Download presentation

Presentation is loading. Please wait.

Published byImogene Harrington Modified over 2 years ago

1
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 5.5 Graphs of Other Trigonometric Functions Objectives: Understand the graph of y = sin x. Graph variations of y = sin x. Understand the graph of y = cos x. Graph variations of y = cos x. Use vertical shifts of sine and cosine curves. Model periodic behavior. Dr.Hayk Melikyan Department of Mathematics and CS melikyan@nccu.edu

2
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 The Graph of y = sinx

3
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Some Properties of the Graph of y = sinx The domain is The range is [ – 1, 1]. The period is The function is an odd function:

4
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Graphing Variations of y = sinx

5
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example: Graphing a Variation of y = sinx Determine the amplitude of Then graph and for Step 1 Identify the amplitude and the period. The equation is of the form y = Asinx with A = 3. Thus, the amplitude is This means that the maximum value of y is 3 and the minimum value of y is – 3. The period is

6
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example: Graphing a Variation of y = sinx (continued) Determine the amplitude of Then graph and for Step 2 Find the values of x for the five key points. To generate x-values for each of the five key points, we begin by dividing the period, by 4. The cycle begins at x 1 = 0. We add quarter periods to generate x-values for each of the key points.

7
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Graphing a Variation of y = sinx (continued) Step 3 Find the values of y for the five key points.

8
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Example: Graphing a Variation of y = sinx (continued) Determine the amplitude of Then graph and for Step 4 Connect the five key points with a smooth curve and graph one complete cycle of the given function. amplitude = 3 period

9
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Amplitudes and Periods

10
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Example: Graphing a Function of the Form y = AsinBx Determine the amplitude and period of Then graph the function for Step 1 Identify the amplitude and the period. The equation is of the form y = A sinBx with A = 2 and The maximum value of y is 2, the minimum value of y is –2. The period of tells us that the graph completes one cycle from 0 to amplitude: period:

11
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Example: Graphing a Function of the Form y = AsinBx (continued) Step 2 Find the values of x for the five key points. To generate x-values for each of the five key points, we begin by dividing the period, by 4. The cycle begins at x 1 = 0. We add quarter periods to generate x-values for each of the key points.

12
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Graphing a Function of the Form y = AsinBx (continued) Step 3 Find the values of y for the five key points.

13
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Graphing a Function of the Form y = AsinBx (continued) Determine the amplitude and period of Then graph the function for Step 4 Connect the five key points with a smooth curve and graph one complete cycle of the given function. amplitude = 2 period

14
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Graphing a Function of the Form y = AsinBx (continued) Determine the amplitude and period of Then graph the function for Step 5 Extend the graph in step 4 to the left or right as desired. We will extend the graph to include the interval amplitude = 2 period

15
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 The Graph of y = Asin(Bx – C)

16
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Example: Graphing a Function of the Form y = Asin(Bx – C) Determine the amplitude, period, and phase shift of Then graph one period of the function. Step 1 Identify the amplitude, the period, and the phase shift. amplitude: period: phase shift:

17
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Example: Graphing a Function of the Form y = Asin(Bx – C) (continued) Step 2 Find the values of x for the five key points.

18
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 Example: Graphing a Function of the Form y = Asin(Bx – C) (continued) Step 3 Find the values of y for the five key points.

19
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 Example: Graphing a Function of the Form y = Asin(Bx – C) (continued) Step 3 (cont) Find the values of y for the five key points.

20
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 Step 4 Connect the five key points with a smooth curve and graph one complete cycle of the function Example: Graphing a Function of the Form y = Asin(Bx – C) (continued) amplitude = 3 period phase shift

21
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21 The graph of y = cosx

22
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22 Some Properties of the Graph of y = cosx The domain is The range is [ – 1, 1]. The period is The function is an even function:

23
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23 Sinusoidal Graphs The graph of is the graph of with a phase shift of The graphs of sine functions and cosine functions are called sinusoidal graphs.

24
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 24 Graphing Variations of y = cosx

25
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 25 Example: Graphing a Function of the Form y = AcosBx Determine the amplitude and period of Then graph the function for Step 1 Identify the amplitude and the period. amplitude: period:

26
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 26 Example: Graphing a Function of the Form y = AcosBx (continued) Determine the amplitude and period of Then graph the function for Step 2 Find the values of x for the five key points.

27
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 27 Example: Graphing a Function of the Form y = AcosBx (continued) Step 3 Find the values of y for the five key points.

28
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 28 Example: Graphing a Function of the Form y = AcosBx (continued) Step 3 Find the values of y for the five key points.

