y x 10 y = 5
x
x
y=3 y=2
5 y-intercept (0, 1) −3
y= y=
1 x 2
−2
y=0 1
−1
1 x 3
10
y=
2
x
3
y
1 x 5
5 y-intercept (0, 1)
y =0 −4
−3
−2
−1
1
2
x 3
4
y
15 10 x
y=2 +4
5
x
x
y=2 0 −10 −5 x y = 2 − 6 −5
5
y 15 y = 2 x (x − 3)
y=2
y=2
(x + 7)
5 x −15
5
−5 6
y=2
−x
15
y y=2
x
4 2 x
−4
−2
4 2 x y = −2
5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5
y (2,4) (1, 2) y=0 x 1 2 3 4 5
(−1, 2)
5 4 3 2 1
−5 −4 −3 −2 −1 −1 −2 −3 −4 −5
y
(1, 2) (0, 1) y=0 1 2 3 4 5
x
y=3
−x
9 8 7 6 5 4 3 2 1
y
y=3
1
−5 −4 −3 −2 −1 −1 −2
18 16 14 12 10 8 6 4 2 −2
−1
x
y=0 x 2 3 4 5
y R
S
1
y=0 x 2
25 20 15 10 5 −5 −4 −3 −2 −1 −5 −10 −15 −20 −25
y y=5
1
x
y=0 x 2 3 4 5
f ( x) −2 g( x )
5 4 3 2 1 −1
−1 −2 −3 −4 −5
y
1
2
x
9
y
5 4 3 2 1
8 7 6 5 4 3
−5 −4 −3 −2 −1 −1
2
−3
−2
1 −5 −4 −3 −2 −1
x 1
2 3 4 5
−4 −5
y
x 1
2 3 4 5
y = ex
y 10 y = 3x
y = 2x
5 y-intercept (0, 1)
x
y=0 −3 −2 −1
1
2
3
y 6 y = 2x
4 y = 3 (2x) −4
2
y= 2
−2
1 x 2 4 x
4
y 6
y = 2x
4 y = 23x 2
1x
y = 25 −4
2
−2
x
4
y 6 y = 2− x
4
y = 2x
2 x −4
−2
2 4 y = −2x
y x 5 y=3 4 3 2 1
y = 2x 1
−3 −2 −1 −1 −2
2
3
x
y 2.5
y = 3x
y = ex
y = 2x
2 1.5 1 0.5 x −2
−1
1
y
y = ax y = loga (x) y=x
x
y 4
(4, 2)
2 (2, 1)
1 −1 (0.25, −1) −1 −2 −3 −4
(8, 3)
f (x) = log2 (x)
3
(4, 1) 1
(0.25, −2)
2
3
4
g (x) = log4 (x)
(8, 1.5)
6
8
x 5
7
y x = –8 (Asymptote)
2 1
−1 −2 f (x) = log2 ( x + 8) − 4
−3 −4 −5 −6 −7 −8
x
(8, 0) 1
−8 −7 −6 −5 −4 −3 −2 −1
(0, −1)
2
3
4
5
6
7
8
9
y
y = ax x y = loga (x) y=x
5 4 3 2 1 −1
−1 −2 −3 −4 −5
y
y 4 3
g ( x)
2
f ( x) 1
2
3
x 4
5
1 g (x) 1 −4 −3 −2 −1 −1 f (x) −2 −3 −4
y 2 1 g ( x) −6 −5 −4−3−2 −1 −1 −2
f ( x) x 1 2 3 4 5 6
x 2
3
4
y
y = ex
e y = In (x) 1 x 1 y=x
e
y 4 3
log2 (x)
2
log3 (x)
1 −1
−1 −2 −3 −4
1
2
3
4
5
6
x
y
y = ex
e y = In (x) 1 x 1 y=x
e
y 4 3
log2 (x)
2
log3 (x)
1 −1
−1 −2 −3 −4
1
2
3
4
5
6
x
7 6 5 4 3 2 1
y
y 4 3 2 g (x) f (x) 1
−1 −1 −2 −3 −4
f (x)
1 x
2 3 4 5 6 7 8
−1
−1
x 1
2
6
7
−4 y 8
8
6
6
4 g (x)
4 2
−6
5
−3
y
−4
4
g (x)
−2
10
−1 −2
3
1
f (x) 2 3 4 5 6 7
f (x)
2 x 8
−3 −2 −1 −2 g (x)
−4 −6 −8
1
2 3 4 5 6
x
y 5 4 3 2
f (x)
1 −1 −2 −3 −4
x 1
2
3
4
5
y
y=
f ( x) = 6 x
1 x +b 10
x x=a
Step 2
Step 1 1
A
B
C
D
E
F
G
h
2
3
4
5
50
100
2
1
3
0.5
4
0.1
5
0.01
6
0.001
7
0.0001
8
0.00001
9
0.000001
Step 3
10
y
y = ex Gradient of the tangent = 1 y=x+1
y
y = ex
5
5
(0, 1) −5 −4 −3 −2 −1 0
10
x 1
2 3 4 5
The value of the function at x = 0 is 1. The gradient of the tangent at this point is 1.
