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Graph Trigonometric Functions with Phase and Vertical Shifts

Learn how to graph trigonometric functions with phase and vertical shifts. Find the amplitude, period, phase shift, and vertical shift of the functions and then plot their graphs. Practice different examples to understand the concept.

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Graph Trigonometric Functions with Phase and Vertical Shifts

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 12–7) CCSS Then/Now New Vocabulary Key Concept: Phase Shift Example 1: Graph Phase Shift Key Concept: Vertical Shift Example 2: Graph Vertical Translations Concept Summary: Graph Trigonometric Functions Example 3: Graph Transformations Example 4: Real-World Example: Represent Periodic Functions Lesson Menu

  3. A.1 B. C.3 D.9 1 __ 3 Find the amplitude of y = 3 cos . Then graph the function. 5-Minute Check 1

  4. A.1 B. C.3 D.9 1 __ 3 Find the amplitude of y = 3 cos . Then graph the function. 5-Minute Check 1

  5. A. B. C. D.3 Find the period of y = sin 3. Thengraph the function. 5-Minute Check 2

  6. A. B. C. D.3 Find the period of y = sin 3. Thengraph the function. 5-Minute Check 2

  7. A.y = tan 40° B.y = 40 tan C.y = tan D.y = tan 40° + Write the equation of a tangent function with a period of 40°. 5-Minute Check 3

  8. A.y = tan 40° B.y = 40 tan C.y = tan D.y = tan 40° + Write the equation of a tangent function with a period of 40°. 5-Minute Check 3

  9. A population of insects can be modeled by the equation y = 12,000 + 8000 sin . What is the greatest number of insects you will find? x __ 6 A. 12,000 insects B. 20,000 insects C. 24,000 insects D. 40,000 insects 5-Minute Check 4

  10. A population of insects can be modeled by the equation y = 12,000 + 8000 sin . What is the greatest number of insects you will find? x __ 6 A. 12,000 insects B. 20,000 insects C. 24,000 insects D. 40,000 insects 5-Minute Check 4

  11. Content Standards F.IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Mathematical Practices 4 Model with mathematics. CCSS

  12. You translated exponential functions. • Graph horizontal translations of trigonometric graphs and find phase shifts. • Graph vertical translations of trigonometric graphs. Then/Now

  13. phase shift • vertical shift • midline Vocabulary

  14. Concept

  15. Graph Phase Shift State the amplitude, period, and phase shift for the function y = 2 sin (θ + 20°). Then graph the function. Since a = 2 and b = 1, the amplitude and period of the function are the same as y = 2 cos . However h = –20, so the phase shift is –20. Because h < 0, the parent graph is shifted to the left. To graph y = 2 sin ( + 20), consider the graph of y = 2 sin . Graph this function and then shift the graph 20 to the left. Example 1

  16. Graph Phase Shift Answer: Example 1

  17. Graph Phase Shift Answer: amplitude: 2; period: 360°; phase shift: 20° left Example 1

  18. A. B.amplitude: 3; period: 1080°; phase shift: 30° C.amplitude: 3; period: 360°; phase shift: –30° D. State the amplitude, period, and phase shift for the function y = 3 sin (θ + 30°). Example 1

  19. A. B.amplitude: 3; period: 1080°; phase shift: 30° C.amplitude: 3; period: 360°; phase shift: –30° D. State the amplitude, period, and phase shift for the function y = 3 sin (θ + 30°). Example 1

  20. Concept

  21. State the amplitude, vertical shift, and equation of the midline,for .Then graph the function. Graph Vertical Translations amplitude: period: vertical shift: k = 3, so the midline is the graph of y = 3. Example 2

  22. Since the amplitude of the function is , draw dashedlines parallel to the midline that are unit above andbelow the midline. Graph Vertical Translations Then draw the cosine curve. Answer: Example 2

