ࡱ>  /} n ] M Ebjbj  l[[[[[ooo8D o Risisisisj $/V:[Ċj:[[isis[O^[is[isis $o^e0R:[t:: !: LESSON 8 THE GRAPHS OF THE TRIGONOMETRIC FUNCTIONS Topics in this lesson: 1.  HYPERLINK \l "Topic1" SINE GRAPHS 2.  HYPERLINK \l "Topic2" COSINE GRAPHS 3.  HYPERLINK \l "Topic3" SINE AND COSINE GRAPHS WITH PHASE SHIFTS 4.  HYPERLINK \l "Topic4" SECANT AND COSECANT GRAPHS 5.  HYPERLINK \l "Topic5" TANGENT GRAPHS 6.  HYPERLINK \l "Topic6" COTANGENT GRAPHS 1. SINE GRAPHS Example Use the Unit Circle to HYPERLINK "http://math.utoledo.edu/~janders/1330/Lectures/Lesson8/Sine2Ccc.gif"graph two cycles of the function  EMBED Equation.3  on the interval  EMBED Equation.3 . Example Use the Unit Circle to HYPERLINK "http://math.utoledo.edu/~janders/1330/Lectures/Lesson8/Sine2Cc.gif"graph two cycles of the function  EMBED Equation.3  on the interval  EMBED Equation.3 . Definition The amplitude of a trigonometric function is one-half of the difference between the maximum value of the function and the minimum value of the function if the function has both of these values. NOTE: The maximum value of a function is the largest y-coordinate on the graph of the function and the minimum value of a function is the smallest y-coordinate on the graph of the function if the graph has both of these values. The sine and cosine functions will have an amplitude. However, the tangent, cotangent, secant, and cosecant functions do not have an amplitude because these functions do not have a maximum value nor a minimum value. Definition The period of a trigonometric function is the distance needed to complete one cycle of the graph of the function. All the trigonometric functions have a period. For the function  EMBED Equation.3 , the amplitude of the function is 1 and the period is  EMBED Equation.3 . Given the function  EMBED Equation.3 , the amplitude of this function is  EMBED Equation.3  and the period is  EMBED Equation.3 . Theorem The sine function is an odd function. That is,  EMBED Equation.3  for all  EMBED Equation.3  in the domain of the function. NOTE: The domain of the sine function is all real numbers. Examples Sketch two cycles of the graph of the following functions. Label the numbers on the x- and y-axes. 1.  EMBED Equation.3  Amplitude =  EMBED Equation.3  = 5 Period =  EMBED Equation.3  =  EMBED Equation.3    EMBED Equation.3  period =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  y 5 x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  x  EMBED Equation.3  NOTE: The first cycle begins at 0. We do not need to label that number. Since the period is  EMBED Equation.3 , the first cycle ends at  EMBED Equation.3 , which is obtained by  EMBED Equation.3 . That is, we add the period of  EMBED Equation.3  to the starting point of 0. The second cycle ends at  EMBED Equation.3 , which is obtained by  EMBED Equation.3 . That is, we add the period of  EMBED Equation.3  to the starting point of  EMBED Equation.3 . Now, the rest of the numbers on the x-axis were obtained in the following manner: The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the starting point of 0. The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . We can check the  EMBED Equation.3  by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . Or, you can obtain the  EMBED Equation.3  by adding the period of  EMBED Equation.3  to the previous  EMBED Equation.3  in the first cycle. Thus,  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . Or, you can obtain the  EMBED Equation.3  by adding the period of  EMBED Equation.3  to the previous  EMBED Equation.3  in the first cycle. Thus,  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . Or, you can obtain the  EMBED Equation.3  by adding the period of  EMBED Equation.3  to the previous  EMBED Equation.3  in the first cycle. Thus,  EMBED Equation.3 . We can check the  EMBED Equation.3  by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth, of the period to the next starting point of  EMBED Equation.3 . The HYPERLINK "http://math.utoledo.edu/~janders/1330/Lectures/Lesson8/5sin3x.gif"graph of two cycles of  EMBED Equation.3  in blue compared with the graph of two cycles of  EMBED Equation.3  in red. 2.  EMBED Equation.3  Amplitude =  EMBED Equation.3  Period =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  period =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3   y  EMBED Equation.3  x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  x  EMBED Equation.3  NOTE: The first cycle begins at 0. We do not need to label that number. Since the period is  EMBED Equation.3 , the first cycle ends at  EMBED Equation.3 , which is obtained by  EMBED Equation.3 . That is, we add the period of  EMBED Equation.3  to the starting point of 0. The second cycle ends at  EMBED Equation.3 , which is obtained by  EMBED Equation.3 . That is, we add the period of  EMBED Equation.3  to the starting point of  EMBED Equation.3 . Now, the rest of the numbers on the x-axis were obtained in the following manner: The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the starting point of 0. The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . We can check the  EMBED Equation.3  by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . Or, you can obtain the  EMBED Equation.3  by adding the period of  EMBED Equation.3  to the previous  EMBED Equation.3  in the first cycle. Thus,  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . Or, you can obtain the  EMBED Equation.3  by adding the period of  EMBED Equation.3  to the previous  EMBED Equation.3  in the first cycle. Thus,  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . Or, you can obtain the  EMBED Equation.3  by adding the period of  EMBED Equation.3  to the previous  EMBED Equation.3  in the first cycle. Thus,  EMBED Equation.3 . We can check the  EMBED Equation.3  by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth, of the period to the next starting point of  EMBED Equation.3 . The HYPERLINK "http://math.utoledo.edu/~janders/1330/Lectures/Lesson8/Sqrt2sin(xdb6).gif"graph of two cycles of  EMBED Equation.3  in blue compared with the graph of two cycles of  EMBED Equation.3  in red. 3.  