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
ө
(cos(ө),sin(ө) )
Output cos(𝑥)=𝑥𝜃 = 𝑦,
sin(x)=𝑦𝜃 = y,
tan(x) ,sec(x), csc(x), cot(x)
y=sin(x)
Input x
ө
•
•
•
•
•
•
•
•
(cos(ө),sin(ө) )
Domain/range
X-intercept
Y-intercept
Other points
Periodic/period
Increase
Decrease
Symmetry (odd)
Output cos(𝑥)=𝑥𝜃 ,sin(x)=𝑦𝜃 ,
tan(x)y,sec(x),
csc(x),
cot(x)
= sin(x)
– using
π/2 for the x-scale
y
y
y
y
x
x
x
x
y = cos(x)
Input x
ө
(cos(ө),sin(ө) )
Output cos(𝑥)=𝑦
y
Domain/range
X-intercept
Y-intercept
Other points
Periodic/period
Increase
Decrease
Symmetry (odd)
x
y = tan(x) and y = cot(x)
Input x
ө
y = tan(x)
(cos(ө),sin(ө) ) restricted/asymptotes:
Range ?
y-intercept
x-intercept
y = cot(x)
y
y
x
x
y = sec(x) and y = csc(x)
Input x
sec(x)
ө
csc(x)
restricted/asymptotes?
range?
(cos(ө),sin(ө) )
y
y
x
x
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
• 3f(x)
f(x) – 4
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
•
y=3(sin(x)
x
𝜋
2
y=sin(x)
y=-2sin(x)
y = 3cos(x)
y
1
x
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
y=cos(2x)
x
y= cos(x/2)
y=cos(x)
Sketch a graph (without a calculator)
• y = 3cos(2x)
• y = - sin(πx)
• 𝑦=
cos
2
𝑥
3
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
𝜋
4
• f(x) = cos(x + )
• 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=
𝑥
cot(
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)