(0, 1) −5 −4 −3 −2 −1 0
Area = 1 1
x
2 3 4 5
The area under the curve up to this point is 1.
Gradient of the tangent = e y = ex
y
10
y = ex
10
5
y
y = ex
5 Area = e
(1, e)
(1, e)
x −5 −4 −3 −2 −1 0
1
x −5 −4 −3 −2 −1 0
2 3 4 5
The value of the function at x = 1 is e. The gradient of the tangent at this point is e.
10
y y = ex
5
1
2 3 4 5
The area under the curve up to this point is e.
Gradient of the tangent = 5 y ≈ 5x − 3.05
≈ (1.609, 5)
10
y
y = ex
5
≈ (1.609, 5) Area = 5 x
x −5 −4 −3 −2 −1 0
1
2 3 4 5
The value of the function at x ≈ 1.609 is 5. The gradient of the tangent at this point is 5.
−5 −4 −3 −2 −1 0
1
2 3 4 5
The area under the curve up to this point is 5.
10
Gradient of the tangent = e
y y = ex
y = ex
5
(1, e) x −4
−2
2
4
y 4 3 2 1 f (x)
x
−4 −3 −2 −1 −1
1
2
3
4
y 2 1
m=1 3 m=1 1
x
−1 −2
(6, In (6) )
(3, In (3) ) 1 (1, 0) 2
−1
m=1 6
y = In (x)
3
4
5
6
7
y
5 x −5
5 −5
10
5 4 3 2 1 −1 −2 −3 −4 −5
y
x 1
2 3 4 5 6 7 8 9 10
P 9000 8000 7000 6000 5000 4000 3000 2000 1000 t 0
1000
2000
3000
4000
5000
y = 4−x
9 8 7 6 5 4 3 2 1
−5 −4 −3 −2 −1 −1 −2
y
y = 4x
1
y=0 x 2 3 4 5
7 6 5 4 3 2 1 −1 −1 −2 −3 −4
y
y 8 6 4 g (x)
2 f (x)
1
2 3 4 5 6 7
x 8
−3 −2 −1 −2 g (x)
−4 −6 −8
f (x) 1
2 3 4 5 6
x
y 4 3 2 1 f (x) −4 −3 −2 −1
x 1 −1
2
3
y
1 (cos (θ ), sin (θ )) θ x −1
0
1
−1
y 2
y = sin (θ )
1 θ −3π
−2π
−π
0
π
2π
3π
−1 −2
y 2
y = cos (θ )
1 θ −3π
−2π
−π
0
π
2π
3π
−1 −2
y 2
y = cos (θ )
1 θ −3π
−2π
−π
0 −1 −2
π
2π
3π
y 2 1 x −3π
−2π
−π
π
2π
3π
π
2π
3π
−1 −2
y 2 1 x −3π
−2π
−π −1 −2
y 2
1 x −3π
−2π
π
−π −1
−2
2π
3π
y
1
x − 2π
− 3π 2
π
−π 2
π 2
3π 2
π
2π
−1
y
1 x − 3π 2
π
−π 2
π 2 −1
π
3π 2
y
1
0
−1
x
π 2
π
3π 2
y y = g (x)
y = f (x)
1
x − 3π 4
−π 2
−π 4
π 4
π 2
−1
y π 2 ,1 1
0
−1
x
π 2
π
3π 2
3π 4
y 5 4 3 2 1 x 0
3π 2
π
π 2 y
1 2 3 1 3 0 −1 3 2 − 3 −1
y = f (x) y = g (x) x π 4
π 2
3π 4
π
5π 4
3π 2
7π 4
y Solutions keep appearing in this direction
x
P Solutions keep appearing in this direction
y 3 2 1 y= sin (x) − 3π 2
x −π
−π 2
π 2
−1
π
3π 2
−2 −3
y 3 2 1 y= 3 sin (x) − 3π 2
x −π
−π 2
−1 −2 −3
π 2
π
3π 2
y
y=3
3 2 1
y= 3 sin (x) − 3π 2
x −π
−π 2
π 2
−1
π
3π 2
−2 −3
y 3 2 1 x − 3π 2
−π
−π 2
−1 −2 −3
π 2
π
3π 2
y (0, 1) − 2, 2 2 2
(−1, 0)
− 2 ,− 2 2 2
3π 4
135°
π 2 90°
π 4 45° 0° 0
π 180° 5π 4
225°
270° 3π 2
2, 2 2 2
315°
7π 4
