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MAC 1114

MAC 1114. Module 4 Graphs of the Circular Functions. Rev.S08. Learning Objectives. Upon completing this module, you should be able to: Recognize periodic functions. Determine the amplitude and period, when given the equation of a periodic function.

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MAC 1114

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

  2. Learning Objectives • Upon completing this module, you should be able to: • Recognize periodic functions. • Determine the amplitude and period, when given the equation of a periodic function. • Find the phase shift and vertical shift, when given the equation of a periodic function. • Graph sine and cosine functions. • Graph cosecant and secant functions. • Graph tangent and cotangent functions. • Interpret a trigonometric model. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

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

  4. Introduction to Periodic Function • 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. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  5. Example of a Periodic Function http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  6. Example of Another Periodic Function http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  7. What is the Amplitude of a Periodic Function? • 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|. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  8. x 0 π/2 π 3π/2 π sin x 0 1 0 −1 0 3sin x 0 3 0 −3 0 How to Graph y = 3 sin(x)? Note the difference between sin x and 3sin x. What is the difference? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  9. How to Graph y = sin(2x)? • The period is 2π/2 = π. The graph will complete one period over the interval [0, π]. • The endpoints are 0 and π, the three middle points are: • Plot points and join in a smooth curve. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

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

  11. Period of a Periodic Function • 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 2π/b. • The graph of y = cos bx will resemble that of y = cos x, with period 2π/b. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  12. x 0 3π/4 3π/2 9π/4 3π 2x/3 0 π/2 π 3π/2 2π cos 2x/3 1 0 −1 0 1 How to Graphy = cos (2x/3) over one period? • The period is 3π. • Divide the interval into four equal parts. • Obtain key points for one period. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  13. How to Graphy = cos(2x/3) over one period? (Cont.) • The amplitude is 1. • Join the points and connect with a smooth curve. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  14. 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, 2π/b. Start with 0 on the x-axis, and lay off a distance of 2π/b. • Step 2 Divide the interval into four equal parts. • Step 3 Evaluate the function for each of the five x-values resulting from Step 2. The points will be maximum points, minimum points, and x-intercepts. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  15. Guidelines for Sketching Graphs of Sine and Cosine Functions Continued • Step 4 Plot the points found in Step 3, and join them with a sinusoidal curve having amplitude |a|. • Step 5 Draw the graph over additional periods, to the right and to the left, as needed. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  16. x 0 π/8 π/4 3π/8 π/2 4x 0 π/2 π 3π/2 2π sin 4x 0 1 0 −1 0 −2 sin 4x 0 −2 0 2 0 How to Graph y = −2 sin(4x)? • Step 1 Period = 2π/4 = π/2. The function will be graphed over the interval [0, π/2] . • Step 2 Divide the interval into four equal parts. • Step 3 Make a table of values http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

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

  18. What is a Phase Shift? • In trigonometric functions, a horizontal translation is called a phase shift. • In the equation • the graph is shiftedπ/2 units to the right. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

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

  20. x π/3 5π/6 4π/3 11π/6 7π/3 x−π/3 0 π/2 π 3π/2 2π sin (x−π/3) 0 1 0 −1 0 How to Graph y = sin (x−π/3) by Using Horizontal Translation or Phase Shift?(Cont.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  21. How to Graph y = 3 cos(x+π/4) by Using Horizontal Translation or Phase Shift? • Find the interval. • Divide into four equal parts. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  22. x −π/4 π/4 3π/4 5π/4 7π/4 x + π/4 0 π/2 π 3π/2 2π cos(x + π/4) 1 0 −1 0 1 3 cos (x + π/4) 3 0 −3 0 3 How to Graph y = 3 cos(x+π/4) by Using Horizontal Translation or Phase Shift? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  23. x 0 π/6 π/3 π/2 2π/3 3x 0 π/2 π 3π/2 2π −2 sin 3x 0 −2 0 2 0 2 − 2 sin 3x 2 0 2 4 2 How to Graph y = 2 − 2 sin 3x by Using Vertical Translation or Vertical Shift? • The graph is translated 2 units up from the graph y = −2 sin 3x. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

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

  25. Further Guidelines for Sketching Graphs of Sine and Cosine Functions • Method 1: Follow these steps. • Step 1 Find an interval whose length is one period 2π/b by solving the three part inequality 0 ≤b(x − d) ≤ 2π. • Step 2 Divide the interval into four equal parts. • Step 3 Evaluate 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.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  26. Further Guidelines for Sketching Graphs of Sine and Cosine Functions (Cont.) • Step 4 Plot the points found in Step 3, and join them with a sinusoidal curve having amplitude |a|. • Step 5 Draw 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 2π/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. The horizontal translation (phase shift) is d units to the right if d > 0 and |d| units to the left if d < 0. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  27. How to Graph y = −1 + 2 sin (4x + π)? • Step 2:Divide the interval. • Step 3 Table • Write the expression in the form c + a sin b(x−d) by rewriting 4x + πas • Step 1 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  28. x −π/4 −π/8 0 π/8 π/4 x + π/4 0 π/8 π/4 3π/8 π/2 4(x + π/4) 0 π/2 π 3π/2 2π sin 4(x + π/4) 0 1 0 −1 0 2 sin 4(x + π/4) 0 2 0 −2 2 −1 + 2sin(4x + π) −1 1 −1 −3 −1 How to Graph y = −1 + 2 sin (4x + π)?(Cont.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  29. How to Graph y = −1 + 2 sin (4x + π)?(Cont.) • Steps 4 and 5 • Plot the points found in the table and join then with a sinusoidal curve. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  30. Let’s Take a Look at Other Circular Functions. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  31. Cosecant Function http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  32. Secant Function http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  33. To Graph Use as a Guide y = a csc bx y = a sin bx y = a sec bx y = cos bx 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. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  34. 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 3 Sketch 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. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  35. How to Graph y = 2 sec(x/2)? • 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. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  36. How to Graph y = 2 sec(x/2)? (Cont.) • Step 2: Sketch the vertical asymptotes. These occur at x-values for which the guide function equals 0, such as x = −3π, x = 3π, x = π, x = 3π. • Step 3: Sketch the graph of y = 2 sec x/2 by drawing the typical U-shaped branches, approaching the asymptotes. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  37. Tangent Function http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  38. Cotangent Function http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  39. 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 1 Determine the period, π/b. To locate two adjacent vertical asymptotes solve the following equations for x: http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  40. Guidelines for Sketching Graphs of Tangent and Cotangent Functions continued • Step 2Sketch 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. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

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

  42. How to Graph y = tan(2x)? (Cont.) • 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. x −π/8 0 π/8 2x −π/4 0 π/4 tan 2x −1 0 1 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  43. How to Graph y = tan(2x)? (Cont.) • 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. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  44. What have we learned? • We have learned to: • Recognize periodic functions. • Determine the amplitude and period, when given the equation of a periodic function. • Find the phase shift and vertical shift, when given the equation of a periodic function. • Graph sine and cosine functions. • Graph cosecant and secant functions. • Graph tangent and cotangent functions. • Interpret a trigonometric model. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  45. 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 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

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