EMBED Equation.3  NOTE: Since the sine function is being multiplied by a negative  EMBED Equation.3 , then the graph will be inverted. Thus, we will need to draw two inverted sine cycles. Amplitude =  EMBED Equation.3  =  EMBED Equation.3  Period =  EMBED Equation.3  = 1  EMBED Equation.3  period =  EMBED Equation.3  =  EMBED Equation.3   y  EMBED Equation.3  x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  1  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  2 x - EMBED Equation.3  NOTE: The first cycle begins at 0. We do not need to label that number. Since the period is 1, the first cycle ends at 1, which is obtained by  EMBED Equation.3 . That is, we add the period of 1 to the starting point of 0. The second cycle ends at 2, which is obtained by  EMBED Equation.3 . That is, we add the period of 1 to the starting point of 1. Now, the rest of the numbers on the x-axis were obtained in the following manner: The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the starting point of 0. The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one- fourth of the period, to the next starting point of  EMBED Equation.3 . We can check the 1 by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . Or, you can obtain the  EMBED Equation.3  by adding the period of 1 to the previous  EMBED Equation.3  in the first cycle. Thus,  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . Or, you can obtain the  EMBED Equation.3  by adding the period of 1 to the previous  EMBED Equation.3  in the first cycle. Thus,  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . Or, you can obtain the  EMBED Equation.3  by adding the period of 1 to the previous  EMBED Equation.3  in the first cycle. Thus,  EMBED Equation.3 . We can check the 2 by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth, of the period to the next starting point of  EMBED Equation.3 . The HYPERLINK "http://math.utoledo.edu/~janders/1330/Lectures/Lesson8/Neg4db7sin2Pix.gif"graph of two cycles of  EMBED Equation.3  in blue compared with the graph of two cycles of  EMBED Equation.3  in red. 4.  EMBED Equation.3  NOTE: Since the sine function is an odd function, then  EMBED Equation.3  =  EMBED Equation.3 . Thus, we have that  EMBED Equation.3  =  EMBED Equation.3  Since the sine function is being multiplied by a negative 8, then the graph will be inverted. Thus, we will need to draw two inverted sine cycles. Amplitude = 8 Period =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3    EMBED Equation.3  period =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  y 8 x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  x  EMBED Equation.3  NOTE: The first cycle begins at 0. We do not need to label that number. Since the period is  EMBED Equation.3 , the first cycle ends at  EMBED Equation.3 , which is obtained by  EMBED Equation.3 . That is, we add the period of  EMBED Equation.3  to the starting point of 0. The second cycle ends at  EMBED Equation.3 , which is obtained by  EMBED Equation.3 . That is, we add the period of  EMBED Equation.3  to the starting point of  EMBED Equation.3 . Now, the rest of the numbers on the x-axis were obtained in the following manner: The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the starting point of 0. The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . We can check the  EMBED Equation.3  by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . Or, you can obtain the  EMBED Equation.3  by adding the period of  EMBED Equation.3  to the previous  EMBED Equation.3  in the first cycle. Thus,  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . Or, you can obtain the  EMBED Equation.3  by adding the period of  EMBED Equation.3  to the previous  EMBED Equation.3  in the first cycle. Thus,  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . Or, you can obtain the  EMBED Equation.3  by adding the period of  EMBED Equation.3  to the previous  EMBED Equation.3  in the first cycle. Thus,  EMBED Equation.3 . We can check the  EMBED Equation.3  by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth, of the period to the next starting point of  EMBED Equation.3 . The HYPERLINK "http://math.utoledo.edu/~janders/1330/Lectures/Lesson8/8sin(Neg7xdb3).gif"graph of two cycles of  EMBED Equation.3  in blue compared with the graph of two cycles of  EMBED Equation.3  in red.  HYPERLINK \l "TopicsList" Back to Topics List 2. COSINE GRAPHS Example Use the Unit Circle to HYPERLINK "http://math.utoledo.edu/~janders/1330/Lectures/Lesson8/Cosine2Ccc.gif"graph two cycles of the function  EMBED Equation.3  on the interval  EMBED Equation.3 . Example Use the Unit Circle to HYPERLINK "http://math.utoledo.edu/~janders/1330/Lectures/Lesson8/Cosine2Cc.gif"graph two cycles of the function  EMBED Equation.3  on the interval  EMBED Equation.3 . For the function  EMBED Equation.3 , the amplitude of the function is 1 and the period is  EMBED Equation.3 . Given the function  EMBED Equation.3 , the amplitude of this function is  EMBED Equation.3  and the period is  EMBED Equation.3 . Theorem The cosine function is an even function. That is,  EMBED Equation.3  for all  EMBED Equation.3  in the domain of the function. NOTE: The domain of the cosine function is all real numbers. Examples Sketch two cycles of the graph of the following functions. 1.  EMBED Equation.3  Amplitude =  EMBED Equation.3  =  EMBED Equation.3  Period =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3    EMBED Equation.3  period =  EMBED Equation.3  =  EMBED Equation.3  y  EMBED Equation.3  x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  x  EMBED Equation.3  Since the period is  EMBED Equation.3 , the first cycle ends at  EMBED Equation.3  and the second cycle ends at  EMBED Equation.3 . The other numbers on the x-axis were obtained by the following:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Check:  EMBED Equation.3   EMBED Equation.3  OR  EMBED Equation.3   EMBED Equation.3  OR  EMBED Equation.3   EMBED Equation.3  OR  EMBED Equation.3  Check:  EMBED Equation.3  The HYPERLINK "http://math.utoledo.edu/~janders/1330/Lectures/Lesson8/Sqrt3cos8x.gif"graph of two cycles of  EMBED Equation.3  in blue compared with the graph of two cycles of  EMBED Equation.3  in red. 2.  EMBED Equation.3  Since the cosine function is being multiplied by a negative 4, then the graph will be inverted. Thus, we will need to draw two inverted cosine cycles. Amplitude = 4 Period =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3    EMBED Equation.3  period =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  y 4 x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  x  EMBED Equation.3  Since the period is  EMBED Equation.3 , the first cycle ends at  EMBED Equation.3  and the second cycle ends at  EMBED Equation.3 . The other numbers on the x-axis were obtained by the following:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Check:  EMBED Equation.3   EMBED Equation.3  OR  EMBED Equation.3   EMBED Equation.3  OR  EMBED Equation.3   EMBED Equation.3  OR  EMBED Equation.3  Check:  EMBED Equation.3  The HYPERLINK "http://math.utoledo.edu/~janders/1330/Lectures/Lesson8/Neg4cos(xdb5).gif"graph of two cycles of  EMBED Equation.3  in blue compared with the graph of two cycles of  EMBED Equation.3  in red. 3.  