(0, −1) Multiples of 45°
π 4
(1, 0) x
2 ,− 2 2 2
y (0, 1)
− 1 , 3 2 2
1 , 3 2 2
π 2π π 2 3 3 − 3, 1 5π π 90° 2 2 60° 120° 6 6 30° 150° (−1, 0) π 180° 0° 0 − 3 ,− 1 2 2
7π 6
− 1 ,− 3 2 2
3, 1 2 2 (1, 0) x
330° 210° 240° 270° 300° 11π 3,− 1 6 2 2 4π 3π 5π 3 3 2 1 ,− 3 2 2 (0, −1)
π Multiples of 30° 6
y 3 y= 2
2 1 x − 3π 2
−π 2
−π
π 2
−1
y= −2 sin (x)
3π 2
π
−2 −3
y 1.5 1 0.5 0 −0.5 −1 −1.5
x π 6
π 3
π 2
2π 3
5π 6
π
7π 6
4π 3
3π 2
5π 3
11π 6
y
1 x − 3π 2
−π
−π 2
π 2 −1
π
3π 2
y
1
x − 3π 2
−π
−π 2
π 2 −1
π
3π 2
y
1
A
E y = cos (x) B π 2
−1
D 3π 2
π
C
x 2π
y D
1
y = −sin (x) A
C π 2
−1
B
π
3π 2
E x 2π
=
=
−
−
π ×1−2×0 2 π3 216 π 2
π3 216
du dv v −u d u dx dx = dx v v2
()
y 1.5 1 0.5 0 − 0.5 −1 −1.5
x π 6
π 3
π 2
2π 3
5π 6
π
7π 6
4π 3
3π 2
5π 3
11π 6
2π
y
1
x − 3π 2
−π
−π 2
π 2 −1
π
3π 2
Water level (metres)
9 (0, 8.5) (12, 8.5)
8 7
(6, 6.3)
6
5 12:00 am
4:00 am
8:00 am Time
12:00 pm
y = f (x)
y
B
f ( b)
Secant f ( b) − f (a)
f ( a) 0
A b−a a
x b
y = f (x)
y
Tangent A 0
a
Gradient of tangent = f ' ( a ) x
2r
30 cm r
r
h
r
y Positive gradient
Negative gradient
x Increasing
Decreasing
zero pos
neg pos neg
pos zero
pos
zero
neg zero
neg
Concave up
m=2
m = −2 m = −1
m=0
m=1
y = x2
m=0 m = −1
m=1 m=2
m = −2
Concave down y = −x2
500 400 300 200 100 −13−12−11−10−9 −8 −7 −6 −5 −4 −3 −2 −1 −100 −200 −300 −400 −500 −600 −700 −800 −900 −1000
y
1 2 3 4 5 6 7 8 9 10 11 12 13
x
y 4 3 2 1 −6 −5 −4 −3 −2 −1
−1
−2 −3 −4
x 1
2
y = ex y 10 y = 3x
y = 2x
5 y-intercept (0,1)
x
y=0 −3
−2
−1
1
2
3
Local maximum 40 (−1, 36)
y f (x) = (x − 5) (x − 2) (x + 3)
20 x −4
2
−2 −20
4 Local minimum 11 , 400 − 3 27
6
y 30 25 20 (−1, 13)
(0, 20)
15
(1, 13)
10 5 (−2, 4) −3 −2 −1
1
(2, 4) 2 3
x
y 5 4 3 2 1 x 0.2 0.4 0.6 0.8
1.0
4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9
8
f ( x)
8 6 x
2 3
3
4
−6 −8
g ( x) − 3 −2 − 1 −2 −4 −6 x
1 2 3 4 5
2
−4
f ( x)
2
x 1
−5 −4 −3 −2 −1 −2
4
−4
4
1
6
−5 −4 −3 −2 −1 −2
f ( x)
−8 − 10 − 12 − 14
−6
− 16
−8
− 18
1
x 2
3 2 1 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7
f ( x)
f ( x) 8 6
x 1
2
4 2 −6 −5 −4 −3 −2 −1 −2
f ( x) 5 4 3 2
A
1 B
C 0.2 0.4 0.6 0.8 1.0
x
1 2 3 4
x
y D A x B C
y
y
(a, f (a)) (a, f (a)) 0
x
x 0
y
y
(a, f (a)) x
x 0
0 (a, f (a))
h cm x cm 3 x cm
14 000
Local maximum (13.3, 14 222.2)