29
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 29 Example: Graphing a Function of the Form y = AcosBx (continued) Determine the amplitude and period of Then graph the function for Step 4 Connect the five key points with a smooth curve and graph one complete cycle of the given function. amplitude = 4 period = 2

30
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 30 Example: Graphing a Function of the Form y = AcosBx (continued) Determine the amplitude and period of Then graph the function for Step 5 Extend the graph to the left or right as desired. amplitude = 4 period = 2

31
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 31 The Graph of y = Acos(Bx – C)

32
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 32 Example: Graphing a Function of the Form y = Acos(Bx – C) Determine the amplitude, period, and phase shift of Then graph one period of the function. Step 1 Identify the amplitude, the period, and the phase shift. amplitude: period: phase shift:

33
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 33 Example: Graphing a Function of the Form y = Acos(Bx – C) (continued) Determine the amplitude, period, and phase shift of Then graph one period of the function. Step 2 Find the x-values for the five key points.

34
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 34 Example: Graphing a Function of the Form y = Acos(Bx – C) (continued) Step 3 Find the values of y for the five key points.

35
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 35 Example: Graphing a Function of the Form y = Acos(Bx – C) (continued) Step 3 (cont) Find the values of y for the five key points.

36
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 36 Example: Graphing a Function of the Form y = Acos(Bx – C) (continued) Step 3 (cont) Find the values of y for the five key points.

37
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 37 Example: Graphing a Function of the Form y = Acos(Bx – C) (continued) Determine the amplitude, period, and phase shift of Then graph one period of the function. Step 4 Connect the five key points with a smooth curve and graph one complete cycle of the given function. amplitude period

38
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 38 Vertical Shifts of Sinusoidal Graphs For sinusoidal graphs of the form and the constant D causes a vertical shift in the graph. These vertical shifts result in sinusoidal graphs oscillating about the horizontal line y = D rather than about the x- axis. The maximum value of y is The minimum value of y is

39
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 39 Example: A Vertical Shift Graph one period of the function Step 1 Identify the amplitude, period, phase shift, and vertical shift. amplitude: period: phase shift: vertical shift: one unit upward

40
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 40 Example: A Vertical Shift Graph one period of the function Step 2 Find the values of x for the five key points.

41
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 41 Example: A Vertical Shift Step 3 Find the values of y for the five key points.

42
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 42 Example: A Vertical Shift Step 3 (cont) Find the values of y for the five key points.

43
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 43 Example: A Vertical Shift Graph one period of the function Step 4 Connect the five key points with a smooth curve and graph one complete cycle of the given function.

44
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 44 Example: Modeling Periodic Behavior A region that is 30° north of the Equator averages a minimum of 10 hours of daylight in December. Hours of daylight are at a maximum of 14 hours in June. Let x represent the month of the year, with 1 for January, 2 for February, 3 for March, and 12 for December. If y represents the number of hours of daylight in month x, use a sine function of the form y = Asin(Bx – C) + D to model the hours of daylight. Because the hours of daylight range from a minimum of 10 to a maximum of 14, the curve oscillates about the middle value, 12 hours. Thus, D = 12.

45
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 45 Example: Modeling Periodic Behavior (continued) A region that is 30° north of the Equator averages a minimum of 10 hours of daylight in December. Hours of daylight are at a maximum of 14 hours in June. Let x represent the month of the year, with 1 for January, 2 for February, 3 for March, and 12 for December. If y represents the number of hours of daylight in month x, use a sine function of the form y = Asin(Bx – C) + D to model the hours of daylight. The maximum number of hours of daylight is 14, which is 2 hours more than 12 hours. Thus, A, the amplitude, is 2; A = 2.

46
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 46 Example: Modeling Periodic Behavior A region that is 30° north of the Equator averages a minimum of 10 hours of daylight in December. Hours of daylight are at a maximum of 14 hours in June. Let x represent the month of the year, with 1 for January, 2 for February, 3 for March, and 12 for December. If y represents the number of hours of daylight in month x, use a sine function of the form y = Asin(Bx – C) + D to model the hours of daylight. One complete cycle occurs over a period of 12 months.

47
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 47 Example: Modeling Periodic Behavior The starting point of the cycle is March, x = 3. The phase shift is

48
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 48 Example: Modeling Periodic Behavior A region that is 30° north of the Equator averages a minimum of 10 hours of daylight in December. Hours of daylight are at a maximum of 14 hours in June. Let x represent the month of the year, with 1 for January, 2 for February, 3 for March, and 12 for December. If y represents the number of hours of daylight in month x, use a sine function of the form y = Asin(Bx – C) + D to model the hours of daylight. The equation that models the hours of daylight is

Similar presentations

OK

9) P = π10) P = π/211) P = π/5; ± π/10 12) P = (2 π)/3; ± π/313) P = π/4; ± π/8 14) P = (3π 2 )/2; ±(3π 2 )/4 15) 16) 17) 18)

9) P = π10) P = π/211) P = π/5; ± π/10 12) P = (2 π)/3; ± π/313) P = π/4; ± π/8 14) P = (3π 2 )/2; ±(3π 2 )/4 15) 16) 17) 18)

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on diode family video Ppt on indian culture and heritage Ppt on save tigers in india downloads Ppt on rlc series circuit Ppt on c programming tutorial Ppt on statistics and probability notes Ppt on railway track maintenance Ppt on care of public property information Ppt on pi in maths what is the range Ppt on poet robert frost