graphs of sine, cosine and tangent functions

Graphs of Trigonometric Functions

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

DAY 1 : OBJECTIVES

1. Define periodic function.2. Define symmetry.3. Differentiate an odd function from an

even function.4. Identify whether the graph of the

function is symmetric with the origin, x – axis, or y – axis.

5. Determine if the given function is an odd or even function.


1. Which among the following is a periodic function?

A. B.

C.D.


A periodic function is a function f such that f(x) = f(x + p),

for every real number x in the domain where p is a

constant.

The smallest positive number p, if there is one, for which

f(x + p) = f(x) for all x, is the period of the function.

This function is

periodic, the

function values

repeat every two

units as we move

from left to right.


Many things in daily life repeat with a predictable pattern,

such as vibrations and simple harmonic motions, rotation

of the earth about its own axis, the rotation of the earth

about the sun, the swinging of the pendulum of a clock,

the vibrations of strings of musical instruments, the

changing of seasons, the rise and fall of tides, the

heartbeat and the circulation of blood through the heart,

and many others.

When a phenomenon such as these results from circular

periodic motion, the circular functions are often used to

mathematically model the data.


A. B.

C.

D.

2. Identify if each graph is symmetric with respect to a

line or to a point.


Symmetry with respect to a

point

A graph is said to be

symmetric with respect to a

point Q if to each point P on

the graph, we can find point P’

on the same graph, such that Q

is the midpoint of the segment

joining P and P’.

Symmetry with respect to the

axis or line

A graph is said to be symmetric

with respect to a line if the

reflection (mirror image) about

the line of every point on the

graph is also on the graph The

line is known as the line of

symmetry.


A.B.

C. D.

3. Which function is symmetric with respect to the x – axis?

To the y – axis? To the origin?


Two points are symmetric with respect to the y – axis

if and only if their x – coordinates are additive

inverses and they have the same y – coordinate.


Two points are symmetric with respect to the x – axis if

and only if their y –coordinates are additive inverses

and they have the same x – coordinate.


Two points are symmetric with respect to the origin if

and only if both their x – and y – coordinates are

additive inverses of each other.

Imagine sticking a pin in

the given figure at the

origin and then rotating

the figure at 1800. Points

P and P1 would be

interchanged. The entire

figure would look exactly

as it did before rotating.


4. Which of the following function is odd?

A. f(x) = 3x2 – 4

B. f(x) = x3 + 5x - 2

C. f(x) = 10x5 + 4x3 - x

D. f(x) = 7x4 – 5x + 8


4. How is an odd function differ from an even function?

A function is an even function when f(-x) = f(x) for all x

in the domain of f. This is a function symmetric with

respect to the y – axis.

A function is an odd function when f(-x) = - f(x) for all x

in the domain of f. This is a function symmetric with

respect to the origin.


SEATWORK # 1

I. Identify whether each graph is symmetric with respect to the x –

axis, the y – axis, the origin or to none of these. (1 point each)

1. 2. 3. 4.

II. Identify if each function is even, odd, or neither. (1 point each)

5. f(x) = 2x2 7. f(x) = 3x2 – 7 9. f(x) = x3 + x

6. f(x) = 2x + 1 8. f(x) = - x4 10. f(x) = -3x7 – 4x5


ASSIGNMENT

I. Identify whether each graph is symmetric with respect to the x –

axis, the y – axis, the origin or to none of these. (1 point each)

1. 2. 4.

II. Identify if each function is even, odd, or neither. (1 point each)

5. f(x) = x2 – 3 7.f(x) = 3x – 7 9. f(x) = x3 + 2x8

6. f(x) = -13x 8. f(x) = - x4 + 9 10. f(x) = 8x7 – x11

3.


OBJECTIVES1. Identify the properties of the basic

sine and cosine functions from its

graph.

2. Find the amplitude and period of a

trigonometric function given its

equation.

3. Graphing sine and cosine functions

with various amplitude and period.


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


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


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


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 x

20x2

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 = 2 sin x

2

1y = sin x

y = sin x


y

x

2

sin xy period: 2 2sin xy

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


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)


Steps in Graphing y = a sin bx and y = a cos bx.

b

2

4

121

bst

a

4

222

bnd

4. Apply the pattern, then graph.

3. Find the intervals.

2. Find the period = .

1. Identify the amplitude = .

4

424

bth

4

323

brd


y = a cos bx

max0min0max

ba

ba

min0max0min

ba

ba

0min0max0

ba

ba

y = a sin bx

0max0min0

ba

ba


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

=


More Examples:

1. Graph y = 3 cos (- 2x).

x4sin2

32. Graph y = .


y

x

2

3

2

32

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 .x

xx

cos

sintan

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


Steps in Graphing y = a tan bx.

1. Determine the period .b

2. Locate two adjacent vertical asymptotes by solving for x:

22

bxandbx

3. Sketch the two vertical asymptotes found in Step 2.

4. Divide the interval into four equal parts.

5. Evaluate the function for the first – quarter point, midpoint,

and third - quarter point, using the x – values in Step 4.

6. Join the points with a smooth curve, approaching the

vertical asymptotes. Indicate additional asymptotes and

periods of the graph as necessary.


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 2tan3

1

22 ,

22

xx

4 ,

4

xxVertical asymptotes:

)2

,0(

3. Plot several points in

1. Period of y = tan x is .

2

. is 2tan of Period xy

x

xy 2tan3

1

8

3

1 0

08

3

18

3

3

1

y

x2

8

3

4

x

4

x

3

1,

8

3

1,

8

3

1,

8

3


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 xy2

1tan3

xx ,Vertical asymptotes:

3. Divide - to into four equal parts.

22

1,

22

1 xx

1. Period of y = tan x is .

2 of Period xy2

1tan3 is

x2

3 0

02

3 3

2

3

xy2

1tan3

y

x2

3

2

3

xx


Graph xy4

1tan2

1. Period is or 4.

41

2. Vertical asymptotes are

24

1

24

1 xandx

22 xandx

3. Divide the interval - 2

to 2 into four equal parts.

y

x

x = - 2 x = 2

x

2 0

0

2 2

3

xy4

1tan2

graphs of sine, cosine and tangent functions

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