Digital Lesson
Graphs of Trigonometric
Functions
HWQ
1. A weight attached to the end of a spring is
pulled down 3 cm below its equilibrium point
and released. It takes 2 seconds for it to
complete one cycle of moving from 3 cm. below
the equilibrium point to 3 cm. above and then
returning to its low point. Find the sinusoidal
function that best represents the position of the
moving weight and the approximate position of
the weight 5 seconds after it is released.
Warm-Up
•
•
•
•
Graph y = tanx on your calculator.
Where does the function seem to be undefined?
Why do you think it is undefined there?
Discuss with your partner.
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3
Graph of the Tangent Function
sin x
To graph y = tan x, use the identity tan x 
.
cos 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 
2
2. range: (–, +)
3. period: 
4. vertical asymptotes:

x  k 
2

2
 3
2
3
2
x

2
period: 
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4
Sketching the Graph of y=tanx
• The standard form is y=atan(bx-c)+d
• For one cycle, determine the endpoints of the
interval by solving:
– Left endpoint: bx-c = - /2
– Right endpoint: bx-c = /2
This interval is bounded by asymptotes.
• Graph the asymptotes, plot the x-intercept,
and sketch the characteristic curve.
Example: Find the period and asymptotes and sketch the graph
 y

1
x


x

of y  tan 2 x
4
4
3
1. Find consecutive vertical
asymptotes by solving for x:


2x   , 2x 
2
2

3
8
 1
 , 
 8 3

2
 1
 , 
 8 3


Vertical asymptotes: x   , x 
4
4
2. Plot several points between
asymptotes (midpoint b/w
asymptotes is x-int)
3. Sketch one branch and repeat.
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x


8
1
1
y  tan 2 x 
3
3
0
0
x
 3 1 
 , 
3
 8
 3
8 8
1 1

3 3
6
Example
• Graph
x
y  2 tan
2
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7
Sketching the Graph of cotx
• The standard form is y=acot(bx-c)+d
• For one cycle, determine the endpoints of the
interval by solving:
– Left endpoint: bx-c = 0
– Right endpoint: bx-c = 
This interval is bounded by asymptotes.
• Graph the asymptotes, plot the x-intercept,
and sketch the characteristic curve.
Graph of the Cotangent Function
cos x
To graph y = cot x, use the identity cot 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
x  
x0

 3
2
2
x
x
2
x  2
9
Example
• Graph
y  3 cot 2 x
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10
Sketching secx and cscx
• These are the reciprocal functions
 cscx=1/sinx
 secx=1/cosx
 You do not graph these functions directly:
 Graph the reciprocal function as a “support”
 Add asymptotes where the support function equals zero
(WHY?)
 Add cscx or secx :they are parabolic curves rising
up/down from the support function
 Remove the support function
Graph of the Secant Function
The graph y = sec x, use the identity sec x 
1
.
cos 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: 2
4. vertical asymptotes:

x  k   k   
2
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4
y  cos x
x



2
2

3
2
2
5 3
2
4
12
Example
Graph
1
y  sec x
4
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13
Graph of the Cosecant Function
1
To graph y = csc x, use the identity csc x 
.
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
4
y  csc x
1. domain : all real x
x  k  k   
2. range: (–,–1]  [1, +)
3. period: 2
4. vertical asymptotes
where sine = 0:
x  k  k   
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x



2
2

3 2
2
5
2
y  sin x
4
14
Example
Graph
y  csc2 x
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15
Homework
4.6 pg 305 7, 9, 31-47 odd, 53,55,67
Quiz Thursday on Graphing Trig. Functions
and Inverse Trig Functions. (4.5/7)
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16