Transcript 4 6 Other Trig Graphs 02

Objectives:
1. To graph tangent
and cotangent as
transformations on
parent functions
2. To find the period
and asymptotes of
trig graphs
Assignment:
• P. 339: 1-6
• P. 339: 7-30 S
– Tangent & Cotangent
• P. 340: 41-48 S
– Tangent & Cotangent
• P. 341: 75
• P. 342: 83, 86
• My First Trig Project
Sneak Peek
You will be able to graph tangent
and cotangent as transformations
on parent functions
Compete the tables below to observe the
behavior of the tan curve close to its
asymptotes.
x
tan x
89°
x
tan x
−89°
89.9°
89.99°
89.999°
 2

−89.9°
−89.99° −89.999°
 2

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’s 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 
1. Start by filling in the needed coordinates on the unit
circle: radian measures, coordinates in the form
y/x.
2. On the coordinate plane, make sure to mark the
scale your axes: Δy = 1; Δx = π/2
3. Since tangent is undefined when the x-coordinate
on unit circle is 0, we’re going to have some
asymptotes.
• At what angle measures will tangent have these
asymptotes? Draw them in!
3. According to your unit circle, what’s the tangent of
x = 0 radians? Plot this as a point on the coordinate
plane:
(angle measure, y/x)
4. Continue working your way around the unit circle.
This will complete one cycle or period of the tan
curve… Or will it?
5. Finish the graph by using angles beyond 2π and
negative angles.
f ( x)  tan x
Click for Trig Tracer
Domain: All x ≠ π/2 + nπ,
(n is an integer)
Range: All Real #s
Period: π (distance btw.
asymptotes)
Looks Like: A Cubic
Zeros: {…, −π, 0, π, …}
Symmetry: Origin
Vertical Asymptotes:
x = π/2 + nπ
y  d  a tan  bx  c 
Click for Transformations
• Amplitude is
undefined
• |a| just
stretches/shrinks the
curves vertically
• Period: π/b
• Vertical asymptotes
affected by b and c
Repeat the previous method to graph y = cot x.
Be sure to label the scale of each axis and
anything else you find to be important.
f ( x)  cot x
Click for Trig Tracer
Domain: All x ≠ nπ, (n is
an integer)
Range: All Real #s
Period: π (distance btw.
asymptotes)
Looks Like: A Cubic
Zeros: {…, −π/2, π/2, …}
Symmetry: Origin
Vertical Asymptotes:
x = nπ
y  d  a cot  bx  c 
Click for Transformations
• Amplitude is
undefined
• |a| just
stretches/shrinks the
curves vertically
• Period: π/b
• Vertical asymptotes
affected by b and c
You will be able to find
the period and
asymptotes of trig
graphs
For y = tan x:
On the unit circle, tan  
y
x
Undefined
when x = 0
(0, 1)


2
Asymptotes at x 
(0, −1)
 

2

2
 n
For y = cot x:
x
On the unit circle, cot  
y
Undefined
when y = 0
(1, 0)
 0
Asymptotes at x  0  n
( −1, 0)
  
Find the period and the asymptotes of each of
the following.
1. y = tan x
2. y = tan (6x)
3. y = tan (x – π/4)
4. y = tan (4x – π/4)
Find the period and the asymptotes of each of
the following.
1. y = cot x
2. y = cot (6x)
3. y = cot (x – π/4)
4. y = cot (4x – π/4)
y = tan x
x

2
First
Asymptote
y = cot x
 n 
x  0  n 
Period
First
Asymptote
Period
y = d + a tan (bx + c)
y = d + a cot (bx + c)


x    phase   n  period
 2b

x  phase  n  period
First
Asymptote
Period
First
Asymptote
Period
y = tan x
x

2
First
Asymptote
y = cot x
 n 
x  0  n 
Period
y = d + a tan (bx + c)

  c
x      n
b
 2b b 
First
Asymptote
Period
First
Asymptote
Period
y = d + a cot (bx + c)
x
c

 n
b
b
First
Asymptote
Period
You will be able to graph tangent
and cotangent as transformations
on parent functions
1. Graph parent function
– Key points: Asymptotes, x-intercepts
2. Perform SRT transformations on parent
…
1. Find asymptotes (SRT)
2. Find x-intercepts: midpoint between
asymptotes
3. Scale vertically with a
Find the values of a, b, c, and d such that
d  a tan bx  c   cot x
Graph each of the following.
1. y = tan (2x)
2. y = cot (x/2)
3. y = (1/2) tan (3x)
4. y = −cot (x + π)
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)
Objectives:
1. To graph tangent
and cotangent as
transformations on
parent functions
2. To find the period
and asymptotes of
trig graphs
Assignment:
• P. 339: 1-6
• P. 339: 7-30 S
– Tangent & Cotangent
• P. 340: 41-48 S
– Tangent & Cotangent
• P. 341: 75
• P. 342: 83, 86
• My First Trig Project
Sneak Peek