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# Class Note for MATH 1330 at UH

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This 11 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 15 views.

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Date Created: 02/06/15

Math 1330 Section 52 Graphs 0f the Sine and Cosine Functions In this section we will graph the basic sine function and the basic cosine function and then graph other sine and cosine functions using transformations Much of what we will do in graphing these problems will be the same as earlier graphing using transformations We ll start with the graph of the basic sine function f x sinx The domain of this function is oo 0 and the range is l 1 We typically graph just one complete period of the graph that is on the interval 0 27139 We ll make a table of values x sinx Then using these ordered pairs we can sketch a graph of the function Next we ll draw in a smooth curve Drawing all of these points is rather tedious We ll ask you to learn the shape of the graph and just graph five basic points the x and y intercepts and the maximum and the minimum Now we ll repeat the process for the basic cosine function fx cosx The domain of this function is 0 0 and the range is l 1 Again we typically graph just one complete period of the graph that is on the interval 0 27139 Here is the table ofvalues for fx cosx x cosx Now we ll graph these ordered pairs For the basic cosine graph you ll need to remember the basic shape and graph the x and y intercepts as well as the maximum and minimum points We have graphed just one period of both the sine and cosine functions However the graphs of these functions continue in both directions fx since AWAA VWVV fx 00506 M Wz W V Now we ll turn our attention to transformations of the basic sine and cosine functions These functions will be ofthe form fx A sinBx C D or gx AcosBx C D We can stretch or shrink sine and cosine functions both vertically and horizontally We can re ect them about the x axis the y axis or both axes and we can translate the graphs either vertically horizontally or both Next we ll see how the values for A B C andD affect the graph of the sine or cosine function For a periodic functionfwith maximum valueM and minimum value m we define the m For a bas1c s1ne or cos1ne functlon the max1mum 11 2 amplitude of the function to be value is l and the minimum value is 1 so the amplitude is 1 The amplitude of the graph will be lAl For functions ofthe form fx A sinx or fx Acosx the maximum and minimum values of the function will be multiplied byA If A gt 1 this will stretch the graph vertically If 0 ltA lt 1 this will shrink the graph vertically IfA lt 0 the graph will be a re ection about the x axis For a periodic function j the period of the graph is the length of the interval needed to draw one complete cycle of the graph For a basic sine or cosine function the period is 27f Sometimes the graph is stretched or shrunk horizontally This means that the length of the interval needed to draw one complete cycle of the graph can be more or less than 27 For functions ofthe form fx sinBx or fx cosBx you can find the period ofthe graph by computing 21 B The graph of a basic sine or cosine function can be shifted either vertically or horizontally A function of the form fx sinx D or fx cosx D will be shifted up D units ifD gt 0 and down D units ifD lt 0 A function ofthe form fx sinx C or fx cosx C will be shifted C units to the right ifC gt 0 and C units to the left ifC lt 0 A function of the form fx sinBx C or fx cosBx C will be shifted units to the right if gt 0 and to the left if lt 0 The number gis called the phase shift It can be helpful to identify the starting and ending points for one period of the graph of a function that has a phase shift To do this solve the equations Bx C 0 and Bx C 27 You will need to identify the transformations required to change a basic sine or cosine function to the desired one You must know the five key points on a basic sine function and the five key points on a basic cosine function Using the information about the amplitude re ections vertical and horizontal stretching or shrinking and vertical and horizontal translations you will be able to correctly plot the translated key points and sketch the desired function Example 1 State the period and amplitude of the periodic function Example 2 State the period and amplitude of the periodic function Example 3 Sketch over one period Example 4 Sketch over one period Example 5 Sketch over one period fx 4sinx fx 2 cosx fx sin j Example 6 Sketch over one period fx 3sin2x Example 7 Sketch over one period fx 2cos2mc Example 8 Sketch over one period fx sin2xl in in nn Example 9 Sketch over one period fx 2cos x Example 10 Sketch over one period fx 3 sin3x l Example 11 Sketch over one period fx 2cos 3 Example 12 Sketch over one period fx isin 2x l Example 13 Consider the graph Write an equation of the form f x AsinBx C D and an equation of the form f x AcosBx C thich could be used to represent the graph Note these answers are not unique Example 14 Consider the graph Write an equation of the form f x AsinBx C D and an equation of the form f x AcosBx C thich could be used to represent the graph Note these answers are not unique 7 3 Example 15 Consider the graph Write an equation of the form f x AsinBx C D and an equation of the form f x A cosBx C D which could be used to represent the graph Note these answers are not unique Example 16 Consider the graph Write an equation of the form f x AsinBx C D and an equation of the form f x A cosBx C D which could be used to represent the graph Note these answers are not unique

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