4.5 Graphs of Trigonometric Functions 2014 Digital Lesson

4.5Graphs of Trigonometric

Functions 2014

Digital Lesson

HWQ

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

2sin , ,3 2

cos tanFind and


Graph of the Sine FunctionTo sketch the graph of y = sin x first locate the key points.These are the maximum points, the minimum points, and the intercepts. To locate the key points, divide the period by 4.

sin x

0x2

23

2

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

23

2

22

32

25

1

x

y = sin x


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.

cos x

0x2

23

2

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

23

2

22

32

25

1

x

y = cos x


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

General Forms for Sine and Cosine

siny a bx c d cosy a bx c d

• a is amplitude

• b represents the speed of the cycle. Period is

• represents the phase shift (horizontal shift)

• d represents the vertical shift.

2b

cb


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.

23

2

y

x

4

2

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

y = 2sin x

21y = sin x

y = sin x


y

x 2

s i n xy period: 2

s i n 2 y xperiod:

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 .

b2

For b 0, the period of y = a cos bx is also .b2

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

c o s xy period: 2

21cos xy

period: 4


y

123

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.

(0, 3)

23( , 0)

( , 0)2

2( , 3)

( , –3)

Start of one cycle End of one cycle

Graph of y 12

sin 3x

x

y

0.5

-0.5

1

-1

Graph of 2cos2

y x

x

y

1

-1

2

-2

Graph of 1 sin 2y x

x

y

1

-1

2

-2

Even and Odd Trig Functions

Remember: if f(-t) = f(t) the function is evenif f(-t) = - f(t) the function is odd

The cosine and secant functions are EVEN.cos(-t)=cos t sec(-t)=sec t

The sine, cosecant, tangent, and cotangent functions are ODD.

sin(-t)= -sin t csc(-t)= -csc ttan(-t)= -tan t cot(-t)= -cot t

(1, 0)(–1, 0)

(0,–1)

(0,1)

x

y

x


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)


y

2

6

x2

65

3

32

6

6

3

2

32

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)32

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

Start: End:

Graph of 3cos 2 4y x

x

y

Homework

• pg. 294 1-11odd, 39-55 odd


y

x

23

23

2

2

Tangent Function

Graph of the Tangent Function

2. range: (–, +) 3. period: 4. vertical asymptotes:

kkx 2

1. domain : all real x kkx

2

Properties of y = tan x

period:

To graph y = tan x, use the identity .xxx

cossintan

At values of x for which cos x = 0, the tangent function is undefined and its graph has vertical asymptotes.


2. Find consecutive vertical asymptotes by solving for x:

4. Sketch one branch and repeat.

Example: Find the period and asymptotes and sketch the graph of xy 2tan

31

22 ,

22

xx

4 ,

4

xxVertical asymptotes:

)2

,0( 3. Plot several points in 1. Period of y = tan x is .

2. i s 2t a n o f P e r i o d xy

xy 2tan31

8

31

8

31

83

31

y

x2

83

4

x4

x

31,

8

31,

8

31,

83


Graph of the Cotangent Function

2. range: (–, +) 3. period: 4. vertical asymptotes:

kkx


Properties of y = cot x y

x

2

2

23

23

2

xy c o t

0xvertical asymptotes xx 2x

To graph y = cot x, use the identity .xxx

sinco sco t

At values of x for which sin x = 0, the cotangent function is undefined and its graph has vertical asymptotes.


23

y

x

2

2

2 325

4

xy c o s

Graph of the Secant Function

2. range: (–,–1] [1, +) 3. period: 4. vertical asymptotes:

kkx 2

1. domain : all real x)(

2 kkx

co s

1secx

x The graph y = sec x, use the identity .

Properties of y = sec x

xy s e c

At values of x for which cos x = 0, the secant function is undefined and its graph has vertical asymptotes.


23

x

2

2

225

y

4

Graph of the Cosecant Function

2. range: (–,–1] [1, +) 3. period:

where sine is zero.

4. vertical asymptotes: kkx


sin

1cscx

x To graph y = csc x, use the identity .

Properties of y = csc x xy c s c

xy s i n

At values of x for which sin x = 0, the cosecant functionis undefined and its graph has vertical asymptotes.

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4.5 Graphs of Trigonometric Functions 2014 Digital Lesson