copyright © 2005 pearson education, inc. chapter 4 graphs of the circular functions

Copyright © 2005 Pearson Education, Inc.

Chapter 4

Graphs of the Circular Functions


4.1

Graphs of the Sine and Cosine Functions

Copyright © 2005 Pearson Education, Inc. Slide 4-3

Periodic Functions (Conceptual View)

Periodic Functions are functions whose values repeat in a regular pattern for every member of their domain

Example:


Periodic Functions (Definition)

A “periodic function” is a function f such that , for every real number x in the

domain of f, every integer n, and some positive real number p. The smallest possible positive value of p is the period of the function.

repeat? tobeginsfunction thebefore distancesmallest theisWhat

22. of period a hasfunction hissay that t We

2nxfxf

PERIODIC ARE FUNCTIONS

CIRCULAR AND RICTRIGINOMET ALL

npxfxf


Sine Function

The periodic nature of the circular sine function can be seen by considering the unit circle and by graphing the (x, sin x) pairs:

.

,2

2,

4

9,0,2,

2

2,

4

7

,1,2

3,

2

2,

4

5,0,

,2

2,

4

3,1,

2,

2

2,

4,0,0

etc


Sine Function f(x) = sin x

Period 2

.

,71.0,4

9,0,2,71.0,

4

7

,1,2

3,71.0,

4

5,0,

,71.0,4

3,1,

2,71.0,

4,0,0

etc

:ionsapproximat decimal as

shown ratios with pairs Ordered

)n2(xsin sin x :because

2 period with Periodic


Notes Concerning the Sine Function

Since the period of the sine function is , all possible values of the sine function will occur in any interval, and in each adjoining interval the exact pattern of values will be repeated

Any interval of values of the sine function is called “one period” of the sine function

We will define the “primary period” of the sine function to be those values in the interval 2,0

2

2

2

2


Sketching the Graph of the Primary Period of the Sine Function

In a rectangular coordinate system, mark the positions of and on the x-axis

Divide this interval on the x-axis into four intervals and label the endpoints of each (quarter points)

For each of these five numbers, the corresponding sine values are 0, 1, 0 -1, 0 (see unit circle)

Plot the (x, sin x) pairs and connect them with a smooth curve

Graph may be extended left and right

20

2

3

2

1

10 2


Cosine Function

The periodic nature of the circular cosine function can be seen by considering the unit circle and by graphing the (x, cos x) pairs:

.

,2

2,

4

9,1,2,

2

2,

4

7

,0,2

3,

2

2,

4

5,1,

,2

2,

4

3,0,

2,

2

2,

4,1,0

etc


Cosine Function f(x) = cos x

:ionsapproximat decimal as

shown ratios with pairs Ordered

)n2(x cos xcos :because

2 period with Periodic

Period 2

.

,71.0,4

9,1,2,71.0,

4

7

,0,2

3,71.0,

4

5,1,

,71.0,4

3,0,

2,71.0,

4,1,0

etc


Notes Concerning the Cosine Function

Since the period of the cosine function is , all possible values of the cosine function will occur in any interval, and in each adjoining interval the exact pattern of values will be repeated

Any interval of values of the cosine function is called “one period” of the cosine function

We will define the “primary period” of the cosine function to be those values in the interval 2,0

2

2

2

2


Sketching the Graph of the Primary Period of the Cosine Function

In a rectangular coordinate system, mark the positions of and on the x-axis

Divide this interval on the x-axis into four intervals and label the endpoints of each (quarter points)

For each of these five numbers, the corresponding cosine values are 1, 0, -1, 0, 1 (see unit circle)

Plot the (x, cos x) pairs and connect them with a smooth curve

Graph may be extended left and right

20

2

3

2

1

10 2


Amplitude of Sine and Cosine Functions

The amplitude of both the sine and the cosine function is defined to be “one half of the difference between the maximum and minimum values of the function”

Since the maximum value of both functions is “1” and the minimum value of both is “-1”, the amplitude of both is:

½[1- (-1)] = ½(2) = 1


Graph of y = a sin x compared with y = sin x

Both graphs will have a period of All the “y” values on y = a sin x, will be multiplied by “a”,

compared with the “y” values on y = sin x The amplitude of y = a sin x will be:

| a | instead of “1” (the graph will be vertically stretched or squeezed depending on whether | a | > 1 or | a | < 1)

If a < 0, the graph of y = a sin x will be inverted with respect to y = sin x

Example: Compare the graphs of:

y = 3 sin x and y = sin x

2


Example: Compare graph of y = 3 sin x with graph of y = sin x. Make a table of values.