Volume (cm3)
12 000 10 000 8000 6000 4000 2000 x (cm) 0
5
10
15
20
Maximum (15, 13 500)
Volume (cm3) 14 000 12 000 10 000 8000 6000 4000 2000
x (cm) 0
5
10
15
20
x x 42 cm x
x x 90 cm
60 cm
w
96 cm
x 3m y
4m
NOT TO SCALE
w
27
d
w h l
13 cm h
r 8 cm
y θ θ
6 cm x
14 cm
9 y θ x
8 x 6 4 2 0 2 −2 −4 −6 −8 −10 −12 −14 −16 −18 −20 −22 −24
10 8 6 4 2 0 −2 −4 −6 −8 −10 −12
t 4
6
4
6
8
10 12 14
x
t 2
8 10 12 14 16
12 9 6 3 0 −3 −6 −9 −12 −15
x
t
4
8
12
16 20 24
6 4 2 0 −2 −4 −6 −8 −10 −12 −14 −16
x
t 1 2 3 4 5 6 7 8 9 10 11
x 4 3 2 1 t 0
1
2
3
4
y
x −2π
− 3π 2
−π
−π 2
π 2
π
3π 2
2π
y
x 2π
− 3π 2
π
−π 2
π 2
π
3π 2
2π
8
f ( x) −3 −2
−1
6
−2
4
−4 −6
2 −5 −4 −3 −2 −1 −2 −4
x 1 2 3 4 5
−8 −10 −12 −14
−6
−16
−8
−18
f ( x) 1
2
3
x
8 6 4 2 0 −2 −4 −6 −8 −10 −11
x
t 2
4
6
8 10 12 14 16
Differentiation x'' ( t) = v' ( t
x' ( t
Displacement x(t
Velocity v(t
∫ v ( t dt = ∫ x' ( t dt
Acceleration a( t
∫ a ( t dt = ∫ v' ( t dt
Integration
Differentiation x'' ( t) = v' ( t
x' ( t
Displacement x(t
Velocity v(t
∫ v ( t dt = ∫ x' ( t dt
Acceleration a( t
∫ a ( t dt = ∫ v' ( t dt
Integration
v 60 40 20 0 −20 −40
t 1
2
3
4
5
6
Tails 1 2 1 2 Heads
1 2 1 2
1 2 1 2
Tails
TT
Heads
TH
Tails
HT
Heads
HH
P X= x 1 3 4 1 2 1 4 0
x 0
1
2
Female
Female
FFF
Male
FFM
Female
FMF
Male
Male
FMM
Female
Female
MFF
Male
MFM
Female
MMF
Male
MMM
Female
Male Male
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
P(x)
x 1
2
3
4
5
6
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
P(x)
x 1
2 3 4 5 6 7 8
0.4
P(x)
0.3 0.2 0.1 0
x 1
2
3
4
5
6
7
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
P(x)
x 1
2
3
4
5
6 7
P(x)
k
x 1
2
3
4
5
6
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
2 P X= x 5
P X= x
1 5
x 1
2
3
4
5
6
0
x 1
2
3
4
5
6
0.5
Distribution A P X= x
0.4 0.3 0.2 0.1 0
x 1
2
3
4
5
6
80 70
Frequency
60 50 40 30 20 10 0 $50 000
$60 000
$70 000 $80 000 Income
$90 000
$100 000
Emma
EEE
0.1
Ben
EEB
0.8
Emma
EBE
0.2
Ben
EBB
Emma 0.9
Emma
BEE
0.8
0.1
0.2
0.8
Ben Emma
BEB BBE
0.2
Ben
BBB
Emma 0.9 Emma 0.9 0.1
0.8
Ben
0.2 Ben
Ben
Spinner A
Spinner B
4
1
4
1
3
2
3
2
X = 1 if the spinner lands on the figure 2 when spun once, and X = 0 otherwise. 5
1
5
4
2
3
4 5
X = 1 if the spinner lands on the figure 3 when spun once, and X = 0 when it lands on the figure 4.
