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Five-Minute Check (over Lesson 4-4)
Then/Now
New Vocabulary
Key Concept: Properties of the Tangent Function
Key Concept: Period of the Tangent Function
Example 1: Graph Horizontal Dilations of the Tangent Function
Example 2: Graph Reflections and Translations of the Tangent Function
Key Concept: Properties of the Cotangent Function
Example 3: Sketch the Graph of a Cotangent Function
Key Concept: Properties of the Cosecant and Secant Functions
Example 4: Sketch Graphs of Cosecant and Secant Functions
Example 5: Sketch Damped Trigonometric Functions
Key Concept: Damped Harmonic Motion
Example 6: Real-World Example: Damped Harmonic Motion
Over Lesson 4-4
A. Describe how the graphs of f(x) = sin x and
g(x) = 2 sin x are related. Then find the amplitude
and period of g(x).
A. The graph of g (x) is the graph of f (x) expanded
horizontally. amplitude = 2, period = 24.
B. The graph of g (x) is the graph of f (x) compressed
horizontally. amplitude = 1, period = π.
C. The graph of g (x) is the graph of f (x) expanded
vertically. amplitude = 2, period = 2π.
D. The graph of g (x) is the graph of f (x) compressed
vertically. amplitude = 0.5, period = 2π.
Over Lesson 4-4
B. Sketch the graphs of f(x) = sin x and
g(x) = 2 sin x on the same coordinate axes.
A.
C.
B.
D.
Over Lesson 4-4
A. State the amplitude, period, frequency, phase
shift, and vertical shift of
A. amplitude = 2; period = 2π;
frequency =
; phase shift =
B. amplitude = 1; period = π;
frequency = ; phase shift =
C. amplitude = 2; period = 2π;
frequency =
; phase shift =
D. amplitude =
frequency =
period = ;
; phase shift =
;
Over Lesson 4-4
B. Graph two periods of
A.
C.
B.
D.
Over Lesson 4-4
Write a sinusoidal function that can be used to
model the initial behavior of a sound wave with
a frequency of 820 hertz and an amplitude of 0.35.
A. y = 0.35 sin 820t
B. y = 0.35 sin 410πt
C. y = 0.35 sin 820πt
D. y = 0.35 sin 1640πt
You analyzed graphs of trigonometric functions.
(Lesson 4-4)
• Graph tangent and reciprocal trigonometric
functions.
• Graph damped trigonometric functions.
• damped trigonometric function
• damping factor
• damped oscillation
• damped wave
• damped harmonic motion
Graph Horizontal Dilations of the Tangent
Function
Locate the vertical asymptotes, and sketch the
graph of y = tan
.
The graph of y = tan
is the graph of y = tan x
expanded horizontally. The period is
or 3. Find two
consecutive vertical asymptotes by solving
bx + c = –
and bx + c =
.
Graph Horizontal Dilations of the Tangent
Function
Create a table listing key points, including the
x-intercept, that are located between the two vertical
asymptotes at
Graph Horizontal Dilations of the Tangent
Function
Sketch the curve through the indicated key points for
the function. Then sketch one cycle to the left on
and one cycle to the right on
.
Graph Horizontal Dilations of the Tangent
Function
Answer:
A. Locate the vertical asymptotes of y = tan 4x.
A. vertical asymptotes:
, n is an odd integer
B. vertical asymptotes:
, n is an odd integer
C. vertical asymptotes:
, n is an integer
D. vertical asymptotes:
, n is an odd integer
B. Sketch the graph of y = tan 4x.
A.
C.
B.
D.
Graph Reflections and Translations of the
Tangent Function
A. Locate the vertical asymptotes, and sketch the
graph of
.
The graph of y = –tan
is the graph of y = tan x
expanded horizontally and then reflected in the x-axis.
The period is
asymptotes.
. Find two consecutive vertical
Graph Reflections and Translations of the
Tangent Function
Multiply.
Create a table listing key points, including the
x-intercept, that are located between the two vertical
asymptotes at x = –2 and x = 2.
Graph Reflections and Translations of the
Tangent Function
Sketch the curve through the indicated key points for
the function. Then repeat the pattern for one cycle to
the left and right of the first curve.
Answer:
Graph Reflections and Translations of the
Tangent Function
B. Locate the vertical asymptotes, and sketch the
graph of
.
The graph of y = –tan
shifted
is the graph of y = tan x
to the left and then reflected in the x-axis.
The period is
asymptotes.
or π. Find two consecutive vertical
Graph Reflections and Translations of the
Tangent Function
Subtract.
Create a table listing key points, including the
x-intercept, that are located between the two vertical
asymptotes at x = – and x = 0.
Graph Reflections and Translations of the
Tangent Function
Sketch the curve through the indicated key points for
the function. Then sketch one cycle to the left and right.
Answer:
Locate the vertical asymptotes of the graph of
y = – tan(3x + π).
A. vertical asymptotes:
n is an odd integer
B. vertical asymptotes:
n is an integer
C. vertical asymptotes:
n is an odd integer
D. vertical asymptotes:
n is an integer
Sketch the Graph of a Cotangent Function
Locate the vertical asymptotes, and sketch the
graph of y = cot 2x.
The graph of y = cot 2x is the graph of y = cot x
compressed horizontally. The period is
or
. Find
two consecutive vertical asymptotes.
2x + 0 = 0
x =0
b = 2, c = 0
Simplify.
2x + 0 = 
x=
Sketch the Graph of a Cotangent Function
Create a table listing key points, including the
x-intercept, that are located between the two vertical
asymptotes at x = 0 and x = .