EMBED Equation.3  NOTE: Since the cosine function is an even function, then  EMBED Equation.3  =  EMBED Equation.3 . Thus, we have that  EMBED Equation.3  =  EMBED Equation.3  Amplitude =  EMBED Equation.3  Period =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3    EMBED Equation.3  period =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  y  EMBED Equation.3  x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  x - EMBED Equation.3  Since the period is  EMBED Equation.3 , the first cycle ends at  EMBED Equation.3  and the second cycle ends at  EMBED Equation.3 . The other numbers on the x-axis were obtained by the following:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Check:  EMBED Equation.3   EMBED Equation.3  OR  EMBED Equation.3   EMBED Equation.3  OR  EMBED Equation.3   EMBED Equation.3  OR  EMBED Equation.3  Check:  EMBED Equation.3  The HYPERLINK "http://math.utoledo.edu/~janders/1330/Lectures/Lesson8/1db2cos(Neg7Pixdb17).gif"graph of two cycles of  EMBED Equation.3  in blue compared with the graph of two cycles of  EMBED Equation.3  in red.  HYPERLINK \l "TopicsList" Back to Topics List 3. SINE AND COSINE GRAPHS WITH PHASE SHIFTS Definition A phase shift for a trigonometric function is a horizontal shift. That is, it is a shift with respect to the x-axis. Thus, the shift is either right or left. NOTE: In order to identify a horizontal shift, hence, a phase shift, the coefficient of the x variable must be 1. If the coefficient is not 1, then you will need to factor out the coefficient. Given the function  EMBED Equation.3 , we may write this function as  EMBED Equation.3  by factoring out b. The amplitude of this function is  EMBED Equation.3  and the period is  EMBED Equation.3 . The phase shift is  EMBED Equation.3  units to the right if  EMBED Equation.3  or is  EMBED Equation.3  units to the left if  EMBED Equation.3 . Similarly, given the function  EMBED Equation.3 , we may write this function as  EMBED Equation.3  by factoring out b. The amplitude of this function is  EMBED Equation.3  and the period is  EMBED Equation.3 . The phase shift is  EMBED Equation.3  units to the right if  EMBED Equation.3  or is  EMBED Equation.3  units to the left if  EMBED Equation.3 . Examples Sketch one cycle of the graph of the following functions. 1.  EMBED Equation.3  NOTE: Since the coefficient of the x variable is not 1, the phase shift is not  EMBED Equation.3  units to the right. Since the coefficient of the x variable is 2, then we will need to factor the 2 out in order to identify the phase shift.  EMBED Equation.3  Amplitude = 3 Period =  EMBED Equation.3  =  EMBED Equation.3  Phase Shift:  EMBED Equation.3  units to the right  y 3 x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  X  EMBED Equation.3  Since the phase shift is  EMBED Equation.3  units to the right, then the cycle starts at  EMBED Equation.3 . Since the period is  EMBED Equation.3 , then this cycle ends at  EMBED Equation.3  obtained by  EMBED Equation.3 . The other numbers on the x-axis were obtained by the following:  EMBED Equation.3  period =  EMBED Equation.3  =  EMBED Equation.3  The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the starting point of the cycle, which is  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . Check:  EMBED Equation.3  2.  EMBED Equation.3  NOTE: Since the coefficient of the x variable is not 1, the phase shift is not  EMBED Equation.3  units to the left. Since the coefficient of the x variable is  EMBED Equation.3 , then we will need to factor the  EMBED Equation.3  out in order to identify the phase shift.  EMBED Equation.3  NOTE: The  EMBED Equation.3  was obtained by  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3 . Since the sine function is an odd function, then  EMBED Equation.3  =  EMBED Equation.3  Since the sine function is being multiplied by a negative 7, then the graph will be inverted. Thus, we will need to draw an inverted sine cycle. Amplitude = 7 Period =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3   Phase Shift:  EMBED Equation.3  units to the right y 7 x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  x  EMBED Equation.3  Since the phase shift is  EMBED Equation.3  units to the right, then the cycle starts at  EMBED Equation.3 . Since the period is  EMBED Equation.3 , then this cycle ends at  EMBED Equation.3  obtained by  EMBED Equation.3 . The other numbers on the x-axis were obtained by the following:  EMBED Equation.3  period =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the starting point of the cycle, which is  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . Check:  EMBED Equation.3  3.  EMBED Equation.3  NOTE: Since the coefficient of the x variable is not 1, the phase shift is not  EMBED Equation.3  units to the left. Since the coefficient of the x variable is 3, then we will need to factor the 3 out in order to identify the phase shift.  EMBED Equation.3  Amplitude =  EMBED Equation.3  Period =  EMBED Equation.3  Phase Shift:  EMBED Equation.3  units to the left  y  EMBED Equation.3  x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  x  EMBED Equation.3  Since the phase shift is  EMBED Equation.3  units to the left, then the cycle starts at  EMBED Equation.3 . Since the period is  EMBED Equation.3 , then this cycle ends at  EMBED Equation.3  obtained by  EMBED Equation.3 . This cycle starts to the left of the y-axis and finishes to the right of the y-axis. We are sketching a graph that goes up, down, and up again. If we allow our sketch to cross the y-axis, our picture will probably contain misinformation about where the actual graph crosses the y-axis. We do not want our picture to have misinformation in it. Our sketches have not been drawn to scale, but all the numbers on the x- and y-axes have been correct. The sketch, that we draw, does not have to cross the y-axis. We know that where one cycle of the graph ends, another cycle begins. In this problem, our first cycle ends at  EMBED Equation.3 . Lets sketch the second cycle that begins at  EMBED Equation.3 . This second cycle will end at  EMBED Equation.3  obtained by  EMBED Equation.3 . That is, will add the period of  EMBED Equation.3  to the starting point of the second cycle, which is  EMBED Equation.3 . The other numbers on the x-axis were obtained by the following:  EMBED Equation.3  period =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the starting point of the second cycle, which is  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . Check:  EMBED Equation.3  4.  