08 How to Graph y = 3 cos(x   /4) by Using Horizontal Translation or Phase Shift? Click link to download other modules. Find the interval. Divide into four equal parts.

22 Rev.S08 How to Graph y = 3 cos(x   /4) by Using Horizontal Translation or Phase Shift? Click link to download other modules. 30−3−303 3 cos (x +  /4) 10−1−101 cos(x +  /4) 22 3  /2  /2 0 x +  /4 7  /45  /43  /4  /4−  /4 x

23 Rev.S08 How to Graph y = 2 − 2 sin 3x by Using Vertical Translation or Vertical Shift? Click link to download other modules − 2 sin 3x 020−2−20−2 sin 3x 22 3  /2  /2 03x3x 2  /3  /2  /3  /6 0x The graph is translated 2 units up from the graph y = −2 sin 3x.

24 Rev.S08 How to Graph y = 2 − 2 sin 3x by Using Vertical Translation or Vertical Shift? (Cont.) Click link to download other modules. Plot the points and connect.

25 Rev.S08 Further Guidelines for Sketching Graphs of Sine and Cosine Functions Click link to download other modules. Method 1: Follow these steps. Step 1Find an interval whose length is one period 2  /b by solving the three part inequality 0  b(x − d)  2 . 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 points that intersect the line y = c (middle points of the wave.)

26 Rev.S08 Further Guidelines for Sketching Graphs of Sine and Cosine Functions (Cont.) Click link to download other modules. 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. Method 2:First graph the basic circular function. The amplitude of the function is |a|, and the period is 2  /b. Then use translations to graph the desired function. The vertical translation is c units up if c > 0 and |c| units down if c 0 and |d| units to the left if d < 0.

27 Rev.S08 How to Graph y = −1 + 2 sin (4x +  )? Click link to download other modules. Write the expression in the form c + a sin b(x − d) by rewriting 4x +  as Step 1 Step 2: Divide the interval. Step 3 Table

28 Rev.S08 How to Graph y = −1 + 2 sin (4x +  )?(Cont.) Click link to download other modules. −1−1−3−3−1−11−1−1 −1 + 2sin(4x +  ) 2−2− sin 4(x +  /4) 0−1−1010 sin 4(x +  /4) 22 3  /2  /2 0 4(x +  /4)  /23  /8  /4  /8 0 x +  /4  /4  /8 0 −  /8−  /4 x

29 Rev.S08 How to Graph y = −1 + 2 sin (4x +  )? (Cont.) Click link to download other modules. Steps 4 and 5  Plot the points found in the table and join then with a sinusoidal curve.

30 Rev.S08 Let’s Take a Look at Other Circular Functions. Click link to download other modules.

31 Rev.S08 Cosecant Function Click link to download other modules.

32 Rev.S08 Secant Function Click link to download other modules.

33 Rev.S08 Guidelines for Sketching Graphs of Cosecant and Secant Functions Click link to download other modules. 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

34 Rev.S08 Guidelines for Sketching Graphs of Cosecant and Secant Functions Continued Click link to download other modules. 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.

35 Rev.S08 How to Graph y = 2 sec(x/2)? Click link to download other modules. Step 1: Graph 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.

36 Rev.S08 How to Graph y = 2 sec(x/2)? (Cont.) Click link to download other modules. Step 2: 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 . Step 3: Sketch the graph of y = 2 sec x/2 by drawing the typical U-shaped branches, approaching the asymptotes.

37 Rev.S08 Tangent Function Click link to download other modules.

38 Rev.S08 Cotangent Function Click link to download other modules.

39 Rev.S08 Guidelines for Sketching Graphs of Tangent and Cotangent Functions Click link to download other modules. 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:

40 Rev.S08 Guidelines for Sketching Graphs of Tangent and Cotangent Functions continued Click link to download other modules. 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.

41 Rev.S08 How to Graph y = tan(2x)? Click link to download other modules. Step 1:The period of the function is  /2. The two asymptotes have equations x = −  /4 and x =  /4. Step 2: Sketch the two vertical asymptotes found. x =   /4. Step 3:Divide the interval into four equal parts. This gives the following key x-values.  First quarter: −  /8  Middle value: 0Third quarter:  /8

42 Rev.S08 How to Graph y = tan(2x)? (Cont.) Click link to download other modules. Step 4:Evaluate the function Step 5: Join the points with a smooth curve, approaching the vertical asymptotes. Indicate additional asymptotes and periods of the graph as necessary. 10−1−1tan 2x  /4 0 −  /4 2x2x  /8 0 −  /8 x

43 Rev.S08 How to Graph y = tan(2x)? (Cont.) Click link to download other modules. 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.

44 Rev.S08 What have we learned? We have learned to: 1. Recognize periodic functions. 2. Determine the amplitude and period, when given the equation of a periodic function. 3. Find the phase shift and vertical shift, when given the equation of a periodic function. 4. Graph sine and cosine functions. 5. Graph cosecant and secant functions. 6. Graph tangent and cotangent functions. 7. Interpret a trigonometric model. Click link to download other modules.

45 Rev.S08 Credit Some of these slides have been adapted/modified in part/whole from the slides of the following textbook: Margaret L. Lial, John Hornsby, David I. Schneider, Trigonometry, 8th Edition Click link to download other modules.