3
1
4
2
3
4 5
3
A
B
X = the numerical result of the figure that the spinner lands on when it is spun once. 2
1
1
2
2
1 1
2 C
X = 1 if the spinner lands on the figure 2 when spun once, and X = 0 otherwise. 5
1
4
2
3
4 5
3 A
G G 1 2
1 2
1 2
1 2
1 2 1 2
B G
1 2
1 2 B
1 2
1 2 B
1 2 1 2 1 2 1 2
G B G B G B G B
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Probability distribution P X= x
x 0
1
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Probability distribution P X= x
x 0
1
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
1
P X= x
0.4096
0.4096
0.1536
0
1
0.0256
0.0016
3
4
2
P X= x
0.96922
0.99757
x
1
0.83692
0.9 0.8 0.7 0.6
0.52822
0.5 0.4 0.3 0.2
0.16807
0.1 0
x 0
1
2
3
4
5
Probability (%)
Graph A
36 32 28 24 20 16 12 8 4 0
0 1 2 3 4 5 6 7 8 9 10 Number of successes
Probability (%)
Probability (%)
36 32 28 24 20 16 12 8 4 0
36 32 28 24 20 16 12 8 4 0
Graph B
0 1 2 3 4 5 6 7 8 9 10 Number of successes
Graph C
0 1 2 3 4 5 6 7 8 9 10 11 Number of successes
P(x)
k
x 1
2
3
4
5
10 9 8 7 6 5 4 3 2 1 0
y
x 1 2 3 4 5 6 7 8 9 10
10 9 8 7 6 5 4 3 2 1 0
y
x 1 2 3 4 5 6 7 8 9 10
y
30
f (x) = x2 + 1
25 20 15 10 5 x 1
30
2
y
3
5
4
30
f (x) = x + 1 2
25
25
20
20
15
15
10
10
5
5
6
y f (x) = x2 + 1
x
x 1
2
3
4
5
6
1
2
3
4
5
6
30
y f (x) = x2 + 1
25 20 15 10 5 x 1
2
3
4
5
6
y 350 300 250 200 150 100 50 x 0
1
2
3
4
5
6
7
y 350 300 250 200 150 100 50 x 0
1
2
3
4
5
6
7
y
y 7
7
6
6
5
5
4
4
3
3
2
2 1
1
x
x 0
1
2
3
4
5
6
7
0
y
1
2
3
4
5
4
5
6
6
7
y 9
5
8 7
4
6
3
5 4
2
3 2
1 0
x 1
2
3
4
5
1 0
x 1
2
3
7
8
9
13 12 11 10 9 8 7 6 5 4 3 2 1
y
x
−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6
16 14 12 10 8 6 4
0
x 2
4
6
8 10 12 14 16
y
x
−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6
f ( x)
2
13 12 11 10 9 8 7 6 5 4 3 2 1
12 11 10 9 8 7 6 5 4 3 2 1
y
x 0 1 2 3 4 5 6 7 8 9 10 11 12
y
f ( x)
6
12
5
10
4
8
3
6
2
4
1
2 x
x 0
1
2
3
4
5
0
6
f ( x)
16 14 12 10 8 6 4 2
x
0
11 10 9 8 7 6 5 4 3 2 1
2
4
6
8 10 12 14 16
10 9 8 7 6 5 4 3 2 1
2
4
6
8
10
12
y
x
0 1 2 3 4 5 6 7 8 9 10
y 16
f ( x)
14 12 10 8 6 4 x 0 1 2 3 4 5 6 7 8 9 10 11
2
x 0
2
4
6
8 10 12 14 16
11 10 9 8 7 6 5 4 3 2 1
y
x 0
1
2
3
4
5
6
7
8
y
2.5 2 1.5 1 0.5
x −1
−0.5
0
0.5
1
1.5
y
y = f (x)
A x a
b
y 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 −5 −4 −3 −2 −1
f (x)
x 1 2 3 4 5 6 7 8 9 10
y 7 6 5 4 3 2 f (x) =
1
x−2
A x
1 −1 −2
2
3
4
5
6
7
8
9
10
11
12
13
14
y
y = f (x)
A b 0
a
x b
y
f (x) = x2
A x 0
a
b
y 10 y = x2 + 2x−3 5
x −5 −4 −3 −2 −1 −5
1
2
3
6
y
4 2 x −6
−4
2
−2 −2
4
6
4 2 −3
−2
4 1
−1 −2
9
y
−4 −6 −8
x 2
5 4 3 B2 1 −5 −4 −3 −2 −1−1 A
−2 −3 −4 −5
y
x 1 2 3 4 5
10
y
y = (x + 2)3 − 1 5
x −4
−3
−2
1
−1
−5
2