EMBED Equation.3  Since the coefficient of the x variable is  EMBED Equation.3 , then we will need to factor the  EMBED Equation.3  out in order to identify the phase shift.  EMBED Equation.3  NOTE: The  EMBED Equation.3  was obtained by  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3 . Since the sine function is an odd function, then  EMBED Equation.3  =  EMBED Equation.3  Amplitude =  EMBED Equation.3  Period =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3   Phase Shift:  EMBED Equation.3  units to the left y  EMBED Equation.3  x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  x - EMBED Equation.3  Since the phase shift is  EMBED Equation.3  units to the left, then the cycle starts at  EMBED Equation.3 . Since the period is  EMBED Equation.3 , then this cycle ends at  EMBED Equation.3  obtained by  EMBED Equation.3 . This cycle starts to the left of the y-axis and finishes to the right of the y-axis. Again, since we are sketching the graph of this function, we do not want our sketch to cross the y-axis because our picture will probably contain misinformation about where the actual graph crosses the y-axis. In this problem, our first cycle ends at  EMBED Equation.3 . So, lets sketch the second cycle that begins at  EMBED Equation.3 . This second cycle will end at  EMBED Equation.3  obtained by  EMBED Equation.3 . That is, will add the period of  EMBED Equation.3  to the starting point of the second cycle, which is  EMBED Equation.3 . The other numbers on the x-axis were obtained by the following:  EMBED Equation.3  period =  EMBED Equation.3  =  EMBED Equation.3  The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the starting point of the second cycle, which is  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . Check:  EMBED Equation.3  5.  EMBED Equation.3  Since the coefficient of the x variable is 4, then we will need to factor the 4 out in order to identify the phase shift.  EMBED Equation.3  Since the cosine function is being multiplied by a negative 2, then the graph will be inverted. Thus, we will need to draw an inverted cosine cycle. Amplitude = 2 Period =  EMBED Equation.3  =  EMBED Equation.3  Phase Shift:  EMBED Equation.3  units to the right  y 2 x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  x  EMBED Equation.3  Since the phase shift is  EMBED Equation.3  units to the right, then the cycle starts at  EMBED Equation.3 . Since the period is  EMBED Equation.3 , then this cycle ends at  EMBED Equation.3  obtained by  EMBED Equation.3 . The other numbers on the x-axis were obtained by the following:  EMBED Equation.3  period =  EMBED Equation.3  =  EMBED Equation.3  The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the starting point of the cycle, which is  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . Check:  EMBED Equation.3  6.  EMBED Equation.3  Since the coefficient of the x variable is  EMBED Equation.3 , then we will need to factor the  EMBED Equation.3  out in order to identify the phase shift.  EMBED Equation.3  NOTE: The  EMBED Equation.3  was obtained by  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3 . Amplitude =  EMBED Equation.3  Period =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  = 6 Phase Shift:  EMBED Equation.3  units to the left  y  EMBED Equation.3  x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  x - EMBED Equation.3  Since the phase shift is  EMBED Equation.3  units to the left, then the cycle starts at  EMBED Equation.3 . Since the period is 6, then this cycle ends at  EMBED Equation.3  obtained by  EMBED Equation.3 . This cycle starts to the left of the y-axis and finishes to the right of the y-axis. Again, since we are sketching the graph of this function, we do not want our sketch to cross the y-axis because our picture will probably contain misinformation about where the actual graph crosses the y-axis. In this problem, our first cycle ends at  EMBED Equation.3 . So, lets sketch the second cycle that begins at  EMBED Equation.3 . This second cycle will end at  EMBED Equation.3  obtained by  EMBED Equation.3 . That is, will add the period of  EMBED Equation.3  to the starting point of the second cycle, which is  EMBED Equation.3 . The other numbers on the x-axis were obtained by the following:  EMBED Equation.3  period =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the starting point of the second cycle, which is  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . Check:  EMBED Equation.3  7.  EMBED Equation.3  Since the coefficient of the x variable is  EMBED Equation.3 , then we will need to factor the  EMBED Equation.3  out in order to identify the phase shift.  EMBED Equation.3  Since the cosine function is an even function, then  EMBED Equation.3  Amplitude = 1 Period =  EMBED Equation.3   Phase Shift:  EMBED Equation.3  units to the right y 1 x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  x  EMBED Equation.3  Since the phase shift is  EMBED Equation.3  units to the right, then the cycle starts at  EMBED Equation.3 . Since the period is  EMBED Equation.3 , then this cycle ends at  EMBED Equation.3  obtained by  EMBED Equation.3 . The other numbers on the x-axis were obtained by the following:  EMBED Equation.3  period =  EMBED Equation.3  =  EMBED Equation.3  The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the starting point of the cycle, which is  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is one-fourth of the period, to the next starting point of  EMBED Equation.3 . Check:  EMBED Equation.3   HYPERLINK \l "TopicsList" Back to Topics List 4. SECANT AND COSECANT GRAPHS Given the function  EMBED Equation.3 , we may write this function as  EMBED Equation.3  by factoring out b. The cosecant function does not have an amplitude, the period is  EMBED Equation.3 , and the phase shift is  EMBED Equation.3  units to the right if  EMBED Equation.3  or is  EMBED Equation.3  units to the left if  EMBED Equation.3 . In order to obtain a sketch of the graph of the cosecant function, you will make use of the sketch of the graph of sine function. First sketch the graph of  EMBED