030303sin x

01010sin x

23/2/20x

122

111

2

1 Amp 36

2

133

2

1 Amp


Effects of Changing a Factor of the “Argument” of a Circular Function

In the circular function “sin x”, “x” is called the argument of the function (The same terminology applies for each of the other circular functions)

If the argument of a circular function is changed from “x” to “bx”, the effect is always that:The original period is changed fromThe original graph is horizontally squeezed if | b | > 1 and horizontally stretched if | b | < 1

These concepts will be discussed and verified on the following slides

b

P toP


Graph of y = sin x compared with y = sin bx

y = sin x will have primary period for (has a period of )

y = sin bx will have a primary period for , but solving this inequality for x:

tells us that it has period:

The graph of y = sin bx will have exactly the same shape* as the graph of y = sin x, except that it will have the period described above.

In effect, the original graph will be horizontally squeezed or stretched depending on whether | b | > 1 or | b | < 1

*The same shape means at the left end of the interval: it will have a value of 0, at its first quarter point a value of 1, at its half point a value of 0, at its three quarter point a value of -1 and at its right end a value of 0

220 x20 bx

bx

20

b

2

b

2,0

bxsin of period gcalculatinfor Formula


Function values for: y = sin x and y = sin 2x(From Unit Circle or Calculator)

x sin x sin 2x x sin x sin 2x

2

4

0

2

4

3

4

5

2

3

4

7

0 0

71. 1

1 0

171.

0

71.

0

1

0

1

0

1

71.

0

x sin x sin 2x x sin x sin 2x


Example: Graph y = sin 2x continued

2 Period

Period

Period


Doing a Quick Sketch of y = sin 4x

Primary period of the graph will occur for:

Divide this interval into four quarters:

Assign to each of these the sine pattern:

Graph:

240 x2

0

x:for x Solved interval) period(Primary

2,

8

3,

4,

8,0

0,1,0,1,0

8

3

48

2

1

10


Periods of Modified Sine and Cosine Functions

For b > 0, the graph of y = sin bx will resemble that of y = sin x, but with period .

For b > 0, the graph of y = cos bx will resemble that of y = cos x, with period .

b

2

b

2


Doing a Quick Sketch of:

Primary period of the graph will occur for:

Find numbers to divide this interval into four quarters:

Assign to each of these the cosine pattern:

Graph:

23

20

x30 x:for x Solved interval) period(Primary

3,

4

9,

2

3,

4

3,0

1,0,1,0,1

4

9

2

3

4

3

1

13

xy3

2cos


Summary Comments about Graphs of y = a sin bx and y = a cos bx

Each of these graphs will, respectively, look like the the graphs of the basic functions, y = sin x and y = cos x, except that:

“a” alters the amplitude of the graph to | a |

“a” inverts the graph, if “a” is negative

“b” , when it is positive, changes the period to:b

2


Guidelines for Sketching Graphs of Sine and Cosine Functions To graph y = a sin bx or y = a cos bx, with b > 0,

follow these steps. Step 1To find the interval for one period, solve the

inequality: (the number at the right end will be the new period) and lay off this interval on the x-axis

Step 2Find numbers to divide the interval into four equal parts.

Step 3Assign the appropriate sine or cosine patterns, multiplied by “a”, to each of these five x-values. (The points will be maximum points, minimum points, and x-intercepts.)

20 bx


Guidelines for Sketching Graphs of Sine and Cosine Functions continued Step 4Plot the points found in Step 3, and join

them with a smooth curve having amplitude |a|.

Step 5Draw the graph over additional periods, to the right and to the left, as needed.


Graph y = 2 sin 4x

Step 1Solve inequality:

Step 2Divide the interval into four equal parts.

