Graphs of Trigonometric Functions

Digital Lesson

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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

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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

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Graph of the Cosine Function

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

10-101cos 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 = cos x

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y

1

123

2

x 32 4

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

Partition the interval [0, 2] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval.

maxx-intminx-intmax

30-303y = 3 cos x20x 2

2

3

(0, 3)

2

3( , 0)( , 0)

2

2( , 3)

( , –3)

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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

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y

x

2

sin xy period: 2 2sin y

period:

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

For b 0, the period of y = a sin bx is .b

2

For b 0, the period of y = a cos bx is also .b

2

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

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

x 2 3 4

cos xy period: 2

2

1cos xy

period: 4

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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)

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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 bx with b > 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:b

2 23

=

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

amplitude = | A|

period = 2 /B.

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

amplitude = | A|

period = 2 /B.

x

y

Amplitude: | A|

Period: 2/B

y = A sin Bx

Starting point: x = C/B

The Graph of y = Asin(Bx - C)

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Example

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

Solution:

Amplitude = |A| = 2

period = 2/B = 2/3

phase shift = C/B = /3

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Example cont.

• y = 2sin(3x- )

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-3

-2

-1

1

2

3

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dcbxa )sin(Amplitude

Period:

2π/b Phase Shift:

c/b

Vertical

Shift