y
y = f (x)
A x 0
a
b
y
y 4
3 2
3
f (x) = x − 1
2 1
x
1 x 1
−1 −1
2
3
4
5
−2 −1
−1
−2 −3
1
2
3
4
f (x) = 0.1x3 −2
5
y
y
4 4 3
f (x) =
2
1 (x−2) (x−3) 20 x
2 f (x) =
x−2
2
−2
1
−2 x
0
1
2
3
4
5
y
y = f (x)
A x 0
a
b
4
6
8
10
x 1
2
2
4
−1
3
4
5
6
7
−2 −3 −4 −5 −6 −7
y y
6 4 2 0 −2
x 6
8 10 12 14 16
−4 −6
y 6 −5
x 0
3 7
y
4 x 0
−3
2
9
y A −2
0
x 3
y 4 3 2 1 −1
x 1
2
3
4
5
6
−2 −3 −4
5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5
y
x 1 2 3 4 5
y 6 4 2 −5 −4 −3 −2 −1 −2 −4 −6
x 1 2 3 4 5
y
3
5
x −4
−1
2
3
y
B x −1 A
3
6
C
y
f(x)
x 0 g(x)
y
y y = f (x)
y = f (x)
−a a
−a
a
x
x
y
y = f (x)
x a
b
c
30
y y = e(2x−3)
20
10 x 0
1
2
3
y 2
1
0
−1
−2
x π 8
π 4
3π 8
π 2
5π 8
3π 7π 4 8
π
9π 8
5π 11π 4 8
3π 13π 7π 15π 2 8 4 8
2π
6
y
4 f (x) = ex 2 x −1.5
−1
−0.5
0
0.5
1
1.5
−2
6
y
4
3x
f (x) = e 2 2
x −1.5
−1
−0.5
0
0.5
1
1.5
−2
6
y
4 f (x) = ex2 2 x −1.5
−1
−0.5
0 −2
0.5
1
1.5
1
0.5
y f (x) = cos x +
x 3 x
−2π − 11π − 5π − 3π − 4π − 7π −π − 5π − 2π − π − π − π 0 2 3 6 3 6 6 3 2 3 6
π 6
π 3
2π 3
π 2
5π 6
π
−0.5
−1 1
0.5
y f (x) = cos x +
x 3 x
−2π − 11π − 5π − 3π − 4π − 7π −π − 5π − 2π − π − π − π 0 3 2 3 3 2 3 6 6 6 6
π 6
π 3
2π 3
π 2
5π 6
π
−0.5
−1
1
0.5
y f (x) = cos x +
x 3 x
−2π − 11π − 5π − 3π − 4π − 7π −π − 5π − 2π − π − π − π 0 3 2 3 3 2 3 6 6 6 6 −0.5
−1
π 6
π 3
π 2
2π 3
5π 6
π
y
y 9 3000
6 3
x −2
1
−1
2
3
x 1
−3 −2 −1
−50
1
2 3 4 5 6
3
y
x −2 −1
2
81 54 27
−100
x −2 −1
−150 y
1
2
3
4
5
y
y 3 2
1
1 x
0
π 2
−1
π
3π 2
x
0
π 3
π 6
−1
−2 −3
y
y 3 2
1
1 0
−1
x π 4
π 2
3π 4
0 −1 −2 −3
x π 2
π
3π 2
1
f (x) = sin x + x 4 G π −π − 3π −2 4 4
F
0.5
0 −0.5
D
y
−1
E π 4
x π 2
g(x) = cos x + x 4
3π 4
1.2
y
1 0.8 f (x) = e
x
0.6
f (x) = e−x
0.4 0.2 x −3π 4
−π 2
−π 4
0 −0.2
π 4
π 2
3π 4
y 10 8 6 4 2 x −12 −10 −8 −6 −4 −2
2
4
6
8
10
12
y 35 30 25 20 15 10 5 −20 −15 −10 −5
y 10 8 6 4 2 x 2
4
6
8
10
12
14
x 5 10 15 20
y 3 2.5 2 1.5 1 0.5 −2 −1.5 −1 −0.5
x 0.5 1 1.5 2
y = −x + 3
6 5 4 3 2 1
−6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6
y
y=
x−1 x
1 2 3 4 5 6
y y = f (x)
0
a
y = g (x)
b
x
y y = f (x) y = g (x)
0
a
b
c d
x
y
y = f (x) − g (x)
0
a
b
c d
x
y 8 y = −x + 2
y = − x2 + 8
6 4 2
x 2 4 6 8
−8 −6 −4 −2 −2 −4 −6 −8
y 8 y = −x + 2
6
y = − x2 + 8
4 2
−8 −6 −4 −2 −2 −4 −6 −8
x 2 4 6 8
y 8 y = −x + 2
6
y = − x2 + 8
4 2
−8 −6 −4 −2 −2 −4 −6 −8
x 6 8
4
y
3
f (x) = 4 cos ( x)
2 1
x π 4
−1 −2
π 2
3π 4
π
g (x) = − 4 cos x −
−3
π 2
−4
4
y
3
f (x) = 4 cos ( x)
2 1 −1 −2 −3 −4
x π 4
π 2
3π 4
π
g (x) = − 4 cos x −
π 2
4
y = −x − 3
y
3 2
y= x+5
1 −6 −5 −4 −3 −2 −1 −1
x 1 2 3 4
−2 −3 −4 y y= 3−x
4 3
y=
2x + 8 3
2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4
x 1 2 3 4 5
y 3 2 1 y = e−x
