MA2.pptglobalizarion on economic landscape

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.

OBJECTIVES
1. 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
-1
0
1
0
sin x
0
x
2

2
3

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
2
3


 2



2
2
3

2

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.
1
0
-1
0
1
cos x
0
x
2

2
3

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
2
3


 2



2
2
3

2

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 .
1
1 

 y

y
1
1

2

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-int
min
x-int
max
3
0
-3
0
3
y = 3 cos x
2

0
x 2

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
3
2

4
y
x
4


2

y = –4 sin x
reflection of y = 4 sin x y = 4 sin x
y = 2 sin x
2
1
y = sin x
y = sin x

y
x

 
2

sin x
y 

period: 2
2
sin x
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 x
y 

period: 2
2
1
cos 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 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
x

2

y = sin x
y = sin(–x)
y = cos (–x)

Steps in Graphing y = a sin bx and y = a cos bx.
b

2







4
1
2
1
b
st

a







4
2
2
2
b
nd

4. Apply the pattern, then graph.
3. Find the intervals.
2. Find the period = .
1. Identify the amplitude = .







4
4
2
4
b
th








4
3
2
3
b
rd


y = a cos bx
max
0
min
0
max 










b
a
b
a
min
0
max
0
min 










b
a
b
a
0
min
0
max
0 










b
a
b
a
y = a sin bx
0
max
0
min
0 










b
a
b
a

2
y
2

6

x
2


6
5
3

3
2
6

6

3

2

3
2
0
2
0
–2
0
y = –2 sin 3x
0
x
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 
2
3
=

More Examples:
1. Graph y = 3 cos (- 2x).
x
4
sin
2
3

2. Graph y = .

y
x
2
3

2
3
2

2


Graph of the Tangent Function
2. range: (–, +)
3. period: 
4. vertical asymptotes:
 



 k
k
x
2


1. domain : all real x
 



 k
k
x
2


Properties of y = tan x
period:
To graph y = tan x, use the identity .
x
x
x
cos
sin
tan 
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:
2
2




 bx
and
bx
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 x
y 2
tan
3
1

2
2
,
2
2




 x
x
4
,
4




 x
x
Vertical asymptotes:
)
2
,
0
(

3. Plot several points in
1. Period of y = tan x is  .
2

.
is
2
tan
of
Period x
y 

x
x
y 2
tan
3
1

8


3
1
 0
0
8

3
1
8
3
3
1

y
x
2

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 x
y
2
1
tan
3



 

 x
x ,
Vertical asymptotes:
3. Divide - to  into four equal
parts.
2
2
1
,
2
2
1 



 x
x
1. Period of y = tan x is .

2
of
Period
 x
y
2
1
tan
3

 is

x
2


3 0
0
2

3
 3
2
3
x
y
2
1
tan
3


y
x
2
3
2
3



x



x

Graph x
y
4
1
tan
2

1. Period is or 4.
4
1

2. Vertical asymptotes are
2
4
1
2
4
1 



 x
and
x

 2
2 

 x
and
x
3. Divide the interval - 2
to 2 into four equal parts.
y
x
x = - 2 x = 2
x 

2
 0
0 
2 2


3
x
y
4
1
tan
2


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