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Analytic Trigonometry Barnett Ziegler Bylean

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CHAPTER 3 Graphs of trig functions

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CH 1 - SECTION 1 Basic graphs

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Why study graphs?

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Assignment Be able to sketch the 6 basic trig functions WITHOUT referencing notes or using a graphing calculator. Be able to answer questions concerning: domain/range x-int/y=int increasing/decreasing symmetry asmptote without notes or calculator.

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Hints for hand graphs X-axis - count by /2 with domain [-2, 3] Y-axis count by 1s with a range of [-5,5]

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Defining trig functions in terms of (x,y) Input x (cos( ),sin() )

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y=sin(x) Domain/range X-intercept Y-intercept Other points Periodic/period Increase Decrease Symmetry (odd) Input x (cos( ),sin() ) Output y = sin(x) using /2 for the x-scale

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y = cos(x) Input x (cos( ),sin() ) Domain/range X-intercept Y-intercept Other points Periodic/period Increase Decrease Symmetry (odd)

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y = tan(x) and y = cot(x) Input x (cos( ),sin() ) y = tan(x) y = cot(x) restricted/asymptotes: Range ? y-intercept x-intercept

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y = sec(x) and y = csc(x) Input x sec(x) csc(x) restricted/asymptotes? range? (cos( ),sin() )

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CHAPTER 3 SECTION 2 Transformations of sin and cos

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Review transformations Given f(x) What do you know about the following f(x-3) f(x + 5) f(3x) f(x/7) f(x) + 6 f(x) 4 3f(x) f(x)/3

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Trigonometric Transformations - dilations Y = Acos(Bx) y = Asin(Bx) Multiplication causes a scale change in the graph The graph appears to stretch or compress

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Vertical dilation : y = Af(x) If the multiplication is external (A) it multiplies the y-co-ordinate (stretches vertically) the x intercepts are stable (y=0), the y intercept is not stable for cosine The height of a wave graph is referred to as the amplitude (direct correlation to physics wave theory) - It is how much impact the x has on the y value - louder sound, harder heartbeat etc. Amplitude is measured from axis to max. and from axis to min.

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Examples of some graphs y=3(sin(x) y=-2sin(x) y=sin(x)

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y = 3cos(x) Scale /2 1

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Horizontal Dilations If the multiplication is inside the function it compresses horizontally against the y-axis the x-intercepts are compressed the y- intercept is stable this affects the period of the function Period the length of the domain interval that covers a full rotation The period for sine and cosine is 2 multiplying the x coordinates speeds up the rotation thereby compressing the period - New period is 2 /multiplier Frequency the reciprocal of the period-

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Examples of some graphs y=cos(x) y=cos(2x) y= cos(x/2)

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Sketch a graph (without a calculator)

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Transformations - Vertical shifts Adding outside the function shifts the graph up or down think of it like moving the x-axis f(x) = sin(x) + 2 g(x) = cos(x) - 4

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Pertinent information affected by shift the amplitude and period are not affected by a vertical shift The x and y intercepts are affected by shift The maximum and minimum values are affected by vertical shift

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Finding max/min values Max/min value for both sin(x) and cos(x) are 1 and -1 respectively Amplitude changes these by multiplying Shift change changes them by adding Ex: k(x)= 4cos(3x -5) 2 the max value is now 4(1)- 2= 2 the min value is now 4(-1) 2 =-6

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Example Graph k(x) = 4 + 2cos( x)

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Summary

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Writing equations Identify amplitude Identify period Identify axis shift

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CHAPTER 3 SECTION 3 Horizontal shifts

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Simple Harmonics f(x) = Asin(Bx + C) or g(x) = Acos(Bx + C) are referred to as Simple Harmonics. These include horizontal shifts referred to as phase shifts The shift is -C units horizontally followed by a compression of 1/B - thus the phase shift is -C/B units The amplitude and period are not affected by the phase shift

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Horizontal shift

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find amplitude, max, min, period and phase shift f(x) = 3cos(2 x /3) y = 2 4sin(x + /5)

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CHAPTER 3 SECTION 6 Tangent/cotangent/secant/cosecant revisited

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Basic graphs asymptotes Period Increasing/decreasing tan(x) cot(x) sec(x) csc(x)

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k + A tan(Bx+C) or k + A cot(Bx+C) No max or min - effect of A is minimal Period is /B instead of 2/B Phase shift is still -C/B and affects the x intercepts and asymptotes k moves the x and y intercepts

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Examples

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k+ Asec(Bx+ C) or k + Acsc(Bx + C) local maxima and minima affected by k and A Directly based on sin and cos so Period is 2 /B Shift is still -C/B

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Examples y = 3 + 2sec(3 x) y = 1 csc (2x + /3)