#### Read Microsoft PowerPoint - TRI_CT10_TrigonometricGraphs.ppt [Compatibility Mode] text version

`Trigonometry Rapid Learning Series - 10Rapid Learning CenterChemistry :: Biology :: Physics :: MathRapid Learning Center Presents ... p gTeach Yourself Trigonometry Visually in 24 Hours1/42http://www.RapidLearningCenter.comGraphs of Trigonometric FunctionsTrigonometry Rapid Learning SeriesWayne Huang, Ph.D. Mark Cowan, Ph.D. Diop El Moctar, Ph.D. Poornima Gowda, Ph.D. Daniel Deaconu, Ph.D. Fabio Mainardi, Ph.D. Theresa Johnson, M.Ed. Jessica Davis, M.S. Wendy Perry, M.A. Cesar Anchiraico, M.S.Rapid Learning Centerwww.RapidLearningCenter.com/© Rapid Learning Inc. All Rights Reserved© Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com1Trigonometry Rapid Learning Series - 10Learning ObjectivesBy completing this tutorial, you will learn concepts of:The basic graphs of the trigonometric functions The general form of the sine and cosine curves Amplitudes and periods of trigonometric functions Transformations3/42Concept MapPrevious Content New ContentTrigonometric Graphsinvolve involveTransformationsof the includeAsymptotesTrigonometric Functions Vertical Shift Sine Si Cosine Tangent4/42Cosecant Secant CotangentAmplitude pHorizontal Shift© Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com2Trigonometry Rapid Learning Series - 10Sine and Cosine Graphs5/42Definition: Key PointsKey points can be used to plot the graph of a function. In drawing the graph of a function, function choose key points that correspond to the: maximum values of a function minimum values of a function x-intercepts of a function (an x-intercept is a point at which the value of a function is equal to zero)6/42© Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com3Trigonometry Rapid Learning Series - 10Graph: y = sin xUse five key points to plot the graph of y = sin x.y = sin x B C A D E xPoint A B C D Ex 0 /2  3/2 3 /2 2y 0 1 0 -1 1 0Type Intercept Maximum Intercept Minimum Mi i Intercept7/42Graph: y = cos xUse five key points to plot the graph of y = cos x.Point A B C D E Type Maximum Intercept Minimum Intercept Maximum C1x 0 /2  3/2 2y 1 0 -1 0 1y = cos x A B D x E8/42© Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com4Trigonometry Rapid Learning Series - 10Definition: AmplitudeAmplitude ­ One-half of the vertical distance between the maxima and minima of a sine or cosine graph.y = a sin xx 2|a|y = a cos xx 2|a|9/42Identifying the AmplitudeThe general forms of the sine and cosine functions are: y = d + a sin(bx ­ c) y = d + a cos(bx ­ c)a, b, c, and d are constants (numbers whose values do not change).The absolute value of &quot;a&quot; is the amplitude of the function.10/42© Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com5Trigonometry Rapid Learning Series - 10Example: AmplitudeFind the equation for the sine graph shown.x2aSolution: Vertical distance between the max and min: 3 ­ (-3) = 3 + 3 = 6 Amplitude: |a| = 6/2 = 3 Equation: y = 3 sin x11/42Identifying the PeriodFundamental Period ­ The smallest interval over which a periodic function repeats itself. y = d + a sin(bx ­ c) y = d + a cos(bx ­ c)The constant &quot;b&quot; is directly related to the period of sine and cosine. period = 2 b12/42© Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com6Trigonometry Rapid Learning Series - 10Example: PeriodyFind the equation for the cosine graph shown.periodxSolution: period = 4 period = 2/b4 = 2/bb = 2/4 b=½ Equation: y = cos(x/2)13/42Transformations14/42© Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com7Trigonometry Rapid Learning Series - 10Transformations - OutlineTransformations on the trigonometric functions include: Vertical t a s at o s e t ca translations Horizontal translations Changes in amplitude and period Phase shifts Exponential