Amplitude, Period, & Phase Shift 6.2 Trig Functions 3 ways we can change our graphs.

Amplitude, Period, & Phase Shift

6.2 Trig Functions

3 ways we can change our graphs

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6. The cycle repeats itself indefinitely in both directions of the x-axis.

Properties of Sine and Cosine Functions

The graphs of y = sin x and y = cos x have similar properties:

3. The maximum value is 1 and the minimum value is –1.

4. The graph is a smooth curve.

1. The domain is the set of real numbers.

5. Each function cycles through all the values of the range over an x-interval of .2

2. The range is the set of y values such that . 11 y


Graph of the Sine Function

To sketch the graph of y = sin x first locate the key points.These are the maximum points, the minimum points, and the intercepts.

0-1010sin x

0x2

2

32

Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period.

y

2

3

2

22

32

2

5

1

1

x

y = sin x

θ tan θ

−π/2 −∞

−π/4 −1

0 0

π/4 1

π/2 ∞

0 θ

tan θ

−π/2 π/2

One period: π

3π/2−3π/2

Vertical asymptotes

where cos θ = 0

Graph of Tangent Function: Periodic

cossin

tan

θ tan θ

0 ∞

π/4 1

π/2 0

3π/4 −1

π −∞

3π/2−3π/2

Vertical asymptotes

where sin θ = 0

Graph of Cotangent Function: Periodic

sin

coscot

π-π −π/2 π/2

cot θ

Cosecant is the reciprocal of sine

One period: 2π

π 2π 3π0

−π−2π−3π

Vertical asymptotes

where sin θ = 0

θ

csc θ

sin θ

Secant is the reciprocal of cosine

One period: 2π

π 3π−2π 2π−π−3π 0θ

sec θ

cos θ

Vertical asymptotes

where cos θ = 0


y

1

123

2

x 32 4

Example: Sketch the graph of y = 3 cos x on the interval [–, 4].

Partition the interval [-π,4] on your x-axis

maxx-intminx-intmax

30-303y = 3 cos x20x 2

2

3

(0, 3)

2

3( , 0)( , 0)

2

2( , 3)

( , –3)


The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function.

amplitude = |a| If |a| > 1, the amplitude stretches the graph vertically.If 0 < |a| < 1, the amplitude shrinks the graph vertically.If a < 0, the graph is reflected in the x-axis.

2

32

4

y

x

4

2

y = – 4 sin xreflection of y = 4 sin x y = 4 sin x

y = 2sin x

2

1y = sin x

y = sin x

Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

10

y

x

2

sin xy period: 2

€

y = sin2x period:

The period of a function is the x interval needed for the function to complete one cycle.

For k 0, the period of y = a sin kx is .

€

2π

kFor k 0, the period of y = a cos kx is also .

€

2π

k

If 0 < k < 1, the graph of the function is stretched horizontally.

If k > 1, the graph of the function is shrunk horizontally.

y

x 2 3 4

cos xy period: 2

2

1cos xy

period: 4

For k 0, the period of y = a tan kx is .

€

k


y

x2

y = cos (–x)

Use basic trigonometric identities to graph y = f (–x)Example 1: Sketch the graph of y = sin (–x).

Use the identity sin (–x) = – sin x

The graph of y = sin (–x) is the graph of y = sin x reflected in the x-axis.

Example 2: Sketch the graph of y = cos (–x).

Use the identity cos (–x) = – cos x

The graph of y = cos (–x) is identical to the graph of y = cos x.

y

x2y = sin x

y = sin (–x)

y = cos (–x)


2

y

2

6

x2

6

53

3

26

6

3

2

3

2

020–20y = –2 sin 3x

0x

Example: Sketch the graph of y = 2 sin (–3x).

Rewrite the function in the form y = a sin kx with k > 0

amplitude: |a| = |–2| = 2

Calculate the five key points.

(0, 0) ( , 0)3

( , 2)2

( , -2)6

( , 0)

3

2

Use the identity sin (– x) = – sin x: y = 2 sin (–3x) = –2 sin 3x

period:

€

π

k2 2

3=


The graph of y = A sin (Kx – C) is obtained by horizontally shifting the graph of y = A sin Kx so that the starting point of the cycle is shifted from x = 0 to x = -C/K. The number – C/K is called the phase shift.

amplitude = | A|

period = 2 /K.

The graph of y = A sin (Kx – C) is obtained by horizontally shifting the graph of y = A sin Kx so that the starting point of the cycle is shifted from x = 0 to x = -C/K. The number – C/K is called the phase shift.

amplitude = | A|

period = 2 /K.

x

y

Amplitude: | A|

Period: 2/B

y = A sin Kx

Starting point: x = -C/K

The Graph of y = Asin(Kx - C)


Example

Determine the amplitude, period, and phase shift of y = 2sin(3x-)

Solution:

Amplitude = |A| = 2

period = 2/K = 2/3

phase shift = -C/K = /3 to the right

€

asin(kx − c)


Example cont.

• y = 2sin(3x- )


€

asin(kx − c) + dAmplitude

Period: 2π/kPhase Shift:

-c/k

Vertical

Shift

17

State the periods of each function:

1.

2. €

y = cos1

2x

€

y = tan2x

4π or 720°

π/2 or 90°

18

State the phase shift of each function:

1.

2. €

y = tan(θ − 45°)

€

y = sin(2x +180°)

Right phase shift 45°

Left phase shift -90°

19

State the amplitude, period, and phase shift of each function:

1.

2.

3.

€

y = 4sinθ

€

y =10tan4θ

4.

5.

6.

€

y = 2cos2θ

€

y = 4 sinθ

2

€

y = tan(2x − π )

€

y = 3cos(θ − 90°)

A = 4, period = 360°,

Phase shift = 0°

A = NONE, period = 45°,

Phase shift = 0°

A = 2, period = 180°,

Phase shift = 0°

A = 4, period = 720°,

Phase shift = 0°

A = NONE, period = 90°,

Phase shift = π/2 Right

A = 3, period = 360°,

Phase shift = 90° Right

20

State the amplitude, period, and phase shift of each function:

1.

2.

€

y =10sin 12θ − 300°( )

€

y = 243sin(15θ − 40°)

A = 10, period = 1080°,

Phase shift = 900° Right

A = 243, period = 24°,

Phase shift = 8/3°

21

Write an equation for each function described:

1.) a sine function with amplitude 7, period 225°, and

phase shift -90°

2.) a cosine function with amplitude 4, period 4π, and phase

shift π/2

3.) a tangent function with period 180° and phase shift 25°

€

y = ±7sin8

5θ +144°

⎛

⎝ ⎜

⎞

⎠ ⎟

€

y = ±4cos1

2θ +

π

4

⎛

⎝ ⎜

⎞

⎠ ⎟

€

y = tan θ + 25°( )


Graph each function:

1.) 2.)

€

y = 3sinθ

€

y = cos(θ + 30°)

Amplitude, Period, & Phase Shift 6.2 Trig Functions 3 ways we can change our graphs.

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