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```Graphs of Trigonometric Functions
Here are the graphs of the six main trigonometric functions along with
their domains and ranges. The windows shown have x-values from
β2π to 2π, and y-values from -3 to 3:
y ο½ sin x
Domain:
Range:
Period:
All Real Numbers
[-1, 1]
2π
y ο½ cos x
Domain:
Range:
Period:
All Real Numbers
[-1, 1]
2π
y ο½ tan x
Domain:
All Real Numbers
π
except 2 ± ππ
Range:
Period:
y ο½ csc x
Domain:
Range:
(-β, β)
π
All Real Numbers
except :0 ± ππ
(-β, -1]
and [1, β)
Period:
y ο½ sec x
Domain:
Range:
Period:
y ο½ cot x
Domain:
Range:
Period:
2π
All Real Numbers
π
except 2 ± ππ
(-β, -1]and[1,β)
2π
All Real Numbers
Except 0 ± ππ
(-β, β)
π
TransformationsThe general form for the sine equation is:
(In our parent graph: y ο½ sin x ;
ο·
ο·
ο·
ο·
y ο½ a sin(bx ο« c ) ο« d
a = b = 1, c = d = 0)
Changing the value of βaβ changes the amplitude of the graph.
βaβ is the amplitude of the graph.
Changing the value of βbβ changes the period of the graph. The
new period is found by taking the original period and dividing by
βbβ
Changing the values of βbβ and βcβ creates phase shift. This
c
moves the graph horizontally. The phase shift is equal to: ο­
b
Changing the value of βdβ creates vertical displacement. If βdβ is
positive, the graph will be moved up. If βdβ is negative, the
graph will be moved down.
Example Set 1:
Find the amplitude, period, phase shift and vertical displacement of
these graphs:
a)
y ο½ 3sin(4 x ο­ 20) ο« 5
b)
yο½
1
1
cos( x ο« 4) ο­ 1
2
2
c)
π
π¦ = tanβ‘(2π₯ β 2 )
Example Set 2:
Give the equation for the sine function that has the following
characteristics:
a)
Amplitude =
Period =
2
b)
Amplitude =
Period =
8π
2π
5
Phase Shift =
5π
½
π
Phase Shift = β 4
9
Example Set 1:
a
3
Amplitude
Period
Phase Shift
5
Vertical
Displacement
+5
Note:
b
½
π
2
c
n/a (See Note)
π
2
π
4
None
4π
-8
-1
We donβt talk about amplitude for tangent, as this graph
goes off to infinity in both directions.
Period Calculations:
a)
2π
4
=
π
2
b)
2π
1
2
= 4π
c)
π
2
=
π
2
Phase Shift Calculations:
a)
c
ο­20
ο­ ο½ο­
ο½5
b
4
b)
c
4
ο­ ο½ ο­ ο½ ο­8
1
b
2
c)
π
βπ = β
β
π
2
2
=
π
4
Example Set 2:
a)
π¦ = 2sinβ‘(5π₯ β
25π
9
)
1
b)
π₯
π
π¦ = 2 sinβ‘(4 + 16)
Calculations to find βbβ:
Recall that the new period is the old period divided by b:
a)
2π
5
=
π=
2π
π
2π
b) 8π =
2π
5
2π
π
2π
π = 8π
1
π=5
π=4
Calculations to find βcβ:
We find βcβ by using the phase shift as well as the value we just found
for βbβ.
c
Remember that the phase shift is equal to: ο­
b
π
a)
ππ = β π
5π
9
π
= β5
π=
β25π
9
b)
π
ππ = β π
π
π
β4 = β 1
4
π
π = 16
```
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