Equation.3  and then locate the x-intercepts of the sketch. These are the locations of the vertical asymptotes of the cosecant function. Draw these vertical asymptotes and then use the sketch of the graph of the sine function to sketch the graph of the cosecant function. Similarly, given the function  EMBED Equation.3 , we may write this function as  EMBED Equation.3  by factoring out b. The secant function does not have an amplitude, the period is  EMBED Equation.3 , and the phase shift is  EMBED Equation.3  units to the right if  EMBED Equation.3  or is  EMBED Equation.3  units to the left if  EMBED Equation.3 . In order to obtain a sketch of the graph of the secant function, you will make use of the sketch of the graph of cosine function. First sketch the graph of  EMBED Equation.3  and then locate the x-intercepts of the sketch. These are the locations of the vertical asymptotes of the secant function. Draw these vertical asymptotes and then use the sketch of the graph of the cosine function to sketch the graph of the secant function. Examples Sketch two cycles of the graph of the following functions. Label the numbers on the y-axis. On the x-axis, only label where the cycles begin and end. 1.  EMBED Equation.3  First, sketch the graph of  EMBED Equation.3 . For this sine function, we have the following: Amplitude = 2 Period =  EMBED Equation.3  =  EMBED Equation.3  Phase Shift: None Since there is no phase shift and the period is  EMBED Equation.3 , then the first cycle starts at 0 and end at  EMBED Equation.3 . The second cycle starts at  EMBED Equation.3  and ends at  EMBED Equation.3 . y 2 x  EMBED Equation.3   EMBED Equation.3  x  EMBED Equation.3  2.  EMBED Equation.3  First, sketch the graph of  EMBED Equation.3 . Since the sine function is an odd function, then  EMBED Equation.3 . Thus, we have that  EMBED Equation.3  =  EMBED Equation.3  Since the sine function is being multiplied by a negative  EMBED Equation.3 , then the graph will be inverted. Thus, we will need to draw two inverted sine cycles. For this sine function, we have the following: Amplitude =  EMBED Equation.3  Period =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  = 8 Phase Shift: None Since there is no phase shift and the period is 8, then the first cycle starts at 0 and end at 8. The second cycle starts at 8 and ends at 16. y  EMBED Equation.3  x 8 16 x - EMBED Equation.3  3.  EMBED Equation.3  First, sketch the graph of  EMBED Equation.3 . Notice the coefficient of the x variable is 1. For this sine function, we have the following: Amplitude =  EMBED Equation.3  Period =  EMBED Equation.3  Phase Shift:  EMBED Equation.3  units to the right Since the phase shift is  EMBED Equation.3  units to the right, then the first cycle starts at  EMBED Equation.3 . Since the period is  EMBED Equation.3 , then this first cycle ends at  EMBED Equation.3 , obtained by  EMBED Equation.3 , and the second cycle ends at  EMBED Equation.3 , obtained by  EMBED Equation.3 . y  EMBED Equation.3  x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  x  EMBED Equation.3  4.  EMBED Equation.3  First, sketch the graph of  EMBED Equation.3 . For this cosine function, we have the following: Amplitude = 12 Period =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Phase Shift: None Since there is no phase shift and the period is  EMBED Equation.3 , then the first cycle starts at 0 and end at  EMBED Equation.3 . The second cycle starts at  EMBED Equation.3  and ends at  EMBED Equation.3 . y 12 x  EMBED Equation.3   EMBED Equation.3  x x  EMBED Equation.3  5.  EMBED Equation.3  First, sketch the graph of  EMBED Equation.3 . Since the cosine function is being multiplied by a negative 1, then the graph will be inverted. Thus, we will need to draw two inverted cosine cycles. For this cosine function, we have the following: Amplitude = 1 Period =  EMBED Equation.3  = 2 Phase Shift: None Since there is no phase shift and the period is 2, then the first cycle starts at 0 and end at 2. The second cycle starts at 2 and ends at 4.  y 1 x 2 4 x x  EMBED Equation.3  6.  EMBED Equation.3  First, sketch the graph of  EMBED Equation.3 . Since the coefficient of the x variable is 7, then we will need to factor the 7 out in order to identify the phase shift. Thus, we have that  EMBED Equation.3  For this cosine function, we have the following: Amplitude = 5 Period =  EMBED Equation.3  Phase Shift:  EMBED Equation.3  units to the left Since the phase shift is  EMBED Equation.3  units to the left, then the first cycle starts at  EMBED Equation.3 . Since the period is  EMBED Equation.3 , then this first cycle ends at  EMBED Equation.3 , obtained by  EMBED Equation.3 . This cycle starts to the left of the y-axis and finishes to the right of the y-axis. Since we are sketching the graph of this function, we do not want our sketch to cross the y-axis because our picture will probably contain misinformation about where the actual graph crosses the y-axis. In this problem, our first cycle ends at  EMBED Equation.3 . So, lets sketch the second cycle that begins at  EMBED Equation.3  as our first cycle. This cycle will end at  EMBED Equation.3  obtained by  EMBED Equation.3 . The next cycle will end at  EMBED Equation.3 , obtained by  EMBED Equation.3 .  y 5 x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  x  EMBED Equation.3   HYPERLINK \l "TopicsList" Back to Topics List 5. TANGENT GRAPHS Example Find the x-intercepts of the graph of  EMBED Equation.3  in the interval  EMBED Equation.3 . NOTE: The interval  EMBED Equation.3  of angles are the angles going one time around the Unit Circle clockwise for the subinterval  EMBED Equation.3  and the angles going one time around the Unit Circle counterclockwise for the subinterval  EMBED Equation.3 . To find the x-intercepts, set y equal to 0:  EMBED Equation.3 . Since  EMBED Equation.3 , then  EMBED Equation.3 . Since a fraction can only equal zero when the numerator of the fraction equals zero, then  EMBED Equation.3 . By Unit Circle Trigonometry, we are looking for angles in the interval  EMBED Equation.3  that intersect the Unit Circle so that the y-coordinate of the point of intersection is 0. Going around the Unit Circle clockwise, these angles are  EMBED Equation.3 . Going around the Unit Circle counterclockwise, these angles are  EMBED Equation.3 . Thus, the x-intercepts of the graph of  EMBED Equation.3  in the interval  EMBED Equation.3  are the points  EMBED Equation.3  and  EMBED Equation.3 . Example Find the vertical