4.5 Graphs of Trigonometric Functions 2014 Digital Lesson
Jan 18, 2018
4.5Graphs of Trigonometric
Functions 2014
Digital Lesson
HWQ
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2sin , ,3 2
cos tanFind and
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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
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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
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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
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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
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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
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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
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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)
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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
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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.
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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
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Graph of the Cotangent Function
2. range: (–, +) 3. period: 4. vertical asymptotes:
kkx
1. domain : all real x 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.
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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.
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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
1. domain : all real x 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.