Graph of Tangent
Today
•
Wed. May 17
MATH 128
3
2
x=
x=
2
3
Trigonometric Graphs
•
•
4
Tangent
Cosecant
•
Choose Base Function
•
Applications
6
,
3
3
,
⇥
,1
x=
⇥
3
x=
2
3
2
⇥
Ryan Hansen
•
Section 8.7–8.8
Sinusoidal Modeling
•
Transformations
•
Equation To Graph
•
Graph To Equation
Graph of Cosecant
x=
x=
2
x=0
6
Harmonic Motion
6
Domain: no odd
multiples of 2
Range:R
3
,
⇥ ⇥
3
3
,
4
⇥
, 1
⇥ ⇥
3
Amplitude
Sinusoidal Graphs
x=2
x=
⇥
,2
•
x-intercepts
multiples of
Amplitude = A
2
2
1.5
1.5
1
Any graph of
sine or cosine
0.5
-7
1
,
6 2
⇥
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
-0.5
1
0.5
-1
2
-1.5
-7
1.5
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
-2
1
-0.5
0.5
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-1
7
-0.5
-1.5
-1
-1.5
-2
-2
Period
Amplitude, Period,
Phase Shift and Vertical Shift
2
Period =
⇥
2
1.5
1
-6
-5
-4
-3
-2
-1
0
-0.5
-1
-1.5
-2
p)] + B
= ±A sin [!x
!p)] + B
best format!
0.5
-7
y = ±A sin [!(x
1
2
3
4
5
6
7
where
Amplitude = A
Phase Shift =
p
>0
2
⇥
=
B
Vertical Shift
Period = T =
Example
Find amplitude, period and phase shift. Then graph.
y=
3 sin( x + 2)
y = ±A sin [! (x
⇤
y = 3 sin
x
Amplitude= A = 3
p)]
2
⇥⇥⌅
2
Phase Shift= p =
2⇡
⇡
=2
Period =
!
Graph
Graph
3
3
2
2
2
1
1
1
2
0
2
y=
sin x
Graph
3
-6
-5
-4
-3
-2
-1 0
1
2
3
4
5
6
-6
-5
-4
-3
-2
-1 0
1
-1
-1
-1
-2
-2
-2
y=
-3
sin ⇡x
Graph
y=
-3
How To Choose
3
3 sin ⇡x
2
3
4
5
6
-3
How To Choose
2
-6
-5
-4
-3
-2
-1 0
sin
(x)
x)
s(
co
1
1
2
3
4
5
6
-1
-3
(x
)
cos
y=
(x)
sin
-2
 ✓
◆
2
3 sin ⇡ x +
⇡
Find A Formula
Find A Formula
Where To Find A and T
3
3
2
Amplitude and Period can be found in several places
2
1
1
-7 -6 -5 -4 -3 -2 -1 0
-1
Red: Amplitude (A)
Blue: Period (T)
-2
-2
-1
0
-1
-2
1
2
3
4
5
6
7
8
-3
-4
-5
-6
Also, Amplitude = (Max-Min)/2
-3
-7
-8
1
2
3
4
5
6
7
8
9 10 11
Find A Formula
Find A Formula
3
1
⇥
1
,0
2
0
1
2
-6
-5
-4
-3
-2
-1
3
2
⇥
1
+ ,0
2
3
4
5
2
1
Applications
6
-6
-5
-4
-3
-2
-1 0
1
2
3
4
5
6
-1
-1
-2
-2
-3
-3
✓
1
, 1
2
◆
✓
9
, 1
2
◆
Spring/Pendulum
Spring/Pendulum
An object attached to a spring is pulled down
a distance of 5 units from equilibrium and then released.
The time for one complete oscillation is 4 seconds.
Find an equation modeling the position of the object at time t .
⇣⇡ ⌘
5 cos
t
2
p(t) =
What is the objects
position at time 3/2?p
5 cos
Period: 29.53 days
What is the frequency
of its oscillation?
5 2
2
=
1
1
=
T
4
Summary
Piston Motion
Amplitude,
Period,
Sinusoidal
Graphs
Where
HowToToFind
Choose
A and T
Phase
and
AmplitudeShift
and Period
can beVertical
found in several Shift
places
11
2
y = ±A sin [!(x Anyp)]
+B
graph of
1
x)
s(
A piston oscillates a total of
75 mm (37.5 above and
below equilibrium) at 3250
rpm.
1.5
0.5
-7
-6
-5
-4
-3
-2
-1
0
-0.5
1
2
3
4
5
6
7
best format! sine or cosine
-1
= ±A sin [!x
-2
Amplitude = A
-6
(x
)
Phase Shift =
-7
2
p
1.5
>0
1
2
⇥
=
B
Vertical Shift
Also,
Don’tAmplitude
forget about homework!!!
= (Max-Min)/2
-5
-4
-3
-2
-1
0
1
2
3
Period = T =
-1
-1.5
-1
-1
Blue: Period (T)
0.5
-0.5
cos
Graph the function that gives the piston’s position as
a function of time (in minutes).
Write a formula for the function.
where
(x)
sin
How long is one cycle of the piston in minutes?
Red:+
Amplitude
!p)]
B (A)
00
-1.5
What is the moon’s intensity
12 days after a full moon?
per second
co
Graph the function. Write a formula.
Full Moon: 100
New Moon: 0
3⇡
4
◆
sin
(x)
Phases of the Moon
✓
-2
4
5
6
7