  23. Since the amplitude of the function is , draw dashedlines parallel to the midline that are unit above andbelow the midline. Answer: vertical shift: +3; midline: y = 3;amplitude:period: 2π Graph Vertical Translations Then draw the cosine curve. Example 2

  24. A.amplitude: 3; period: 2π; vertical shift: –2; midline: y = –2 • B.amplitude: 3; period: 2π; vertical shift: 2; midline: y = 2 • vertical shift: –2; midline: y = –2 • vertical shift: 2; midline: y = –2 ; ; State the amplitude, period, vertical shift and equation of the midline for y = 3 sin θ – 2. Example 2

  25. A.amplitude: 3; period: 2π; vertical shift: –2; midline: y = –2 • B.amplitude: 3; period: 2π; vertical shift: 2; midline: y = 2 • vertical shift: –2; midline: y = –2 • vertical shift: 2; midline: y = –2 ; ; State the amplitude, period, vertical shift and equation of the midline for y = 3 sin θ – 2. Example 2

  26. Concept

  27. State the amplitude, period, phase shift, and vertical shift for Then graph the function. b = 2, so the period is or π. h = so the phase shift is right. Graph Transformations The function is written in the form y = a cos [b(θ – h) + k].Identify the values of a, b, and k. a = 3, so the amplitude is |3| or 3. k = 4, so the vertical shift is 4 units up. Example 3

  28. Graph Transformations Graph the function. Step 1The vertical shift is 4. Graph the midline y = 4. Example 3

  29. Graph Transformations Step 2The amplitude is 3. Draw dashed lines 3 units above and below the midline at y = 1 and y = 7. Example 3

  30. Graph Transformations Step 3The period is π, so the graph is compressed. Graph y = 3 sin 2θ + 4 using the midline as a reference. Example 3

  31. Step 4Shift the graph to the right. Graph Transformations Answer: Example 3

  32. Step 4Shift the graph to the right. Graph Transformations Answer: Example 3

  33. State the amplitude, period, phase shift and vertical shift for . ; vertical shift: 2 A. B. C. D. ; vertical shift: –2 ; vertical shift: –2 ; vertical shift: Example 3

  34. State the amplitude, period, phase shift and vertical shift for . ; vertical shift: 2 A. B. C. D. ; vertical shift: –2 ; vertical shift: –2 ; vertical shift: Example 3

  35. Represent Periodic Functions WAVE POOL The height of water in a wave pool oscillates between a maximum of 10 feet and a minimum of 6 feet. The wave generator pumps 3 waves per minute. Write a sine function that represents the height of the water at time t seconds. Then graph the function. Step 1 Write the equation for the midline, and determine the vertical shift. The midline lies halfway between the maximum and the minimum. Since the midline is y = 8, the vertical shift is k = 8. Example 4

  36. Represent Periodic Functions Step 2 Find the amplitude. |a| = |10 – 8| or 2 Find the difference between the midline value and the maximum value. Step 3 Find the period. Since there are 3 waves per minute, there is one wave every 20 seconds. So, the period is 20 seconds. Example 4

  37. | b | = Represent Periodic Functions Solve for |b|. Simplify. Step 4 Write an equation for the function. h = a sin b (t – 0) + 8 Write the equation for sine relating height h and time t. Example 4

  38. Represent Periodic Functions Simplify. Answer: Example 4

  39. Represent Periodic Functions Simplify. Answer: Example 4

  40. A. B. C. D. WAVE POOL The height of water oscillates between a maximum of 8 feet and a minimum of 4 feet, and the wave generator pumps 5 waves per minute. Determine the correct sine function that represents the height of the water at time t seconds. Example 4

  41. A. B. C. D. WAVE POOL The height of water oscillates between a maximum of 8 feet and a minimum of 4 feet, and the wave generator pumps 5 waves per minute. Determine the correct sine function that represents the height of the water at time t seconds. Example 4

  42. End of the Lesson

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