Name ________________________________________ Date __________________ Class__________________
LESSON
11-2
Reteach
Graphs of Other Trigonometric Functions
Transformations of the tangent function change the period and/or asymptotes of the graph.
For y = atanbx:
• the period is
π
|b|
,
Note: x is measured in radians.
• the asymptotes are located at x =
π
2| b |
+
πn
|b|
, where n is an
integer.
1
Use the graph of f(x) = tanx to sketch the graph of g ( x ) = tan x .
2
Step 1 Find b to identify the period.
1
π
π
There will be one full
b = , and
=
= 2π , so the period is 2π.
1
2
|b|
cycle between −π and π.
2
Step 2
Step 3
Step 4
The intercepts shift
Use the period to identify the x-intercepts.
2π radians right
The first x-intercepts of both f(x) and g(x) occur at 0.
and occur at integer
Because the period is 2π, the intercepts occur at 2π n.
multiples of 2π or the
For example, −2π, 0, and 2π are x-intercepts.
even multiples of π.
Identify the asymptotes.
π
πn
1
b = , so the asymptotes occur at x =
+
= π + 2π n .
1
1
2
2
2
2
1
Graph f(x) = tanx and g ( x ) = tan x
2
on the same plane.
The x-intercepts of g(x) are also x-intercepts of f(x).
The asymptotes are different. Because the period of
g(x) is 2π, there are fewer cycles of g(x) in the same
interval as f(x).
Complete to graph g(x) = tan2x. Use the interval from −π to π.
π
= _____________
|b|
2. Find the x-intercepts of g(x).
1. Find the period of g(x).
_________________________________________
3. Find the asymptotes of g(x).
_________________________________________
4. Sketch the graph of f(x) = tanx.
Then graph g(x) = tan2x.
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
11-14
Holt McDougal Algebra 2
Name ________________________________________ Date __________________ Class__________________
LESSON
11-2
Reteach
Graphs of Other Trigonometric Functions (continued)
Transformations of the cotangent function are similar to transformations of the tangent
function.
For y = acotbx:
• the period is
This is the same as for tangent transformations.
π
|b|
,
• the asymptotes are located at x =
πn
|b|
, where n is an integer.
Use the graph of f(x) = cotx to sketch the graph of g(x) = cot2x.
Step 1
Find b to identify the period.
b = 2, and
Step 2
π
|b|
=
π
|2|
=
π
2
, so the period is
π
2
There will be one full
.
cycle between 0 and
π
2
.
.
The first x-intercept of g(x) occurs at
From −π to π the x-intercepts are −
Identify the asymptotes.
π 1
⋅
2 2
, or
π
4
The x-intercepts shift
π
.
3π π π
3π
, − , , and
.
4
4 4
4
b = 2, so the asymptotes occur at x =
Step 4
2
Use the period to identify the x-intercepts.
The first x-intercept of f(x) is
Step 3
π
πn
|2|
=
πn
2
2
radians right.
These are the
.
multiples of
Graph g(x) = cot2x.
π
2
.
• Sketch the asymptotes.
• Plot the intercepts.
• Then use f(x) = cotx as a guide to
sketch the graph of g(x) = cot2x.
Complete to graph g ( x ) = cot
5. Find the period of g(x).
π
|b|
1
x . Use the interval from −2π to 2π.
2
= ________________
6. Find the x-intercepts of g(x). ________________
7. Where are the asymptotes of g(x)?
_________________________________________
8. Sketch the graph of g ( x ) = cot
1
x.
2
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
11-15
Holt McDougal Algebra 2
Practice C
4. Period: π; asymptotes:
1. Period: 1; x-intercepts: n; asymptotes:
1
+n
2
5. a. 6
πn
2
b. 46.2 ft
Reteach
3π
3π 3π n
; x-intercepts:
+
;
2
4
2
3π n
asymptotes:
2
2. Period:
1.
π
π
3. x = −
π
2. −π , − , 0, , π
2
2
2
3π π π 3π
,− , ,
4
4 4 4
4.
3. Period: 8π; asymptotes: 2π + 4π n
5. 2π
6. −π, π
7. x = −2π, 0, 2π
8.
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
A45
Holt McDougal Algebra 2