6.4

Precalculus: Enhanced with Graphing Utilities Seventh Edition Chapter 6 Trigonometric Functions Copyright © 2017, 2013, 2009 Education, Inc. All Rights Reserved Section 6.4 Graphs of the Sine and Cosine Functions Objectives Graph Functions of the Form Using Transformations Graph Functions of the Form Using Transformations Determine the Amplitude and Period of Sinusoidal Functions Graph Sinusoidal Functions Using Key Points Find an Equation for a Sinusoidal Graph Properties 1 Graph of Sine Function (1 of 2) Graph of Sine Function (2 of 2) Properties 2 Properties of the Sine Function The domain is the set of all real numbers. The range consists of all real numbers from ?1 to 1, inclusive. The sine function is an odd function, as the symmetry of the graph with respect to the origin indicates. The sine function is periodic, with period The x-intercepts are the y-intercept is 0. The maximum value is 1 and occurs at the minimum value is ?1 and occurs at Objective 1 Graph Functions of the Form y = A Sine of Left Parenthesis Omega x Right Parenthesis Using Transformations Example 1: Graphing Functions of the Form y = A Sine of Left Parenthesis Omega x Right Parenthesis Using Transformations using transformations. Use the graph to determine the domain and the range of the function. Solution: The domain of is the set of all real numbers, or Example 2: Graphing Functions of the Form y = A Sine of Left Parenthesis Omega x Right Parenthesis Using Transformations (1 of 2) using transformations. Use the graph to determine the domain and the range of the function. Solution: The figures illustrate the steps. Example 2: Graphing Functions of the Form y = A Sine of Left Parenthesis Omega x Right Parenthesis Using Transformations (2 of 2) The figures illustrate the steps. The domain of is the set of all real numbers, or Graph of Cosine Function (1 of 2) Graph of Cosine Function (2 of 2) Properties 3 Properties of the Cosine Function The domain is the set of all real numbers. The range consists of all real numbers from ?1 to 1, inclusive. The cosine function is an even function, as the symmetry of the graph with respect to the y-axis indicates. The cosine function is periodic, with period The x-intercepts are the y-intercept is 1. The maximum value is 1 and occurs at the minimum value is ?1 and occurs at Objective 2 Graph Functions of the Form y = A Cosine of Left Parenthesis Omega x Right Parenthesis Using Transformations Example 3: Graphing Functions of the Form y = A Cosine of Left Parenthesis Omega x Right Parenthesis Using Transformations (1 of 4) using transformations. Use the graph to determine the domain and the range of the function. Example 3: Graphing Functions of the Form y = A Cosine of Left Parenthesis Omega x Right Parenthesis Using Transformations (2 of 4) Solution: Example 3: Graphing Functions of the Form y = A Cosine of Left Parenthesis Omega x Right Parenthesis Using Transformations (3 of 4) Example 3: Graphing Functions of the Form y = A Cosine of Left Parenthesis Omega x Right Parenthesis Using Transformations (4 of 4) The domain of is the set of all real numbers, or The range is Notice that the period of the function because the compression of the original period by the original factor of Sinusoidal Graphs Objective 3 Determine the Amplitude and Period of Sinusoidal Functions Theorem the amplitude and period of are given by Example 4: Finding the Amplitude and Period of a Sinusoidal Function Determine the amplitude and period of Solution: Comparing note that A = 5 and Objective 4 Graph Sinusoidal Functions Using Key Points Example 5: Graphing a Sinusoidal Function Using Key Points (1 of 4) using key points. Solution: Step 1: Determine the amplitude and period of the sinusoidal function. Comparing note that A = 2 and so the amplitude is A = 2 and the period is Because the amplitude is 2, the graph of will lie between ?2 and 2 on the y-axis. Because the period is one cycle will begin at x = 0 and end at Example 5: Graphing a Sinusoidal Function Using Key Points (2 of 4) Step 2: Divide the interval into four subintervals of the same length. Divide the interval into four subintervals, each of length The endpoints of the subintervals are These values represent the x-coordinates of the five key points on the graph. Example 