grphs of trig fnctns

7/31/2019 Grphs of Trig Fnctns

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Graphs of TrigonometricFunctions

Digital Lesson


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Copyright by Houghton Mifflin Company, Inc. All rights reserved. 2

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 theintercepts.

0-1010sin x

0 x2

2

3 2

Then, connect the points on the graph with a smooth curvethat extends in both directions beyond the five points. Asingle cycle is called a period .

y

2

3

2

223

2

25

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 theintercepts.

10-101cos x

0 x2

2

3 2

Then, connect the points on the graph with a smooth curvethat extends in both directions beyond the five points. Asingle cycle is called a period .

y

2

3

2

223

2

25

1

1

x

y = cos x


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y

1

12

3

2

x 3 2 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.

max x-intmin x-intmax

30-303 y = 3 cos x 2 0 x 2

2

3

(0, 3)

23 ( , 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 distancebetween 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

3

2

4

y

x

4

2

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

y = 2sin x

21 y = sin x

y = sin x


7/13Copyright by Houghton Mifflin Company, Inc. All rights reserved. 7

y

x 2

sin x y period: 22sin y

period:

The period of a function is the x interval needed for thefunction 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 x y period: 22

1cos x y

period: 4



y

x 2

y = cos ( x)

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

Use the identitysin ( x) = sin x

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

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

Use the identitycos ( x) = cos x

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

y

x

2

y = sin x

y = sin ( x)

y = cos ( x)



2

y

2

6

x2

6

5

3

3

2

6

6

3

2

3

2

020 20 y = 2 sin 3 x

0 x

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

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 (

3 x) =

2 sin 3 x period :

b

2 23

=


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Example

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

Solution: Amplitude = |A| = 2period = 2 /B = 2 /3

phase shift = C/B = /3



Example cont.

y = 2sin(3x- )

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

-3

-2

-1

1

2

3



d cbxa )sin(Amplitude

Period:2 /b Phase Shift:

c/b

VerticalShift

grphs of trig fnctns

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