asymptotes of the graph of  EMBED Equation.3  in the interval  EMBED Equation.3 . NOTE: The interval  EMBED Equation.3  of angles are the angles going one time around the Unit Circle clockwise for the subinterval  EMBED Equation.3  and the angles going one time around the Unit Circle counterclockwise for the subinterval  EMBED Equation.3 . Since  EMBED Equation.3 , then the vertical asymptotes will occur where the denominator of this fraction is equal to zero. Thus, we want to solve the equation  EMBED Equation.3  in the interval  EMBED Equation.3 . By Unit Circle Trigonometry, we are looking for angles in the interval  EMBED Equation.3  that intersect the Unit Circle so that the x-coordinate of the point of intersection is 0. Going around the Unit Circle clockwise, these angles are  EMBED Equation.3 . Going around the Unit Circle counterclockwise, these angles are  EMBED Equation.3 . Thus, the vertical asymptotes of the graph of  EMBED Equation.3  in the interval  EMBED Equation.3  are  EMBED Equation.3  and  EMBED Equation.3 . Example Sketch two cycles of the graph of  EMBED Equation.3  using the x-intercepts and vertical asymptotes of the function.  y x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The function  EMBED Equation.3  does not have an amplitude and the period of the function is  EMBED Equation.3 . In general, given the function  EMBED Equation.3 , we may write this function as  EMBED Equation.3  by factoring out b. The tangent function does not have an amplitude, the period is  EMBED Equation.3 , and the phase shift is  EMBED Equation.3  units to the right if  EMBED Equation.3  or is  EMBED Equation.3  units to the left if  EMBED Equation.3 . Theorem The tangent function is an odd function. That is,  EMBED Equation.3  for all  EMBED Equation.3  in the domain of the function. A sketch of the graph of the tangent function can be obtained from the x-intercepts and vertical asymptotes of the function. Two consecutive vertical asymptotes are a distance of the period from each other. The x-coordinates of the x-intercepts are midway between two consecutive vertical asymptotes. That is, the x-coordinates of the x-intercepts are the midpoint of two consecutive vertical asymptotes. So to sketch the graph of a tangent function, you only need to know where the first vertical asymptote is located. Then use the period to find the next consecutive vertical asymptote. For an unshifted tangent function, the first two consecutive vertical asymptotes are symmetric about the y-axis. For a shifted tangent function, first shift one vertical asymptote for the unshifted graph. Once you have the vertical asymptotes, you find the x-coordinates of the x-intercepts by finding the midpoint of two consecutive vertical asymptotes. Recall the midpoint of the numbers a and b is the number  EMBED Equation.3 . Examples Sketch two cycles of the graph of the following functions. 1.  EMBED Equation.3  Amplitude: None Period =  EMBED Equation.3  Phase Shift: None   EMBED Equation.3 period =  EMBED Equation.3  y x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is the period, to the starting point of  EMBED Equation.3 , which is where the first vertical asymptote to the right of the y-axis crosses the x-axis. The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we found the midpoint of  EMBED Equation.3  and  EMBED Equation.3 . Of course, you could also use the fact that two consecutive x-intercepts are a distance of the period from each other. Since the first x-intercept is 0 and the period is  EMBED Equation.3 , then the second x-intercept is  EMBED Equation.3 . 2.  EMBED Equation.3  Since the tangent function is being multiplied by a negative  EMBED Equation.3 , then the graph will be inverted. Thus, we will need to draw two inverted tangent cycles. Amplitude: None Period =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Phase Shift: None  EMBED Equation.3 period =  EMBED Equation.3  =  EMBED Equation.3   y x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is the period, to the starting point of  EMBED Equation.3 , which is where the first vertical asymptote to the right of the y-axis crosses the x-axis. The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we found the midpoint of  EMBED Equation.3  and  EMBED Equation.3 . Of course, you could also use the fact that two consecutive x-intercepts are a distance of the period from each other. Since the first x-intercept is 0 and the period is  EMBED Equation.3 , then the second x-intercept is  EMBED Equation.3 . 3.  EMBED Equation.3  NOTE: Since the tangent function is an odd function, then  EMBED Equation.3  =  EMBED Equation.3 . Thus, we have that  EMBED Equation.3  Since the tangent function is being multiplied by a negative  EMBED Equation.3 , then the graph will be inverted. Thus, we will need to draw two inverted tangent cycles. Amplitude: None Period =  EMBED Equation.3  = 1 Phase Shift: None  EMBED Equation.3 period =  EMBED Equation.3   y x  EMBED Equation.3   EMBED Equation.3  1  EMBED Equation.3  The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add 1, which is the period, to the starting point of  EMBED Equation.3 , which is where the first vertical asymptote to the right of the y-axis crosses the x-axis. The 1 was obtained by  EMBED Equation.3 . That is, we found the midpoint of  EMBED Equation.3  and  EMBED Equation.3 . Of course, you could also use the fact that two consecutive x-intercepts are a distance of the period from each other. Since the first x-intercept is 0 and the period is 1, then the second x-intercept is 1. 4.  