6.3 Graphing Trig Functions
Last section we analyzed graphs,
now we will graph them.
Graph: y = sin θ - 1
First, look at y = sin θ
1
-1
Since the – 1 is on the
outside that means we
are shifting DOWN
ONE unit
Graph: y = cos θ + 2
First, look at y = cos θ
Since the + 2 is on the
outside that means we
are shifting UP TWO
units
1
-1
Graph: y = 4sin 2θ
First, look at y = sin θ
Amplitue = 4
Period = 360/2 = 180
Phase Shift = 0°
1
-1
I will change the period first
Then change the
amplitude
Graph: y = -2cos (θ + 90°)
First, look at y = cos θ
Amplitue = 2
Period = 360/1 = 360
Phase Shift = Left 90°
1
-1
I will change the amplitude
first
Then change the
phase shift
Graph: y = 2tan( θ +45)
First, look at y = 2tan x
Asymptotes are still 90° + 180k°
1
Since 2 in front
changes the
“amplitude”?? Then
each output is doubled
-1
We’re not done, go to next slide
Graph: y = 2tan( θ +45)
Continued
1
-1
Now let’s shift 45° to
the right
Graph: y = sin (

2
Θ
270
y
0 90 180

1
0.7
0
+ 90°)
-0.7
See if you can graph this without graphing
each step.
360 450
-1
-0.7
540 630 720
0 0.7
1
Amplitude = 1
Period = 360/½ = 720
Phase Shift = 180° Left
(0,1)
(4π,1)
(π,0)
(2π,-1)
(3π,0)
(5π,0)
Graph:
 
y  tan 12 x
Θ
0 90
y
0
1
180
270
UD
-1
360 450
0
1
See if you can graph this without graphing
each step.
Amplitude = 1
540 630 720
Period = 180/½ = 360
UD -1 0
Phase Shift = 0°
Graph: y = 3cos (θ - 90°)
FIX THIS!!!
First, look at y = cos θ
Amplitue = 3
Period = 360/1 = 360°
Phase Shift = 90°
1
-1
I will change the period first
Then change the
amplitude
Graph: y = cot (θ – 90°)
FIX THIS!!!
Cot 0 = Does Not Exist
Amplitue = none
Period = 180/1 = 180°
Phase Shift = 90° Right
1
-1
I will change the period first
Then change the
amplitude
Graph: y = sin x + cos x
Best approach - table
θ
cos θ
sin θ sum
0°
1
0
1
45°
.71
.71
1.4
90°
0
1
1
135°
-.71
.71
0
180°
-1
0
-1
225°
-.71
-.71
-1.4
270°
0
-1
-1
315°
.71
-.71
0
360°
1
0
1
Period = 360
Graph: y = cos 2x – cos x
Best approach - table
θ
cos 2θ
cos θ
-
0°
1
1
0
45°
0
.71
-.71
90°
-1
1
-1
135°
0
-.71
.71
180°
1
-1
2
225°
0
-.71
.71
270°
-1
0
-1
315°
0
.71
-.71
360°
1
1
0
Period = ???
Graph: y = tan (
x
2
-

8
)
Amplitude = 1
Period = 180/½ = 360


Phase Shift = π/4 right
Graph: y = 2sin x + 3cos x
Best approach - table
θ
3cos θ
2sin θ
sum
0°
0
3
3
45°
1.4
2.1
3.5
90°
2
0
2
135°
1.4
-2.1
-.7
180° 0
-3
-3
225° -1.4
-2.1
-3.5
270° -2
0
-2
315°
2.1
.7
3
3
-1.4
360° 0
Period = 360???