You analyzed graphs of trigonometric functions. (Lesson 4-4)

LEQ: How do we 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.

B.

C.

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 . Find two consecutive vertical

asymptotes.

Graph Reflections and Translations of the

Tangent Function

Create a table listing key points, including the x-intercept, that are located between the two vertical asymptotes at x = –2 and x = 2.

Multiply.

Graph Reflections and Translations of the

Tangent Function

Answer:

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.

Graph Reflections and Translations of the

Tangent Function

B. Locate the vertical asymptotes, and sketch the

graph of .

The graph of y = –tan is the graph of y = tan x

shifted to the left and then reflected in the x-axis.

The period is or π. Find two consecutive vertical

asymptotes.

Graph Reflections and Translations of the

Tangent Function

Create a table listing key points, including the x-intercept, that are located between the two vertical asymptotes at x = – and x = 0.

Subtract.

Graph Reflections and Translations of the

Tangent Function

Answer:

Sketch the curve through the indicated key points for the function. Then sketch one cycle to the left and right.

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.

2x + 0 = 0 b = 2, c = 0 2x + 0 =

The graph of y = cot 2x is the graph of y = cot x

compressed horizontally. The period is or . Find

two consecutive vertical asymptotes.

x = 0 Simplify. 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.

B.

C.

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 is the graph of y = csc x

shifted 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

A.

B.

C.

D.

B. Sketch the graph of y = csc

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 = is the product of the functions

y = 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 = x2 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 = x2 cos 3x is the product of the functions y = x2 and y = cos 3x. Therefore, the damping factor is f(x) = x2.

The amplitude is decreasing as x approaches 0 from both directions.

Sketch Damped Trigonometric Functions

Answer: f(x) = x2; The amplitude is decreasing as xapproaches 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

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 wthas 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.

Damped Harmonic Motion

You can use the value of the frequency to find w.

Multiply each side by 2π.

= frequency

Sample Answer: y = 0.95e–1.3t cos 400πt

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.

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 ofy = 0.95e–1.3t cos 400πt to oscillate within the interval –0.38 ≤ y ≤ 0.38.

Answer: about 0.7 second

Damped Harmonic Motion

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