EMBED Equation.3  Since the coefficient of the x variable is 3, then we will need to factor the 3 out in order to identify the phase shift. Thus, we have that  EMBED Equation.3  Amplitude: None Period =  EMBED Equation.3  Phase Shift:  EMBED Equation.3  units to the right  EMBED Equation.3 period =  EMBED Equation.3  =  EMBED Equation.3  For the unshifted tangent graph the first two consecutive vertical asymptotes are symmetric about the y-axis separated by a distance of the period, which is  EMBED Equation.3  for this function. Thus, the first vertical asymptote to the right of the y-axis for the unshifted tangent graph is  EMBED Equation.3 . Lets shift this vertical asymptote  EMBED Equation.3  units to the right. Thus, the first vertical asymptote for the shifted graph is  EMBED Equation.3 , obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is the amount of the shift, to the starting point of  EMBED Equation.3 , which is the first unshifted vertical asymptote to the right of the y-axis crosses the x-axis.  y x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is the period, to the starting point of  EMBED Equation.3 , which is where the first shifted vertical asymptote to the right of the y-axis crosses the x-axis. The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is the period, to the starting point of  EMBED Equation.3 , which is where the second vertical asymptote crosses the x-axis. The  EMBED Equation.3  was obtained by  EMBED Equation.3   EMBED Equation.3 . That is, we found the midpoint of  EMBED Equation.3  and  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3   EMBED Equation.3 . That is, we found the midpoint of  EMBED Equation.3  and  EMBED Equation.3 . Of course, you could also use the fact that two consecutive x-intercepts are a distance of the period from each other. Since the first x-intercept, that we found using the midpoint formula, is  EMBED Equation.3  and the period is  EMBED Equation.3 , then the second x-intercept is  EMBED Equation.3 . 5.  EMBED Equation.3  Since the coefficient of the x variable is  EMBED Equation.3 , then we will need to factor the  EMBED Equation.3  out in order to identify the phase shift. Thus, we have that  EMBED Equation.3  NOTE: The  EMBED Equation.3  was obtained by  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3 . Amplitude: None Period =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Phase Shift:  EMBED Equation.3  units to the left  EMBED Equation.3 period =  EMBED Equation.3  =  EMBED Equation.3  For the unshifted tangent graph the first two consecutive vertical asymptotes are symmetric about the y-axis separated by a distance of the period, which is  EMBED Equation.3  for this function. Thus, the first vertical asymptote to the left of the y-axis for the unshifted tangent graph is  EMBED Equation.3 . Lets shift this vertical asymptote  EMBED Equation.3  units to the left. Thus, the first vertical asymptote for the shifted graph is  EMBED Equation.3 , obtained by  EMBED Equation.3 . That is, we subtract  EMBED Equation.3 , which is the amount of the shift, to the starting point of  EMBED Equation.3 , which is the first unshifted vertical asymptote to the right of the y-axis crosses the x-axis. Note that we subtract the amount of the shift because we are moving to the left.  y x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we subtract  EMBED Equation.3 , which is the period, to the starting point of  EMBED Equation.3 , which is where the first shifted vertical asymptote to the left of the y-axis crosses the x-axis. The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we subtract  EMBED Equation.3 , which is the period, to the starting point of  EMBED Equation.3 , which is where the second vertical asymptote crosses the x-axis. The  EMBED Equation.3  was obtained by  EMBED Equation.3   EMBED Equation.3 . That is, we found the midpoint of  EMBED Equation.3  and  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3   EMBED Equation.3 . That is, we found the midpoint of  EMBED Equation.3  and  EMBED Equation.3 . Of course, you could also use the fact that two consecutive x-intercepts are a distance of the period from each other. Since the first x-intercept, that we found using the midpoint formula, is  EMBED Equation.3  and the period is  EMBED Equation.3 , then the second x-intercept is  EMBED Equation.3   EMBED Equation.3 .  HYPERLINK \l "TopicsList" Back to Topics List 6. COTANGENT GRAPHS Example Find the x-intercepts of the graph of  EMBED Equation.3  in the interval  EMBED Equation.3 . NOTE: The interval  EMBED Equation.3  of angles are the angles going one time around the Unit Circle clockwise for the subinterval  EMBED Equation.3  and the angles going one time around the Unit Circle counterclockwise for the subinterval  EMBED Equation.3 . To find the x-intercepts, set y equal to 0:  EMBED Equation.3 . Since  EMBED Equation.3 , then  EMBED Equation.3 . Since a fraction can only equal zero when the numerator of the fraction equals zero, then  EMBED Equation.3 . By Unit Circle Trigonometry, we are looking for angles in the interval  EMBED Equation.3  that intersect the Unit Circle so that the x-coordinate of the point of intersection is 0. Going around the Unit Circle clockwise, these angles are  EMBED Equation.3 . Going around the Unit Circle counterclockwise, these angles are  EMBED Equation.3 . Thus, the x-intercepts of the graph of  EMBED Equation.3  in the interval  EMBED Equation.3  are the points  EMBED Equation.3  and  EMBED Equation.3 . Example Find the vertical asymptotes of the graph of  EMBED Equation.3  in the interval  EMBED Equation.3 . NOTE: The interval  EMBED Equation.3  of angles are the angles going one time around the Unit Circle clockwise for the subinterval  EMBED Equation.3  and the angles going one time around the Unit Circle counterclockwise for the subinterval  EMBED Equation.3 . Since  EMBED Equation.3 , then the vertical asymptotes will occur where the denominator of this fraction is equal to zero. Thus, we want to solve the equation  EMBED Equation.3  in the interval  EMBED Equation.3 . By Unit Circle Trigonometry, we are looking for angles in the interval  EMBED Equation.3  that intersect the Unit Circle so that the y-coordinate of the point of intersection is 0. Going around the Unit Circle clockwise, these angles are  EMBED Equation.3 . Going around the Unit Circle counterclockwise, these angles are  EMBED Equation.3 . Thus, the vertical asymptotes of the graph of  EMBED Equation.3  in the interval  EMBED Equation.3  are  EMBED Equation.3  and  EMBED Equation.3 . NOTE: The vertical line given by the equation  EMBED Equation.3  is the y-axis. Example Sketch two cycles of the graph of  EMBED Equation.3  using the x-intercepts and vertical asymptotes of the function.  