Transcript 4.5

Digital Lesson
4.5
Graphs of Trigonometric
Functions 2014
HWQ
2

sin   ,
 ,
3
2
Find cos  and tan 
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2
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. To locate the key points, divide the period by 4.
x
0

2
3
2
2
sin x
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 = sin x
y
3

2



2
2
3
2
2
5
2
x
1
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3
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.

3
x
0
2
2
2
cos x
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 = cos x
y
3

2



2
2
3
2
2
5
2
x
1
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4
Properties of Sine and Cosine Functions
The graphs of y = sin x and y = cos x have similar properties:
1. The domain is the set of real numbers.
2. The range is the set of y values such that  1  y  1 .
3. The maximum value is 1 and the minimum value is –1.
4. The graph is a smooth curve.
5. Each function cycles through all the values of the range
over an x-interval of 2  .
6. The cycle repeats itself indefinitely in both directions of the
x-axis.
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5
General Forms for Sine and Cosine
y  a sin  bx  c   d
y  a cos  bx  c   d
• a is amplitude
2
• b represents the speed of the cycle. Period is
b
•
c represents the phase shift (horizontal shift)
b
• d represents the vertical shift.
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.
y
y = 2sin x

2
y=
1
sin x
2
y = – 4 sin x
reflection of y = 4 sin x
3
2 2
x
y = sin x
y = 4 sin x
 4
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7
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 2 .
b
For b  0, the period of y = a cos bx is also 2 .
b
If 0 < b < 1, the graph of the function is stretched horizontally.
y
y  cos x
1
period: 2
y  cos x
2
3
4
2
x
period: 4
y
If b > 1, the graph of the function
is shrunk horizontally.
y  s in x
y  s in 2 x
period: 2
period:
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x
2
8
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.
Start of one cycle
(0, 3)
End of one cycle
y
(2  , 3)
2
1
 2
3
( 3, 0)
2
(
, 0)
2
3
4
x
( , –3)
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9
1
Graph of y  sin 3x
2
y
1
0.5
x
-0.5
-1


Graph of y  2cos  x  
2

y
2
1
x
-1
-2
Graph of y  1  sin  2 x   
y
2
1
x
-1
-2
Even and Odd Trig Functions
Remember:
if f(-t) = f(t) the function is even
if f(-t) = - f(t) the function is odd
The cosine and secant functions are EVEN.
cos(-t)=cos t
sec(-t)=sec t
(0,1) y
(–1, 0)
(1, 0)
(0,–1)
The sine, cosecant, tangent, and cotangent functions are
ODD.
sin(-t)= -sin t
csc(-t)= -csc t
tan(-t)= -tan t
cot(-t)= -cot t
x
Use basic trigonometric identities to graph y = f (–x)
Example 1: Sketch the graph of y = sin (–x).
The graph of y = sin (–x) is the graph of y = sin x reflected in
the x-axis.
y = sin (–x)
y
Use the identity
sin (–x) = – sin x
y = sin x
x
2
Example 2: Sketch the graph of y = cos (–x).
The graph of y = cos (–x) is identical to the graph of y = cos x.
y
Use the identity
x
cos (–x) = cos x
2
y = cos (–x)
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14
Example: Sketch the graph of y = 2 sin (–3x).
Rewrite the function in the form y = a sin bx with b > 0
y = 2 sin (–3x) = –2 sin 3x
Use the identity sin (– x) = – sin x:
amplitude: |a| = |–2| = 2
Calculate the
five key points.
Start:
x
0
y = –2 sin 3x
0
 
2
6 3 2 3
–2
0
2
0
(
, 2)
y
2

   2
6
6 3 2 3
(0, 0)
 2
End:
(
,-2)
(
, 0) 2
3 ( , 0)
5
6
x
3
6
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15
Graph of y  3cos  2 x  4 
y
x
Homework
• pg. 294 1-11odd, 39-55 odd
Graph of the Tangent Function
sin x
To graph y = tan x, use the identity tan x 
.
co s x
At values of x for which cos x = 0, the tangent function is
undefined and its graph has vertical asymptotes.
y
Properties of y = tan x
1. domain : all real x

x  k   k   
2
2. range: (–, +)
3. period: 
4. vertical asymptotes:

x  k   k   
2

2
 3
2
3
2
x

2
period:
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18
Example: Find the period and asymptotes and sketch the graph
 y

1
x


x

of y  tan 2 x
4
4
3
1. Period of y = tan x is .

 P e r io d o f y  t a n 2 x is .
3
 1
2


, 

8
2
 8 3
x
2. Find consecutive vertical
 1
asymptotes by solving for x:
 3 1 
 , 
, 



 8 3
8
3

2x   , 2x 
2
2


Vertical asymptotes: x   , x 
4
4


3
3. Plot several points in (0, )

2
8
88
1
1
1

y  tan 2 x 
4. Sketch one branch and repeat.
3
3
3
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19
Graph of the Cotangent Function
co s x
To graph y = cot x, use the identity co t x 
.
sin x
At values of x for which sin x = 0, the cotangent function is
undefined and its graph has vertical asymptotes.
y
Properties of y = cot x
y  cot x
1. domain : all real x
x  k  k   
2. range: (–, +)
3. period: 
4. vertical asymptotes:
x  k  k   
vertical asymptotes
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
3
2



2
2
x  0
x
3
2
2
x  2
20
Graph of the Secant Function
The graph y = sec x, use the identity sec x 
1
.
co s x
At values of x for which cos x = 0, the secant function is undefined
and its graph has vertical asymptotes.
y
y  sec x
Properties of y = sec x
1. domain : all real x

x  k  (k   )
2
2. range: (–,–1]  [1, +)
3. period: 
4. vertical asymptotes:

x  k   k   
2
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


2
2
y  cos x
x
3
2
2
5 3 
2
 4
21
Graph of the Cosecant Function
To graph y = csc x, use the identity csc x 
1
.
sin x
At values of x for which sin x = 0, the cosecant function
is undefined and its graph has vertical asymptotes.
y
Properties of y = csc x
y  csc x
1. domain : all real x
x  k  k   
2. range: (–,–1]  [1, +)
3. period: 
4. vertical asymptotes:
x  k  k   
where sine is zero.
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


2
2
x
3 2 
2
5
2
y  s in x
 4
22