y = e2x 1
−3 −2 −1
5 4 3 2 1
2
x
3
y y = 4 − x2
−5 −4 −3 −2 −1 1 2 3 4 5 −1 −2 −3 −4 y = 1 − x2 −5
x
y y = sin ( x)
1
x π 4
3π 4
π 2
−1
y = cos ( x)
y y = sin ( 2 x)
1 0.5
y = cos ( x) π 6
−0.5
x
π 3
π 2
−1
y 4 y = x4 3 y = x2
2 1
x −2
−1
1
2
y 80
f (x)
40
g (x) x
1 2 3 4 5 6 7 8 9
−1
y
y = sin ( 8 x)
1 0.5
y = cos ( 8 x)
0
π 32
−0.5
x
3π 32
π 16
−1
4 3 2 1
y A x
B π y y = 6x2 + k
A B x
y
g (x)
35 30 Q
25
f (x)
20 15 10
P
5 0
x 2
4
6
8
10
y
f (x) 1
A
2
3
4
5 x
C D
B g (x)
y f (x) A p
x
C B
q
D
r g(x)
8
y
6 4 T3
T2
2
T1 x 2
4
6
8
6
y
4 2 x 1
2
3
4
5
−2 −4 −6
y
3.9 m 0
5.4 m
5.1 m
4.7 m
4.4 m
5.3 m
4.1 m x
1
2
3
4
5
6
6
13 12 11 10 9 8 7 6 5 4 3 2 1 0
y
A1
A2
A3
A4
x
1 2 3 4 5 6 7 8 9
82 m 87 m 89 m
43 m 43 m 43 m
41 m
99 m
y 1 f ( x)
0.5
x 0.5
1
1.5
2
y
f ( c) x 0 a
c
b
120 000
y
100 000 80 000 60 000 40 000 20 000 x 0
1
2
3
4
5
$ / kg 6 5
(0, 5) R' (x)
4 3
(10 000, 4)
C' (x)
2
(10 000, 1.5)
1 x 0
2000
4000
6000
8000
10 000
x' (t)
Displacement x(t)
Differentiation
Velocity v( t )
x'' (t) = v' (t)
Acceleration a( t )
∫ a(t) d t = ∫ v' (t) d t
∫ v (t) d t = ∫ x' (t) d t Integration
4 v (t)
2 (2.5, 0) 2 −2
−4
t 4
x (t) 15 (0, 12) 10 5 (3, 0) −5
(2.5, −0.5) 10
t 15
y 6 4
v(t) = − t2 + 4 t
2 −6 −4 −2 −2 −4 −6
t 2
4
6
y
4
5
3
4
2
3
1
2 1 x −1 −2
y
1
2
f (x) = 2x − 1
3
4
−1 −2 −3 −4
x 1 f (x) = 3x3 − 1
2
4 3 2
y
y
6 5
f (x) = (x − 2)2
4
1 −1 −2
x 1
2
3
4
2 1
−3 −4
f (x) = (x−3) (x−4)
3
−1
x 1
2
3
4
5
6
7
y C −1
0
x 4
6 D
y 2
h (x) = cos x −
−π
−
3π 4
−
π 4
π 2
π k (x) = sin x − 4
1
−
H
G
I
π 4
π 4
F
x 3π 4
π 2
π
−1
−2
y 320 p(x) 240 q(x) 160 80 x 2 4 6
8 10 12 14 16
9
y
y
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1 0
x 1
2
3
4
5
1 0
x 1
2
3
4
5
6
7
8
9
13 12 11 10 9 8 7 6 5 4 3 2 1
y
−5 −4 −3 −2 −1 0 1 2 3 4 5
x
13 12 11 10 9 8 7 6 5 4 3 2 1
y
x −7−6−5−4−3−2−1 0 1 2 3 4 5 6 7
12
y
10 8 6 4 2 0
x 2
4
6
8
10
12
14
y f(x) A s
x
C B
t
D
u g(x)
5
f ( x)
4 3 2 1
−1 −2
x 1
2
3
4
Opposite
e
e
Opposite us en ot yp H
us en ot yp H
θ Adjacent θ Adjacent θ
f mm
11 mm 25°
A
7 C 25
x B
30° 2 45°
3
2
1
45° 1
60° 1
30° 2 45°
3
2
1
45°
60°
1
1
Angle of elevation
Angle of depression
A
D x 50° 50° x
B
C
B θ θ A
Horizontal
B Dir
ect
θ
dis
tan
ce
Dir
ect
θ Horizontal
A
dis
tan
ce
Vertical
Vertical
B
A
A
13°
B
16° C 4000 m
Airport
Angle of elevation Angle of depression
Horizontal
7 θ
24
25
θ
17
15 8
A
13
12
θ B
5
C
f mm 29°
7 cm
f cm
12 mm 30°
4 cm 12 mm
25° g cm
58° f mm
bm 41° 15 m hm 42° 10 m
14 12
x
50
x 5
x
x
12
θ
8
θ 5
6 B
5 4
C
x
A
12 cm
44°
16.68 cm
a cm
B α
D
σ
θ C
A
30 m
θ 23 m
Pole 2
25 m
Pole 1 8m A
B
C
60° 10
x
30°
30°
x
3
45°
14
x
x
4 3
45° y
y
8 3 θ
4 2
12
4
θ
43° hm 70 m
30 m
2.36 m
