Analytic Trigonometry Barnett Ziegler Bylean

Graphs of trig functions Chapter 3

Basic graphs Ch 1 - section 1

Why study graphs?

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.

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

Defining trig functions in terms of (x,y) Input x (cos(ө),sin(ө) ) ө Output cos(𝑥)= 𝑥 𝜃 =𝑦, sin(x)= 𝑦 𝜃 =y, tan(x) ,sec(x), csc(x), cot(x)

y=sin(x) Domain/range X-intercept Y-intercept Other points Input x Domain/range X-intercept Y-intercept Other points Periodic/period Increase Decrease Symmetry (odd) (cos(ө),sin(ө) ) ө Output y = sin(x) – using π/2 for the x-scale Output cos(𝑥)= 𝑥 𝜃 ,sin(x)= 𝑦 𝜃 , tan(x) ,sec(x), csc(x), cot(x)

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

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

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

Transformations of sin and cos Chapter 3 – section 2

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

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

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.

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

y = 3cos(x) 1 Scale π/2

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 2π – multiplying the x – coordinates speeds up the rotation thereby compressing the period - New period is 2π /multiplier Frequency – the reciprocal of the period-

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

Sketch a graph (without a calculator) y = 3cos(2x) y = - sin(πx) 𝑦= cos⁡ 𝑥 3 2

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

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

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

Example Graph k(x) = 4 + 2cos(𝛑x)

Summary

Writing equations Identify amplitude Identify period Identify axis shift

Horizontal shifts Chapter 3 – section 3

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

Horizontal shift f(x) = cos(x + 𝜋 4 ) g(x) = cos(2x – 𝜋 4 )

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

Tangent/cotangent/secant/cosecant revisited Chapter 3 – section 6

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

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

Examples y = 3 + 2tan(3x) y = cot( 𝑥 2 − 𝜋 3 )

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 2π/B Shift is still -C/B

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