Sketch the Graph of a Cotangent Function
Following the same guidelines that you used for the
tangent function, sketch the curve through the
indicated key points that you found. Then sketch one
cycle to the left and right of the first curve.
Answer:
A. Locate the vertical asymptotes of
A. vertical asymptotes:
n is an odd integer
B. vertical asymptotes:
n is an integer
C. vertical asymptotes: x = nπ, n is an odd integer
D. vertical asymptotes: x = nπ, n is an integer
B. Sketch the graph of
A.
C.
B.
D.
Sketch Graphs of Cosecant and Secant
Functions
A. Locate the vertical asymptotes, and sketch the
graph of y = –sec 2x .
The graph of y = –sec 2x is the graph of y = sec x
compressed horizontally and then reflected in the
x-axis. The period is
or . Two vertical
asymptotes occur when bx + c =
and bx + c =
Therefore, two asymptotes are 2x + 0 =
or x = –
and 2x + 0 =
or x =
.
.
Sketch Graphs of Cosecant and Secant
Functions
Create a table listing key points that are located
between the asymptotes at x =
and x =
.
Sketch Graphs of Cosecant and Secant
Functions
Graph one cycle on the interval. Then sketch one
cycle to the left and right.
Answer:
Sketch Graphs of Cosecant and Secant
Functions
B. Locate the vertical asymptotes, and sketch the
graph of
.
The graph of y = csc
shifted
is the graph of y = csc x
units to the left. The period is
or 2. Two
vertical asymptotes occur when bx + c = – and
bx + c = . Therefore, two asymptotes are x +
or x =
and x +
=  or x =
.
= –
Sketch Graphs of Cosecant and Secant
Functions
Create a table listing key points, including the relative
maximum and minimum, that are located between the
two vertical asymptotes at x =
and x =
.
Sketch Graphs of Cosecant and Secant
Functions
Sketch the curve through the indicated key points for
the function. Then sketch one cycle to the left and
right.
Answer:
A. Locate the vertical asymptotes of y = csc
A. x = nπ, n is an odd integer
B.
n is an integer
C.
n is an odd integer
D. x = nπ, n is an integer
B. Sketch the graph of y = csc
A.
C.
B.
D.
Sketch Damped Trigonometric Functions
A. Identify the damping factor f(x) of
.
Then use a graphing calculator to sketch the
graphs of f(x), –f(x), and the given function in the
same viewing window. Describe the behavior of
the graph.
The function y =
y=
is the product of the functions
and y = sin x, so f(x) =
.
Sketch Damped Trigonometric Functions
The amplitude of the function is decreasing as
x approaches 0 from both directions.
Sketch Damped Trigonometric Functions
Answer: f(x) =
; The amplitude is decreasing as x
approaches 0 from both directions.
Sketch Damped Trigonometric Functions
B. Identify the damping factor f(x) of y = x 2 cos3x.
Then use a graphing calculator to sketch the
graphs of f(x), –f(x), and the given function in the
same viewing window. Describe the behavior of
the graph.
The function y = x 2 cos 3x is the product of the
functions y = x 2 and y = cos 3x. Therefore, the
damping factor is f(x) = x 2.
The amplitude is decreasing as x approaches 0 from
both directions.
Sketch Damped Trigonometric Functions
Answer: f(x) = x 2; The amplitude is decreasing as x
approaches 0 from both directions.
Identify the damping factor f(x) of y = 4x sin x.
A. f(x) = 4x
B. f(x) =
C. f(x) = sin x
D. f(x) = 4x sin x
Damped Harmonic Motion
A. MUSIC A guitar string is plucked at a distance
of 0.95 centimeter above its rest position, then
released, causing a vibration. The damping
constant for the string is 1.3, and the note
produced has a frequency of 200 cycles per
second. Write a trigonometric function that models
the motion of the string.
The maximum displacement of the string occurs when
t = 0, so y = ke–ct cos t can be used to model the
motion of the string because the graph of y = cos wt
has a y-intercept other than 0.
Damped Harmonic Motion
The maximum displacement occurs when the string is
plucked 0.95 centimeter. The total displacement is the
maximum displacement M minus the minimum
displacement m, so k = M – m = 0.95 – 0 or 0.95 cm.
You can use the value of the frequency to find w.
= frequency
Multiply each side by 2π.
Damped Harmonic Motion
Write a function using the values of k, w, and c.
y = 0.95e–1.3t cos 400πt is one model that describes
the motion of the string.
Sample Answer:
y = 0.95e–1.3t cos 400πt
Damped Harmonic Motion
B. MUSIC A guitar string is plucked at a distance
of 0.95 centimeter above its rest position, then
released, causing a vibration. The damping
constant for the string is 1.3, and the note
produced has a frequency of 200 cycles per
second. Determine the amount of time t that it
takes the string to be damped so that
–0.38 ≤ y ≤ 0.38.
Use a graphing calculator to determine the value of
t when the graph of y = 0.95e–1.3t cos 400πt is
oscillating between y = –0.38 and y = 0.38.
Damped Harmonic Motion
From the graph, you can see that it takes
approximately 0.7 second for the graph of
y = 0.95e–1.3t cos 400πt to oscillate within the interval –
0.38 ≤ y ≤ 0.38.
Answer: about 0.7 second
MUSIC Suppose another string on the guitar was
plucked 0.3 centimeter above its rest position with
a frequency of 64 cycles per second and a
damping constant of 1.4. Write a trigonometric
function that models the motion of the string y as
a function of time t.
A. y = 0.3e–2.8t cos 64πt
B. y = 0.6e–0.07t cos 32πt
C. y = 0.15e–5.6t cos 256πt
D. y = 0.3e–1.4t cos 128πt