damping15/42Vertical TranslationVertical Translation ­ An upward or downward shift in the graph of a function. Function Upward Shift Downward Shift16/42© Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com8Trigonometry Rapid Learning Series - 10Identifying Vertical Translationsy = d + a sin(bx ­ c) y = d + a cos(bx ­ c)The constant &quot;d&quot; determines the magnitude and direction of the vertical shift of a sine or cosine graph.y ydxdx17/42Example: Vertical TranslationPlot the graph of y = 1 + sin x.Solution: We begin by plotting y = sin x. Since d = 1, shift y = sin x upward by 1 unit.yy = 1 + sin x1x18/42© Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com9Trigonometry Rapid Learning Series - 10Horizontal TranslationHorizontal Translation ­ A shift to the left or right of the graph of a function. Function Shift to the Left Shift to the Right19/42Identifying Horizontal Translationsy = d + a sin(bx ­ c) y = d + a cos(bx ­ c)The ratio c/b is called the phase shift. The ratio c/b determines the magnitude and direction of the horizontal translation.y yc/b x c/b20/42x© Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com10Trigonometry Rapid Learning Series - 10Example: Horizontal TranslationPlot the graph of y = cos(x + /5). Solution: y = cos(x + /5) y = cos(x ­ (-/5)) b = 1 , c = -/5 c/b = -/5/5yxSince c/b is negative, shift y = cos x to the left by /5 units.21/42Example: Multiple TransformationsPlot the graph of y = 1 + 2cos(2x + /3).Solution: Amplitude: |a| = 2 Period: 2/b = 2/2 =  Horizontal Shift: c/b = (-/3)/2 = -/6 Vertical Shift: d = 1y/6x22/42© Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com11Trigonometry Rapid Learning Series - 10Example: Exponential DecayDraw the graph of y = 3e-0.1x sin x. Solution: y = a sin bx a = 3e-0.1x b=1y3e-0.1x x -3e-0.1xAmplitude: |a| = 3e-0.1x  plot 3e-0.1x and -3e-0.1x Period: 2/b = 2/1 = 2 *Remove the exponential curve.23/42Tangent and Cotangent Graphs24/42© Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com12Trigonometry Rapid Learning Series - 10Definition: AsymptoteAsymptote ­ Any line that a function approaches closely without ever intersecting. Vertical asymptote ­ A vertical line that a function approaches closely without ever intersecting. As a function approaches a vertical asymptote it will either become:orincreasingly positive25/42increasinglynegativeAsymptotes of Tangent and CotangentThe function y = tan x has a vertical asymptote at each odd integer multiple of /2 (..., -3/2, -/2, /2, 3/2, ...). The function y = cot x has a vertical asymptote at each multiple of  (..., -3, -2, -, , 2, 3,...).y yxxAsymptotes of Tangent26/42Asymptotes of Cotangent© Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com13Trigonometry Rapid Learning Series - 10How-to: Graph y = tan xTo graph y = tan x: 1. Draw a pair of consecutive vertical asymptotes. t t 2. Tabulate and plot 7 key points. 3 key points will be located between the left asymptote and the midpoint 1 key point will be the midpoint between the asymptotes 3 key points will be located between the midpoint and the right asymptote27/42Graph: y = tan xUse key points to plot the graph of y = tan x.x A B C D E F G H I28/42y Undef. Undef -10381 -1256 -14.10 0 14.10 1256 10381 Undef.Type asymptote normal normal normal intercept normal normal normal asymptote-/2 /2 C B D E x y F H G-/2 /2 -1.5707 -1.57 -1.5 0 1.5 1.57 1.5707 /2© Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com14Trigonometry Rapid Learning Series - 10Graph of y = a · tan(bx)Consider the function y = a · tan(bx).  The period of this function will be /b.  