Download - x - intercepts
a
a
y
b2
b2
2
b2
3b2
b2
2
b2
3
x- intercepts
0
0cos abxay
b2
4
b2
4 x
a
a
y
b2
b2
2b2
3b2
b2
2
b2
3 0
0cos abxay
b2
4b2
4
x
x- intercepts
3
3
x
y
4
2
4
3 4
2
4
3
0
232cos3 baxy
2
2
x
y
6
3
2
3
2
6
3
2
3
2
0
323cos2 baxy
x
y
0
b2
b2
2b2
3
b2
4b2
5
0sin abxay
a
a
x
y
0b2
b2
3
0sin abxay
a
a
0sin abxay
b2
2b2
4b2
5
Period = 2π/|b| = 2π/3
2
-2
π/6 5π/6 π/2
Draw one full period of 2sin(3x–π/2)
2π/3
This is the graph of 2sin(3x). Now click to see the phase shift and to get 2sin(3x–π/2)
π/3
Amplitude = |a| = 2
Phase shift = -c/b = π/6
Graph one full period of sin(x–π /2) –1/2
a =1, b =1,c = – π/2 and d = –1/2
Amplitude = |a| =1
Period = 2π/b = 2π
Phase shift = – c/b = π/2
Vertical translation: 1/2 units down
2
y = sin(x)
y = sin(x–π /2)
y = sin(x–π /2) –1/2
1
1/2
–1
–3/2Phase shift
π/2 units right
Vertical translation½ units down
Section 5.7 Question 43
Graph one full period of 2sin(3x–π /2) +1
a =2, b =3,c = – π/2 and
d = 1
Amplitude = |a| =2
Period = 2π/b = 2π/3
Phase shift = – c/b = π/6
Vertical translation:
1 unit up
2
3
–1
–2
π/6
π/2 2π/3
x
y
π/3
This is the graph of 2sin(3x). Now click to see the phase shift , vertical translation and to get 2sin(3x–π/2)+1
1
Graph one full period of sin(x+π /6)
a =1, b =1,c = π/6
Amplitude = |a| =1
Period = 2π/b = 2π
Phase shift = – c/b
= –π/6
1
–1
π/2 π
3π/2 2π
x
y
–π/6
Section 5.7 Question 18
This is the graph of sin(x). Now click to see the phase shift and to get sin(x+π/6)
Graph one full period of cos(2x–π/3)
a =1, b = 2,c = – π/3
Amplitude = |a| =1
Period = 2π/b = π
Phase shift = – c/b
= π/6
1
–1
x
y
π/2ππ/6 π/4 3π/4
Section 5.7 Question 20
This is the graph of cos(2x). Now click to see the phase shift and to get cos(2x-π/3)
7π/6
Graph one full period of (1/2)sin(πx/3)
a =1/2, b = π/3
Amplitude = |a| =1/2
Period = 2π/b = 6
1/2
–1/2
3/2
9/2 6
x
y
3
Section 5.5 Question 38
x
y
0 3
3
2 3
43
5
322
33
2
3sin3 bbaxy
3
3
2
–2
1
–1
2
2
32
In [0 , π] ,
In [π , 2π] ,
0≤ sinx ≤ 2sinx
2sinx ≤ sinx ≤ 0
Graph one full period of 2sinx and sinx
x
y
b4
b4
b4
2b4
2
a
a
0tan abxay
x
y
b4
b4
b4
2b4
2 a
a
0tan abxay
x
y
b4
b4
3b4
2b4
4a
a
0cot abxay
x
y
b4
b4
2
a
a
0cot abxay
b4
3b4
4
x
y
2
2
2
–2
Draw one full period of y = 2tan(x/2)
a = 2 and b = 1/2 , 4b = 2
Asymptotes:
Lets draw asymptotes
Mark 2 and –2 on the y-axis
and ±π/4b = ±π/2 on the x-axis
x = ±2π/4b = ± 2π/2 = ± π
Now we can draw the graph
Section 5.6 Question 29
