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Analytic Trigonometry Barnett Ziegler Bylean
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### Transcript of Analytic Trigonometry Barnett Ziegler Bylean. CHAPTER 3 Graphs of trig functions.

• Slide 1
• Analytic Trigonometry Barnett Ziegler Bylean
• Slide 2
• CHAPTER 3 Graphs of trig functions
• Slide 3
• CH 1 - SECTION 1 Basic graphs
• Slide 4
• Why study graphs?
• Slide 5
• 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.
• Slide 6
• Hints for hand graphs X-axis - count by /2 with domain [-2, 3] Y-axis count by 1s with a range of [-5,5]
• Slide 7
• Defining trig functions in terms of (x,y) Input x (cos( ),sin() )
• Slide 8
• 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
• Slide 9
• y = cos(x) Input x (cos( ),sin() ) Domain/range X-intercept Y-intercept Other points Periodic/period Increase Decrease Symmetry (odd)
• Slide 10
• y = tan(x) and y = cot(x) Input x (cos( ),sin() ) y = tan(x) y = cot(x) restricted/asymptotes: Range ? y-intercept x-intercept
• Slide 11
• y = sec(x) and y = csc(x) Input x sec(x) csc(x) restricted/asymptotes? range? (cos( ),sin() )
• Slide 12
• CHAPTER 3 SECTION 2 Transformations of sin and cos
• Slide 13
• 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
• Slide 14
• Trigonometric Transformations - dilations Y = Acos(Bx) y = Asin(Bx) Multiplication causes a scale change in the graph The graph appears to stretch or compress
• Slide 15
• 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.
• Slide 16
• Examples of some graphs y=3(sin(x) y=-2sin(x) y=sin(x)
• Slide 17
• y = 3cos(x) Scale /2 1
• Slide 18
• 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-
• Slide 19
• Examples of some graphs y=cos(x) y=cos(2x) y= cos(x/2)
• Slide 20
• Sketch a graph (without a calculator)
• Slide 21
• 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
• Slide 22
• 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
• Slide 23
• 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
• Slide 24
• Example Graph k(x) = 4 + 2cos( x)
• Slide 25
• Summary
• Slide 26
• Writing equations Identify amplitude Identify period Identify axis shift
• Slide 27
• CHAPTER 3 SECTION 3 Horizontal shifts
• Slide 28
• 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
• Slide 29
• Horizontal shift
• Slide 30
• find amplitude, max, min, period and phase shift f(x) = 3cos(2 x /3) y = 2 4sin(x + /5)
• Slide 31
• CHAPTER 3 SECTION 6 Tangent/cotangent/secant/cosecant revisited
• Slide 32
• Basic graphs asymptotes Period Increasing/decreasing tan(x) cot(x) sec(x) csc(x)
• Slide 33
• 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
• Slide 34
• Examples
• Slide 35
• 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
• Slide 36
• Examples y = 3 + 2sec(3 x) y = 1 csc (2x + /3)