Step 3Assign pattern of sine values (0, 1, 0, -1, 0) multiplied by “a” (-2) to each of these five numbers:

30, , , ,

8 4 8 2

20

4

2

4

4

4

0

240

x

x

x

0,

8,2,

8,0,

4,2,

8,0,0

2 is period new The


Graph y = 2 sin 4x continued

Step 4Plot the points and join them with a smooth curve

28

3

480

negative is a"" because inverted is curve Sine2 |-2| | a | is Amplitude


Homework

4.1 Page 141 All: 1 – 32

MyMathLab Assignment 4.1 for practice

MyMathLab Homework Quiz 4.1 will be due for a grade on the date of our next class meeting


4.2

Translations of the Graphs of the Sine and Cosine Functions


Horizontal Translations

The horizontal movement of a graph to the left or right of its original position is called a “horizontal translation”

In circular functions, a horizontal translation is called a phase shift.

A phase shift in a circular function occurs when a term is added or subtracted from an argument:

When a positive number “d” is added to an argument the phase shift is “d” units to the left

When a positive number “d” is subtracted from an argument, the phase shift is “d” units to the right


Example of a Phase Shift in the Sine Function

Compared with , the graph of

will be moved units to

the left.

The amplitude will still be “1” and the period will still be

xy sin

2sin

xy

2

2

2 be shift will Phase


Sketch the graph of

The primary period of this function will occur when:

Solving for x gives:

2sin

xy

2

20 x

2

3

2

32

420

x

x

x

Shift Phase

2

22

3 :Period


Sketch the graph of

Divide this interval: into quarter points:

Assign pattern of sine values (0, 1, 0, -1, 0) to each of these five numbers:

Connect the points with a smooth curve:

2sin

xy

2

3

2

x

0,

2

3,1,,0,

2,1,0,0,

2

2

3,,

2,0,

2


Sketch the graph of

Points to be connected with smooth curve:

2

3

2

1

10 2

2

2sin

xy

xy sin

0,

2

3,1,,0,

2,1,0,0,

2

2sin

xy

left units 2

isshift Phase :Note


Graph y = sin (x /3)

Find the interval for one primary period.

Divide the interval into four equal parts.

Assign pattern of sine values (0, 1, 0, -1, 0) to each of these five numbers:

0 237

3 3

x

x

5 4 11 7, , , ,

3 6 3 6 3

0,3

7,1,

6

11,0,

3

4,1,

6

5,0,

3


Graph y = sin (x /3) continued

Plot:

0,3

7,1,

6

11,0,

3

4,1,

6

5,0,

3

3

7

6

11

3

4

6

5

3

xy sin

3sin

xy

right to3

3

shift Phase :Note


Graph

Note: There will be both an amplitude change and a phase shift.

Find the interval of primary period:

Divide into four equal parts.

Assign pattern of cosine values, (1, 0, -1, 0, 1), multiplied by 3, to each of these five numbers (3, 0, -3, 0, 3):

3cos4

y x

0 247

4 4

x

x

3 5 7, , , ,

4 4 4 4 4

3,

4

7,0,

4

5,3,

4

3,0,

4,3,

4


Graph continued

Plot:

3cos4

y x

3,

4

7,0,

4

5,3,

4

3,0,

4,3,

4

4

7

4

5

4

3

44

4cos3

xy

xy cos3

left to4

4

shift Phase :Noteπ

-

xy cos

2 period have All


Vertical Translations

The vertical movement of a graph up or down from its original position is called a “vertical translation”

A vertical translation of a circular function occurs when a term is added to, or subtracted from, a basic function

When a positive number “c” is added to a basic function the vertical translation is “c” units up

When a positive number “c” is subtracted from a basic function the vertical translation is “c” units down

In both these cases the effect is to move the “x-axis” from y = 0 to y = c


Example of a Vertical Translation in the Sine Function

Compared with , the graph of

will be moved unit up

The amplitude will still be “1” and the period will still be

There will be no phase shift

xy sinxy sin1 1

2


Sketch the graph of

The primary period of this function will occur when:

Divide this interval into quarter points:

Assign pattern of sine values (0, 1, 0, -1, 0), with “1” added to each, to each of these five numbers:

Connect the points with a smooth curve:

1,2,0,2

3,1,,2,

2,1,0

2,

2

3,,

2,0

xy sin1

20 x


Sketch the graph of

Points to be connected with smooth curve:

2

3

2

1

10 2

xy sin1xy sin

2

xy sin1

axis"- xnew" the torelative measured is amplitude andunit one

up movedbeen has axis,- x theincluding graph, wholeeffect theIn

1,2,0,2

3,1,,2,

2,1,0


Analyzing Functions of the Form:

Before analyzing transformations, these functions should be written in the exact form shown here, including having the argument factored as shown.