5: Graphing a Sinusoidal Function Using Key Points (3 of 4) Step 3: Use the endpoints of the subintervals from Step 2 to obtain five key points on the graph. To obtain the y-coordinates of the five key points of multiply the y-coordinates of the five key points for The five key points are Example 5: Graphing a Sinusoidal Function Using Key Points (4 of 4) Step 4: Plot the five key points and draw a sinusoidal graph to obtain the graph of one cycle. Extend the graph in each direction to make it complete. Summary Steps for Graphing a Sinusoidal Function of the Form Using Key Points Step 1: Determine the amplitude and period of the sinusoidal function. Step 2: Divide the interval into four subintervals of the same length. Step 3: Use the endpoints of these subintervals to obtain five key points on the graph. Step 4: Plot the five key points, and draw a sinusoidal graph to obtain the graph of one cycle. Extend the graph in each direction to make it complete. Example 6: Graphing a Sinusoidal Function Using Key Points (1 of 5) using key points. Solution: Since the sine function is odd, use the equivalent form: Step 1: Comparing note that so the amplitude is and the period is Example 6: Graphing a Sinusoidal Function Using Key Points (2 of 5) Because the amplitude is 4, the graph of will lie between ?4 and 4 on the y-axis. Because the period is 8, one cycle will begin at x = 0 and end at x = 8. Step 2: Divide the interval  into four subintervals, each of   as follows: Example 6: Graphing a Sinusoidal Function Using Key Points (3 of 5) Step 3: Since multiply the y-coordinates of the five key points in the figure below by A = ?4. The five key points are Example 6: Graphing a Sinusoidal Function Using Key Points (4 of 5) Step 4: Plot the five key points and draw a sinusoidal graph to obtain the graph of one cycle. Extend the graph in each direction to make it complete. Example 6: Graphing a Sinusoidal Function Using Key Points (5 of 5) Plot the five key points and draw a sinusoidal graph to obtain the graph of one cycle. Extend the graph in each direction to make it complete. Show the graph of one period using a graphing utility. Example 7: Graphing a Sinusoidal Function Using Key Points (1 of 5) using key points. Use the graph to determine the domain and the range of Solution: Begin by graphing the function Comparing note that A = ?4 and The amplitude is and the period is The graph of will lie between ?4 and 4 on the y-axis. One cycle will begin at x = 0 and end at x = 2. Example 7: Graphing a Sinusoidal Function Using Key Points (2 of 5) Divide the interval into four subintervals, each of length The x-coordinates of the five key points are Since multiply the y-coordinates of the five key points of as shown by A = ?4. Example 7: Graphing a Sinusoidal Function Using Key Points (3 of 5) By doing this, the five key points on the graph of can be obtained. Example 7: Graphing a Sinusoidal Function Using Key Points (4 of 5) Plot these five points, and fill in the graph of the cosine function as shown. Extend the graph in each direction to obtain the graph of A vertical shift down 2 units gives the graph of as shown. Example 7: Graphing a Sinusoidal Function Using Key Points (5 of 5) The domain of is the set of all real numbers, or The range of Objective 5 Find an Equation for a Sinusoidal Graph Example 8: Finding an Equation for a Sinusoidal Graph (1 of 2) Find an equation for the graph shown. Example 8: Finding an Equation for a Sinusoidal Graph (2 of 2) Solution: The graph has the characteristics of a sine function. Do you see why? The y value of 0 occurs at x = 0. The maximum value, 2, occurs at x = 1. So the equation can be viewed as a sine function with A = 2 and period T = 4. The sine function whose graph is shown is Example 9: Finding an Equation for a Sinusoidal Graph (1 of 2) Find an equation for the graph shown. Example 9: Finding an Equation for a Sinusoidal Graph (2 of 2) Solution: The graph is sinusoidal, with amplitude The period is 8, so Since the graph passes through the origin, it is easier to view the equation as a sine function, but note that the graph is actually the reflection of a sine function about the x-axis (since the graph is decreasing near the origin). This requires that A = ?3. The sine function is