Graph one full period of (3/2)csc(3x)
a =3/2, b = 3
Period = 2π/b = 2π/3
π/6 π/3π/2
2π/3
Section 5.6 Question 34
3/2
–3/2
Graph one full period of (1/3)tanx
a =1/3, b =1→4b = 4
Period = π/|b| = π
π/4–π/4
1/3
–1/3
Section 5.6 Question 22
–π/2 π/2
Graph one full period of 2cscx
x
ya =2, b = 1
Period = 2π/b = 2π
2
–2
π/2 π3π/2
2π
Section 5.6 Question 28
Graph one full period of -3sec(2x/3)
x
ya = –3 , b = 2/3
Period = 2π/b = 3π
3
–3
3π/4 3π/2 9π/4 3π
Section 5.6 Question 36
x
y
12
6
12
6
3
–3
Draw one full period of y = –3tan(3x)
a = –3 and b = 3 , 4b = 12
Asymptotes:
Lets draw asymptotes Mark 3 and –3 on the y-axis
and ±π/4b = ±π/12 on the x-axis
x = ±2π/4b = ± 2π/12 = ± π/6Period = π/b = π/2
Now we can draw the graph
Section 5.6 Question 30
x
y
8
8
3
0
1/2
–1/2
Draw one full period of y = (1/2)cot(2x)
a = 1/2 and b = 2 , 4b = 8
Asymptotes:
Lets draw asymptotes Mark 1/2 and –1/2 on the y-axis
and π/8, 2π/8, 3π/8 and 4π/8 on the
x-axis
x = π/b = π/2 and x = 0(y-axis)Period = π/b = π/2
Now we can draw the graph
Section 5.6 Question 32
8
28
4
Graph one full period of 3/2sin(x /4+3π /4)
a =3/2, b = 1/4,c = 3π/4
Amplitude = |a| = 3/2
Period = 2π/b = 8π
Phase shift = – c/b = –3π
2π
6π 8π
3/2
–3/2
–3π x
y
4π
This is the graph of 3/2sin(x/4). Now click to see the phase shift and to get 3/2sin(x/4+3π/4)
Graph one full period of sec(x − π/2 )+1
x
y
a = 1 , b = 1, c = −π/2, d = 1
Period = 2π/|b| = 2π
Phase shift = −c/b = π/2
Vertical translation :
(d =) 1 unit up
1
–1
π
3π/2 2π
Section 5.7 Question 50
π/2
2
cos(x)
sec(x) sec(x − π/2 )
Click to shift π/2 unit to right
Click to shift 1 unit up
sec(x − π/2 )+1
| | | |
Graph one full period of csc(x/3-π/12)+4
x
y
a = 1, b = 1/3, c = π/3, d = 4
Period = 2π/|b| = 6π
Phase shift = -c/b = π/4
Vertical translation: 4 unit up
1
–1
3π/2 3π
9π/2
6π
Section 5.7 Question 48
| | | |π/4
5
sin(x/3)
csc(x/3−π/12)
csc(x/3)
csc(x/3-π/12)+4
3
1/2sin(3x) ≥ 0 1/2sin(3x) ≥ 0
1/2sin(3x) ≤ 01/2sin(3x) ≤ 0
Sketch the graph of y = |(1/2)sin(3x)|
Click to see y = |(1/2)sin(3x)|
π/3 2π/3-π/3-2π/3
-1/2
1/2
Ran
ge o
f 1/
2sin
(3x)
Ran
ge o
f |1
/2si
n(3
x)|
0
One period of 1/2sin(3x)
One period of |1/2sin(3x)|
Section 5.5 Question 48
Sketch the graph of y = cos2(x)
1
π/2-π/2 π-π π/4 3π/2-3π/2 -π/4
Section 5.5 Question 65
1/2
Sketch the graph of y = sin|x|
π 2π-π-2π
-1
1
0
Section 5.5 Question 68
y = sin|x| =
sin(x) if x ≥ 0
−sin(x) if x ≤ 0