If the constant &quot;a&quot; is positive, the graph of a period positive will be increasing. (Figure 1)  If the constant &quot;a&quot; is negative, the graph of a period will be decreasing. (Figure 2)Figure 1 Figure 2 g29/42Example: Tangent TransformationDraw the graph of y = 2 tan(x/2). Solution: a=2 Period: /b = /(1/2) = 2xb=½yConsecutive asymptotes: y p x = - and  Behavior: a &gt; 0  increasing interval (-, )30/42-© Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com15Trigonometry Rapid Learning Series - 10How-to: Graph y = cot xTo graph y = cot x: 1. Draw a pair of consecutive vertical asymptotes. t t 2. Tabulate and plot 7 key points. 3 key points will be located between the left asymptote and the midpoint 1 key point will be the midpoint between the asymptotes 3 key points will be located between the midpoint and the right asymptote31/42Graph: y = cot xUse key points to plot the graph of y = cot x.x A B C D E F G H I32/42y Undef. 999 99.9 9.97 0 -24.0 -627 -1687 Undef.Type asymptote normal normal normal intercept normal normal normal asymptote0  C B D E F x G H y0 0.001 0.01 0.1 /2 3.1 3.14 3.141 © Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com16Trigonometry Rapid Learning Series - 10Graph of y = a cot(bx)Consider the function y = a · cot(bx). The period will be /b. If the constant &quot;a&quot; is positive, the graph of a period positive will be decreasing. If the constant &quot;a&quot; is negative, the graph of a period will be increasing.Figure 1Figure 233/42Example: Cotangent TransformationDraw the graph of y = -3 cot(x/6). Solution: a = -3 b = 1/6 Period: /b = /(1/6) = 6 Consecutive asymptotes: x = 0 and 60 Behavior: a &lt; 0  increasing interval (0, 6) 6 y3x34/42© Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com17Trigonometry Rapid Learning Series - 10Secant and Cosecant Graphs35/42Graph: y = csc xUse five key points to plot the graph of y = csc x.  Use the fact that csc x is the reciprocal of sin x.x 0 /2  3/2 2 sin x 0 1 0 -1 0 y = csc x Undefined/ Asymptote 1 Minimum Undefined/ Asymptote -1 Maximum Undefined/ asymptotey min x max36/42© Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com18Trigonometry Rapid Learning Series - 10Example: Cosecant TransformationDraw the graph of y = csc(2x ­ /3). Solution: y = csc(bx ­ c) b=2 c = /3 Period: 2/b= 2/2 =  Horizontal shift: c/b = (/3)/2 = /6 Vertical asymptotes: x = 2/3, 7/6, 5/3, 13/637/42y /6xGraph: y = sec xUse five key points to plot the graph of y = sec x.  Use the fact that sec x is the reciprocal of cos x.x 0 /2  3/2 2 cos x 1 0 -1 0 1 y = sec x 1 Minimum Undefined/ Asymptote -1 Maximum Undefined/ Asymptote 1 minimumy min xminmax38/42© Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com19Trigonometry Rapid Learning Series - 10Example: Secant TransformationDraw the graph of y = -1 + 2 sec x. Solution: y = d + a · sec x a=2 d = -1x 1 yAmplitude: |a| = |2| = 2 Vertical shift: d = -1  down 1 unit Vertical asymptotes: x = /2, 3/2, 5/2, 7/239/42Learning SummaryThe vertical shift of a trigonometric function is determined by the value of d.The period of a sine or cosine function is 2/b.The period of a tangent or cotangent function is /b.The amplitude is determined by |a| in the general form equations of sine and cosine.40/42The horizontal shift of a trigonometric function is determined by the ratio c/b.© Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com20Trigonometry Rapid Learning Series - 10CongratulationsYou have successfully completed the core tutorialGraphs of Trigonometric FunctionsRapid Learning CenterRapid Learning CenterChemistry :: Biology :: Physics :: MathWhat's N t Wh t' Next ... Step 1: Concepts ­ Core Tutorial (Just Completed) Step 2: Practice ­ Interactive Problem Drill Step 3: Recap ­ Super Review Cheat Sheet Go for it!42/42http://www.RapidLearningCenter.com© Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com21`

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Microsoft PowerPoint - TRI_CT10_TrigonometricGraphs.ppt [Compatibility Mode]