2 1 Graphs of Sine and Cosine Functions

2. 1 Graphs of Sine and Cosine Functions Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 1

Learning 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. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 2

The Graph of y = sinx Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 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: Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 4

Graphing Variations of y = sinx 1. Identify the amplitude and the period. 2. Find the values of x for the five key points—the three x-intercepts, the maximum point, and the minimum point. Start with the value of x where the cycle begins and add quarter-periods—that is, —to find successive values of x. 3. Find the values of y for the five key points by evaluating the function at each value of x from step 2. 4. Connect the five key points with a smooth curve and graph one complete cycle of the given function. 5. Extend the graph in step 4 to the left or right as desired. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 5

Example 1: Graphing a Variation of y = sinx (1 of 4) Then graph Determine the amplitude of Step 1: Identify the amplitude and the period. The equation is of the form 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 Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 6

Example 1: Graphing a Variation of y = sinx (2 of 4) Determine the amplitude of Then graph 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. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 7

Example 1: Graphing a Variation of y = sinx (3 of 4) Step 3: Find the values of y for the five key points. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 8

Example 1: Graphing a Variation of y = sinx (4 of 4) Determine the amplitude of Then graph Step 4: Connect the five key points with a smooth curve and graph one complete cycle of the given function. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 9

Amplitudes and Periods Amplitudes and periods The graph of y = A sin Bx, B > 0, has Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 10

Example 2: Graphing a Function of the Form y = Asin. Bx (1 of 5) 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 sin. Bx 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 Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 11

Example 2: Graphing a Function of the Form y = Asin. Bx (2 of 5) 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. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 12

Example 2: Graphing a Function of the Form y = Asin. Bx (3 of 5) Step 3: Find the values of y for the five key points. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 13

Example 2: Graphing a Function of the Form y = Asin. Bx (4 of 5) 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. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 14

Example 2: Graphing a Function of the Form y = Asin. Bx (5 of 5) 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 Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 15

The Graph of y = Asin(Bx − C) The graph of is obtained by horizontally shifting the graph of y = A sin Bx so that the starting point of the circle is shifted from x = 0 to If the shift is to the right. If the shift is to the left. The number is called the phase shift. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 16

Example 3: Graphing a Function of the Form y = Asin(Bx − C) (1 of 5) 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. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 17

Example 3: Graphing a Function of the Form y = Asin(Bx − C) (2 of 5) Step 2: Find the values of x for the five key points. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 18

Example 3: Graphing a Function of the Form y = Asin(Bx − C) (3 of 5) Step 3: Find the values of y for the five key points. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 19

Example 3: Graphing a Function of the Form y = Asin(Bx − C) (4 of 5) Step 3: Find the values of y for the five key points. [continued] Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 20

Example 3: Graphing a Function of the Form y = Asin(Bx − C) (5 of 5) Step 4: Connect the five key points with a smooth curve and graph one complete cycle of the function Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 21

The Graph of y = cosx Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 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: Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 23

Sinusoidal Graphs The graphs of sine functions and cosine functions are called sinusoidal graphs. The graph of is the graph of with a phase shift of Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 24

Graphing Variations of y = cosx The Graph of y = A cos Bx The graph of y = A cos Bx, B > 0, has Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 25

Example 4: Graphing a Function of the Form y = Acos. Bx (1 of 6) Determine the amplitude and period of Then graph the function for Step 1: Identify the amplitude and the period. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 26

Example 4: Graphing a Function of the Form y = Acos. Bx (2 of 6) Determine the amplitude and period of Then graph the function for Step 2: Find the values of x for the five key points. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 27

Example 4: Graphing a Function of the Form y = Acos. Bx (3 of 6) Step 3: Find the values of y for the five key points. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 28

Example 4: Graphing a Function of the Form y = Acos. Bx (4 of 6) Step 3: Find the values of y for the five key points. [continued] Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 29

Example 4: Graphing a Function of the Form y = Acos. Bx (5 of 6) 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. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 30

Example 4: Graphing a Function of the Form y = Acos. Bx (6 of 6) Determine the amplitude and period of Then graph the function for Step 5: Extend the graph to the left or right as desired. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 31

The Graph of y = Acos(Bx − C) The graph of is obtained by horizontally shifting the graph of y = A cos Bx so that the starting point of the circle is shifted from x = 0 to If the shift is to the right. If the shift is to the left. The number is called the phase shift. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 32

Example 5: Graphing a Function of the Form y = Acos(Bx − C) (1 of 6) 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. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 33

Example 5: Graphing a Function of the Form y = Acos(Bx − C) (2 of 6) 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. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 34

Example 5: Graphing a Function of the Form y = Acos(Bx − C) (3 of 6) Step 3: Find the values of y for the five key points. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 35

Example 5: Graphing a Function of the Form y = Acos(Bx − C) (4 of 6) Step 3: Find the values of y for the five key points. [continued] Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 36

Example 5: Graphing a Function of the Form y = Acos(Bx − C) (5 of 6) Step 3: Find the values of y for the five key points. [continued] Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 37

Example 5: Graphing a Function of the Form y = Acos(Bx − C) (6 of 6) 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. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 38

Vertical Shifts of Sinusoidal Graphs For sinusoidal graphs of the form 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 Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 39

Example 6: A Vertical Shift (1 of 5) Graph one period of the function Step 1: Identify the amplitude, period, phase shift, and vertical shift. phase shift: 0 vertical shift: one unit upward Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 40

Example 6: A Vertical Shift (2 of 5) Graph one period of the function Step 2: Find the values of x for the five key points. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 41

Example 6: A Vertical Shift (3 of 5) Step 3: Find the values of y for the five key points. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 42

Example 6: A Vertical Shift (4 of 5) Step 3: Find the values of y for the five key points. [continued] Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 43

Example 6: A Vertical Shift (5 of 5) 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. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 44

Graphing the Sum of Two Trigonometric Functions We can graph the sum of two (or more) trigonometric functions using a method that involves adding y-coordinates. For example, to graph y = sin x + cos x, we begin by graphing in the same rectangular coordinate system. Then we select several values of x at which we add the y-coordinates of sin x and cos x. Plotting the points (x, sin x + cos x) and joining them with a smooth curve gives the graph of the desired function. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 45

Example 7: Graphing the Sum of Two Trigonometric Functions (1 of 3) Use the method of adding y-coordinates to graph Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 46

Example 7: Graphing the Sum of Two Trigonometric Functions (2 of 3) Use the method of adding y-coordinates to graph Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 47

Example 7: Graphing the Sum of Two Trigonometric Functions (3 of 3) Use the method of adding y-coordinates to graph Plot the nine points from the table and connect with a smooth curve. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 48

Example 8: Modeling Periodic Behavior (1 of 5) 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 to model the hours function of the form 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. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 49

Example 8: Modeling Periodic Behavior (2 of 5) 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 to model the hours function of the form 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. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 50

Example 8: Modeling Periodic Behavior (3 of 5) 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 to model the hours function of the form of daylight. One complete cycle occurs over a period of 12 months. Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 51

Example 8: Modeling Periodic Behavior (4 of 5) The starting point of the cycle is March, x = 3. The phase shift is Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 52

Example 8: Modeling Periodic Behavior (5 of 5) 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 to model the hours of daylight. The equation that models the hours of daylight is Copyright © 2018, 2014 Pearson Education, Inc. All Rights Reserved Slide - 53