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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.

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analytic trigonometry

Analytic Trigonometry

Barnett Ziegler Bylean

assignment
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
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
Defining trig functions in terms of (x,y)

Input x

(cos(ө),sin(ө) )

ө

Output cos()=,

sin(x)=,

tan(x) ,sec(x), csc(x), cot(x)

y sin x
y=sin(x)

Input x

  • Domain/range
  • X-intercept
  • Y-intercept
  • Other points
  • Periodic/period
  • Increase
  • Decrease
  • Symmetry (odd)

(cos(ө),sin(ө) )

ө

Output cos()=,sin(x)= , tan(x) ,sec(x), csc(x), cot(x)

Output

y = sin(x) – using π/2 for the x-scale

y cos x
y = cos(x)

Input x

(cos(ө),sin(ө) )

Output cos()=

ө

Domain/range

X-intercept

Y-intercept

Other points

Periodic/period

Increase

Decrease

Symmetry (odd)

y tan x and y cot x
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
y = sec(x) and y = csc(x)

Input x

sec(x) csc(x)

restricted/asymptotes?

range?

(cos(ө),sin(ө) )

ө

review transformations
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
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
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
Examples of some graphs

y=3(sin(x)

y=sin(x)

y=-2sin(x)

y 3cos x
y = 3cos(x)

1

Scale π/2

horizontal dilations
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 graphs1
Examples of some graphs

y=cos(2x)

y= cos(x/2)

y=cos(x)

sketch a graph without a calculator
Sketch a graph (without a calculator)
  • y = 3cos(2x)
  • y = - sin(πx)
transformations vertical shifts
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
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
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
Example
  • Graph k(x) = 4 + 2cos(𝛑x)
writing equations
Writing equations
  • Identify amplitude
  • Identify period
  • Identify axis shift
simple harmonics
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
Horizontal shift
  • f(x) = cos(x + )
  • g(x) = cos(2x – )
find amplitude max min period and phase shift
find amplitude, max, min, period and phase shift
  • f(x) = 3cos(2x – π/3)
  • y = 2 – 4sin(πx + π/5)
basic graphs
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
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
Examples
  • y = 3 + 2tan(3x)
  • y = cot()
k asec bx c or k acsc bx c
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
examples1
Examples
  • y = 3 + 2sec(3πx)
  • y = 1 – csc (2x + π/3)