Compared with the basic functions: “a” changes the amplitude from 1 to | a | if “a” is negative, the graph is inverted “b” changes the period from “c” causes a vertical translation of “c” units “d” causes a phase shift (horizontal translation) of “d”

units (If “d” is positive, then “x – d” gives a phase shift “d” units to the right, and “x + d” gives a phase shift “d” units left)

)(cosor )(sin dxbacydxbacy

xyxy cosor sin

b

2 to2


Analyze y = 2 2 sin 3x compared with y = sin x

The graph will have an amplitude of 2 and will be inverted

Its period will be It will have a vertical translation of +2

(2 units up) It will have no phase shift

3

2


Further Guidelines for Sketching Graphs of Sine and Cosine Functions To graph y = c + a sin b(x - d) or y = c + a cos b(x – d),

with b > 0, follow these steps. Step 1To find the interval for one period, solve the

inequality: and lay off this interval on the x-axis (the number at the right end will be the new period)

Step 2Find numbers to divide the interval into four equal parts.

Step 3Assign the appropriate sine (0, 1, 0, -1, 0) or cosine (1, 0, -1, 0, 1) patterns, multiplied by “a”, and with “c” added, to each of these five x-values.

20 dxb


Guidelines for Sketching Graphs of Sine and Cosine Functions continued Step 4Plot the points found in Step 3, and join

them with a smooth curve having amplitude |a|.

Step 5Draw the graph over additional periods, to the right and to the left, as needed.


Sketch the graph of xy 3sin22

230 x

3

20

x

236

2

2 03

2

4

:PointsQuarter and Endpoints

2,4,2,0,2

:added 2 with 2,-by

multiplied 0), 1,- 0, 1, (0,pattern Sine3

2,

2,

3,

6,0


Analyze y = 1 + 2 sin (4x + ) compared with y = sin x

First factor the coefficient of x from argument:

Period:

Phase Shift:

Amplitude:

Vertical Translation:

44sin21

xy

24

2

4

2

1


Graph y = 1 + 2 sin (4x + )

240 x

x4

44

x


4,

8,0,

8,

4

:added 1- with 2,by

multiplied 0), 1,- 0, 1, (0,pattern Sine

1,3,1,1,1

80

8

4

4

1

1

3


Homework

4.2 Page 152 All: 1 – 12, 17 – 22, 27 – 28, 31 – 34, 39 – 46




4.3

Graphs of Other Circular Functions


Graphs of Cosecant and Secant Functions

Since these functions are reciprocals of the sine and cosine functions, they also have period

The graph of one period can be done over any interval of the domain that has this length, but we will call the interval the primary period

At the values of the domain where sine and cosine have values of 0, cosecant and secant will be undefined, these values of the domain establish vertical asymptotes, shown on the graphs as vertical dashed lines (asymptotes are not actually part of the graph, but they show where a function is undefined)

Graphs will be continuous between these vertical asymptotes

2

2,0


Graphs of Cosecant and Secant Functions

You will recall that values of the cosecant and secant functions (ranges) will be so no portion of their graphs will be in the interval

Cosecant and secant functions will have value 1 and -1 when sine and cosine functions have these values

Other values of cosecant and secant functions can be found by reciprocating sine and cosine values

,11,

1,1


Cosecant Function

xy sin


Secant Function

xy cos


Reference Graphs for Cosecant and Secant

By reciprocal identities already learned

To sketch the graphs of cosecant or secant with these arguments, we can first sketch the corresponding sine and cosine graphs with those same arguments as references

Where the sine and cosine graphs have value of 0 draw vertical asymptotes to show places where the cosecant and secant functions are undefined

