Transcript 4 6 Other Trig Graphs 01

Objectives:
1. To graph cosecant
and secant as
transformations on
parent functions
2. To find the period
and asymptotes of
trig graphs
Assignment:
• P. 339: 7-30 S
– Secant & Cosecant
• P. 340: 41-48 S
– Secant & Cosecant
• P. 340: 51
• P. 341: 76
• P. 342: 81, 82, 84
You will be able to graph cosecant
and secant as transformations on
parent functions
Graph:
f ( x) 
2x 1
x
Vertical Asymptotes:
• Your graph can never cross one!
• If x = a is a vertical asymptote, then interesting
things happen really close to a:
– f (x) could approach +∞ or −∞
– Think of vertical asymptotes as black holes that
attract values near a
Vertical Asymptote:
The line x = a is a vertical asymptote of the graph of
f (x) if f (x) →+∞ or f (x) →−∞ as x → a.
Since the graphs of y = sin x and y = cos x were
generated by rotating a point around the unit
circle, they are often referred to as circular
functions. Here’re some more:
y  d  a tan  bx  c 
y  d  a csc  bx  c 
y  d  a cot  bx  c 
y  d  a sec  bx  c 
The first thing that you need to realize about the
graphs of y = tan x, y = cot x, y = csc x, and
y = sec x is the fact that they all have vertical
asymptotes. Explain why this is so.
1. Start by filling in the needed coordinates on the unit
circle: radian measures, coordinates in the form
1/y.
2. On the coordinate plane, make sure to mark the
scale your axes: Δy = 1; Δx = π/2
3. Since cosecant is undefined when the y-coordinate
on unit circle is 0, we’re going to have some
asymptotes.
• At what angle measures will cosecant have these
asymptotes? Draw them in!
3. According to your unit circle, what’s the cosecant of
x = 0 radians? Plot this as a point on the coordinate
plane:
(angle measure, 1/y)
4. Continue working your way around the unit circle.
This will complete one cycle or period of the csc
curve… Or will it?
5. Finish the graph by using angles beyond 2π and
negative angles.
Compete the tables below to observe the
behavior of the cosecant curve close to its
asymptotes.
x
csc x
1°
x
csc x
−1°
0.1°
0.01°
0.001°
0

−0.1°
−0.01°
−0.001°
0

f ( x)  csc x
Click for Trig Tracer
Domain: All x ≠ nπ, (n is
an integer)
Range: (−∞, −1] U [1, ∞)
Period: 2π (distance btw.
2 asymptotes)
Looks Like: Parabolas
Zeros: None
Symmetry: Origin
Vertical Asymptotes:
x = nπ
y  d  a csc  bx  c 
Click for Transformations
• |a| is the distance
from the x-axis
• A “Hill” of sin x is a
“Valley” of csc x, and
vice versa
• Period: 2π/b
• Asymptotes at the
zeros of sin x
Repeat the previous method to graph y = sec x.
Be sure to label the scale of each axis and
anything else you find to be important.
f ( x)  sec x
Click for Trig Tracer
Domain: All x ≠ π/2 + nπ,
(n is an integer)
Range: (−∞, −1] U [1, ∞)
Period: 2π (distance btw.
2 asymptotes)
Looks Like: Parabolas
Zeros: None
Symmetry: y-axis
Vertical Asymptotes:
x = π/2 + nπ
y  d  a sec  bx  c 
Click for Transformations
• |a| is the distance
from the x-axis
• A “Hill” of cos x is a
“Valley” of sec x, and
vice versa
• Period: 2π/b
• Asymptotes at the
zeros of cos x
You will be able to find
the period and
asymptotes of trig
graphs
For y = csc x:
1
On the unit circle, csc  
y
Undefined
when y = 0
(1, 0)
 0
Asymptotes at x  0  n
( −1, 0)
  
For y = sec x:
On the unit circle, sec  
1
x
Undefined
when x = 0
(0, 1)


2
Asymptotes at x 
(0, −1)
 

2

2
 n
Find the period and the asymptotes of each of
the following.
1. y = csc x
2. y = csc (6x)
3. y = csc (x – π/4)
4. y = csc (4x – π/4)
Find the period and the asymptotes of each of
the following.
1. y = sec x
2. y = sec (6x)
3. y = sec (x – π/4)
4. y = sec (4x – π/4)
y = sec x
x

2
First
Asymptote
y = csc x
 n 
x  0  n 
½ Period
y = d + a sec (bx + c)



x    phase   n 
b
 2b

First
Asymptote
½ Period
First
Asymptote
½ Period
y = d + a csc (bx + c)
x  phase  n 
First
Asymptote

b
½ Period
y = sec x
x

2
First
Asymptote
y = csc x
 n 
x  0  n 
½ Period
y = d + a sec (bx + c)

  c
x      n
b
 2b b 
First
Asymptote
½ Period
First
Asymptote
½ Period
y = d + a csc (bx + c)
x
c

 n
b
b
First
Asymptote
½ Period
You will be able to graph cosecant
and secant as transformations on
parent functions
1. Graph parent
– Key points: Asymptotes, Relative min/max
2. Perform SRT transformations
…
1. Graph corresponding sin or cos curve
– x-intercepts = asymptotes
– “Hills” become “Valleys
– “Valleys” become “Hills”
x
Tangent graph
x
Cotangent graph
Graph each of the following.
1. y = csc (2x)
2. y = sec (x/2)
3. y = (1/2) csc (3x)
4. y = −sec (x + π)
Find the values of a, b, c, and d such that
d  a csc  bx  c   sec x
Objectives:
1. To graph cosecant
and secant as
transformations on
parent functions
2. To find the period
and asymptotes of
trig graphs
Assignment:
• P. 339: 7-30 S
– Secant & Cosecant
• P. 340: 41-48 S
– Secant & Cosecant
• P. 340: 51
• P. 341: 76
• P. 342: 81, 82, 84