y x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The function  EMBED Equation.3  does not have an amplitude and the period of the function is  EMBED Equation.3 . In general, given the function  EMBED Equation.3 , we may write this function as  EMBED Equation.3  by factoring out b. The cotangent function does not have an amplitude, the period is  EMBED Equation.3 , and the phase shift is  EMBED Equation.3  units to the right if  EMBED Equation.3  or is  EMBED Equation.3  units to the left if  EMBED Equation.3 . Theorem The cotangent function is an odd function. That is,  EMBED Equation.3  for all  EMBED Equation.3  in the domain of the function. Like the tangent function, a sketch of the graph of the cotangent function can be obtained from the x-intercepts and vertical asymptotes of the function. Two consecutive vertical asymptotes are a distance of the period from each other. The x-coordinates of the x-intercepts are midway between two consecutive vertical asymptotes. That is, the x-coordinates of the x-intercepts are the midpoint of two consecutive vertical asymptotes. So to sketch the graph of a cotangent function, you only need to know where the first vertical asymptote is located. Then use the period to find the next consecutive vertical asymptote. For an unshifted cotangent function, the first vertical asymptote is the y-axis. For a shifted cotangent function, first shift one vertical asymptote for the unshifted graph. Once you have the vertical asymptotes, you find the x-coordinates of the x-intercepts by finding the midpoint of two consecutive vertical asymptotes. Examples Sketch two cycles of the graph of the following functions. 1.  EMBED Equation.3   Amplitude: None Period =  EMBED Equation.3  Phase Shift: None y x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we found the midpoint of 0 and  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we found the midpoint of  EMBED Equation.3  and  EMBED Equation.3 . Of course, you could also use the fact that two consecutive x-intercepts are a distance of the period from each other. Since the first x-intercept, that we found using the midpoint formula, is  EMBED Equation.3  and the period is  EMBED Equation.3 , then the second x-intercept is  EMBED Equation.3 . 2.  EMBED Equation.3  NOTE: Since the cotangent function is an odd function, then  EMBED Equation.3  =  EMBED Equation.3 . Thus, we have that  EMBED Equation.3  Since the cotangent function is being multiplied by a negative 15, then the graph will be inverted. Thus, we will need to draw two inverted cotangent cycles. Amplitude: None Period =  EMBED Equation.3  =  EMBED Equation.3  Phase Shift: None  y x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we found the midpoint of 0 and  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we found the midpoint of  EMBED Equation.3  and  EMBED Equation.3 . Of course, you could also use the fact that two consecutive x-intercepts are a distance of the period from each other. Since the first x-intercept, that we found using the midpoint formula, is  EMBED Equation.3  and the period is  EMBED Equation.3 , then the second x-intercept is  EMBED Equation.3 . 3.  EMBED Equation.3  Since the coefficient of the x variable is  EMBED Equation.3 , then we will need to factor the  EMBED Equation.3  out in order to identify the phase shift. Thus, we have that  EMBED Equation.3  NOTE: The  EMBED Equation.3  was obtained by  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3 . NOTE: Since the cotangent function is an odd function, then  EMBED Equation.3  =  EMBED Equation.3 . Thus, we have that  EMBED Equation.3  Since the cotangent function is being multiplied by a negative  EMBED Equation.3 , then the graph will be inverted. Thus, we will need to draw two inverted cotangent cycles. Amplitude: None Period =  EMBED Equation.3  Phase Shift:  EMBED Equation.3  units to the right For the unshifted cotangent graph the first vertical asymptote is the y-axis. Lets shift this vertical asymptote  EMBED Equation.3  units to the right. Thus, the first vertical asymptote for the shifted graph is  EMBED Equation.3 , obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is the amount of the shift, to the starting point of 0.  y x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is the period, to the starting point of  EMBED Equation.3 , which is where the first shifted vertical asymptote to the right of the y-axis crosses the x-axis. The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we add  EMBED Equation.3 , which is the period, to the starting point of  EMBED Equation.3 , which is where the second vertical asymptote crosses the x-axis. The  EMBED Equation.3  was obtained by  EMBED Equation.3   EMBED Equation.3 . That is, we found the midpoint of  EMBED Equation.3  and  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we found the midpoint of  EMBED Equation.3  and  EMBED Equation.3 . Of course, you could also use the fact that two consecutive x-intercepts are a distance of the period from each other. Since the first x-intercept, that we found using the midpoint formula, is  EMBED Equation.3  and the period is  EMBED Equation.3 , then the second x-intercept is  EMBED Equation.3 . 4.  EMBED Equation.3  Since the coefficient of the x variable is  EMBED Equation.3 , then we will need to factor the  EMBED Equation.3  out in order to identify the phase shift. Thus, we have that  EMBED Equation.3  Amplitude: None Period =  EMBED Equation.3  =  EMBED Equation.3  Phase Shift:  EMBED Equation.3  units to the left For the unshifted cotangent graph the first vertical asymptote is the y-axis. Lets shift this vertical asymptote  EMBED Equation.3  units to the right. Thus, the first vertical asymptote for the shifted graph is  EMBED Equation.3 , obtained by  EMBED Equation.3 . That is, we subtract  EMBED Equation.3 , which is the amount of the shift, to the starting point of 0. Note that we subtract the amount of the shift because we are moving to the left.  y x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we subtract  EMBED Equation.3 , which is the period, to the starting point of  EMBED Equation.3 , which is where the first shifted vertical asymptote to the left of the y-axis crosses the x-axis. The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we subtract  EMBED Equation.3 , which is the period, to the starting point of  EMBED Equation.3 , which is where the second shifted vertical asymptote to the left of the y-axis crosses the x-axis. The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we found the midpoint of  EMBED Equation.3  and  EMBED Equation.3 . The  EMBED Equation.3  was obtained by  EMBED Equation.3 . That is, we found the midpoint of  EMBED Equation.3  and  EMBED Equation.3 . Of course, you could also use the fact that two consecutive x-intercepts are a distance of the period from each other. Since the first x-intercept, that we found using the midpoint formula, is  EMBED Equation.3  and the period is  EMBED Equation.3 , then the second x-intercept is  EMBED Equation.3 .  HYPERLINK \l "TopicsList" Back to Topics List     Copyrighted by James D. 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