Reciprocate the values of the sine and cosine functions and sketch the graphs of cosecant and secant

dxbdxbs

dxbdxb

cos

1ec and

sin

1csc


Graph Primary Period of

23sec

xy

23cos

xy

2

230

x

3

2

20

x

4360 x

763 x

6

7

2

x


6

7,,

6

5,

3

2,

2

1) 0, 1,- 0, (1,pattern Cosine

function reference

ingreciprocatby 2

3secGraph

xy

6

36

7


Graphing

First graph the reference function:

Note that functions boxed in red are reciprocals Also note that both functions have undergone

the same vertical stretching or squeezing and, the same vertical translation Therefore, to complete the graph is it only

necessary to attached the U-shaped cosecant curve to the reference curve

dxbacy csc

dxbacy sin


Graphing

First graph the reference function:

Note that functions boxed in red are reciprocals Also note that both functions have undergone

the same vertical stretching or squeezing and, the same vertical translation Therefore, to complete the graph is it only

necessary to attached the U-shaped secant curve to the reference curve

dxbacy sec

dxbacy cos


Sketch Graph: xy2

1sec31


Visualizing and Graphing the Tangent Function

1

2

1

2

3

2tan

3

21

23

7.13

2tan

2

1

3

2cos

2

3

3

2sin

3

2


Observations About the Tangent Function and Graph of y = tan x

The tangent will be undefined at and every odd multiple of it (the graph will have vertical asymptotes at these values of the domain)

The graph will be continuous and all possible values of tangent will be obtained as x varies between

Based on this last observation, the period of the tangent function will be and will be called the primary period

Since there are no maximum or minimum values for tangent, amplitude is not defined

2

2,

2

2,

2


TangentFunction

Endpoints &

Quarter Points:

Tangent

Values:

2,

4,0,

4,

2

,1,0,1,


Cotangent Function

This function is the reciprocal of the tangent function

At places where the tangent has value of 0, cotangent will be undefined and the graph will have vertical asymptotes at those values

At odd multiples of where the tangent is undefined, the value of cotangent is 0

Using these facts and taking reciprocals of tangent values the graph of cotangent can be established as follows:

2


Cotangent Function as a Reciprocal of Tangent

The period is: Primary period: Endpoints and Quarter Points: Cotangent values:

4

4

3 0

,0

,

4

3,

2,

4,0

,1,0,1,

xy tan

xy cot


Cotangent Function


Sketching Graphs of Transformed Tangent Functions

To graph Determine the primary period by solving for x:

Find endpoints and quarter points Assign pattern of tangent values

multiplied by “a” with “c” added Note: Analysis of phase shift, period change,

vertical stretching or squeezing, and vertical translation will be the same as for sine and cosine

dxbacy tan

22

dxb

,1,0,1,


Sketch xy 2tan31


Sketching Graphs of Transformed Cotangent Functions

To graph Determine the primary period by solving for x:

Find endpoints and quarter points Assign pattern of tangent values

multiplied by “a” with “c” added Note: Analysis of phase shift, period change,

vertical stretching or squeezing, and vertical translation will be the same as for sine and cosine

dxbacy cot

dxb0

,1,0,1,


Sketch

42

1cot31

xy


Homework

4.3 Page 165

All: 1 – 6,

Even: 8 – 46




4.4

Harmonic Motion


Simple Harmonic Motion

The position of a point oscillating about an equilibrium position at time t is modeled by either

where a and are constants, with The amplitude of the motion is |a|, the period is and the frequency is

( ) cos or ( ) sins t a t s t a t

0. 2.

2


Example

Suppose that an object is attached to a coiled spring such the one shown (on the next slide). It is pulled down a distance of 5 in. from its equilibrium position, and then released. The time for one complete oscillation is 4 sec.

a) Give an equation that models the position of the object at time t.

b) Determine the position at t = 1.5 sec. c) Find the frequency.


Example continued

When the object is released at t = 0, distance the object of the object from its equilibrium position 5 in. below equilibrium.

We use a = 5

( ) coss t t

24, or

2


Example continued

The motion is modeled by

b) After 1.5 sec, the position is

Since 3.54 > 0, the object is above the equilibrium position.

c) The frequency is the reciprocal of the period,

of ¼.

5cos .2t

(1.5) 5cos (1.5) 2.54 in.2

s

copyright © 2005 pearson education, inc. chapter 4 graphs of the circular functions

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