1.25 m x
68
37°
46° w y
3.38°
1.22° 1600 m
5
30° 30°
A
B 62°
O
C
24 cm
A E 60°
m
B n C 5 7 D
28° 62° h
102°
L
16° 12°
B a
c
A
b
C
a 69° 18
33°
B
9
A 56°
17
x C
15 cm 80° θ 35 cm
62° 42°
10 cm
33°
6 cm
10 cm 6 cm 33°
Possibility 1
Possibility 2
B 30 cm
15 cm
15 cm x
19° A
D
C
a 58°
13
a 63°
15
28°
28°
a
4.5 39°
104° 25°
310
25° a
a
17 45°
27° a
28°
18.3
48°
16 39°
57°
x 20
9 11.2
x
62°
x
11
18
34°
x
58
93
6.58
41°
x
x 116°
3.69
287
275
A
c
63° B
88° C
18 3.52 cm
110° 5.52 cm
x
C
b
a
x
A
B
D c 18 cm 58° 75° 47° θ 42 cm
B
20 cm
30 cm 20 cm x
31° A
D
C
J 68° 25 m
K 28 m x M
L
A
303 m 31°
B
47°
h
C
Pole
Matt
34° Jenny
100 m
52°
58° xm
hm
D
h a
35°
12° A
B
67 m
C
A
c
b
55° B
87° a = 14 m
C
S
18° T
26° 526 km
P
W
B 5 mm
a
45° A
y 10
15
z x
30
D
C
10 9 8 7 6 5 4 3 2 1 0
y
x 1 2 3 4 5 6 7 8 9 10
A
c b
B a
C
C b
A
h
D
a
B
x
10
32°
4
x 7
6
9
C a
13
A
33° 30
B
32 25 w 19
θ 16 cm
10 cm d x
A c b B a C
B 31.05 cm 38.44 cm A
C
C
C 17
A
19.4
a
26°
B
23
a
25°
A
C
C a
131 21 a
A
42° 14
B
B
23.3
A
54° 261
B
16
q
24° 26
53°
23
24
c y
y
9
42°
63.2
7 46°
45.1
51°
21 67°
w
12.3
z 18
54° 15.2
A
7 cm B 9 cm
C
35
D
A
130°
18
x
C
B
25
14
19
12 w
w
35 18.1
5
A
9
7
11.2
A
7
5.2 6
35 25 9.43
B 15
7.8 A
45 4
8
56 40
C
B 6
2m
2.6 m 3.1 m
x
x 82 cm
31 cm P
S x 15 cm
15 cm
Q
20 cm
R
7m
B
19 m
A 127°
82°
14 m
D
C
C
b
a h
A
x
D
B
c−x
c
Q P
R R
4 km
Q
7 km 85° S
4.5 km
40° P
15
C
9 5
B
15 13
A
ym
xm
hm
hm
115 m Path A Path B 24° 46°
21°
5° 3° 308°
130 m 337°
N
B A
C x+2 x2 + 5x + 7 A
B
x+3
P 9 15
5 M
N 6
7
E D 10 cm B A
H
15 cm
C 5 cm
N
L
M
A c
B
b
a
C
B
c
A
a
h b−x
D
x
C
N NW NNW
NNE
WNW
NE ENE
W
E WSW
SW
ESE SSW
SSE
SE
S Starting direction
Direction of rotation
N S
E W Angle rotated by
North
West
A
East 53° B
South
North
West
A
127° East B
South
N
O
32° A
Starting direction
Direction of rotation
N S
E W Angle rotated by
N x y P
49°
S
20 km
P
170°
8.4 km 60° L
80° Q
P 8.4 km
x
170° y
60° 80°
L
Q
P 8.4 km
60° 10°
60° L
170°
80° Q
P 8.4 km
60° 10°
60° L
170°
80°
30° Q
N
N
P
30° O P
O
65°
N
N A
30° O
O
41° A
N
Sydney b 42 km Plane
57 km
N Z d km 10.7 km Y
X b km
335°
A
B 37°
48°
C
End
x km
N
N
5.9 km
a Start
32° 2.2 km
B
18 km
26° A
G
N
x
121°
C
N
55 km
31° R T 312°
N Start 14°
b 12 km
x km 49°
N 16 km
a E
W
S
Checkpoint
N
N
S
B
145°
916 m
P
b
C a
b h C a
73° 6m
2.2 m
A b cm
B
35° 48 cm
C
θ 8 cm
12 cm 101° 7 cm
7 cm 101° 12 cm
50°
44° 13.1 m
12 m
4m
123°
7.4 m
5.5 m
50°
575 m
5.7 m
3m
685 m
101°
9m
44°
50 m
80 m
39°
29° 25°
22 cm 33 cm
107° 24.7 cm
11.6 cm
23.2 cm
17 cm
54°
46°
11.6 cm 79°
155 mm
70°
A b cm 35°
B
C
60 cm
b cm
47° 19.8 cm
52° b cm
19.41 cm
b cm 107° 12.7 cm
5.6 cm A = 7.3 cm2 6.2 cm
x
Radius
113° 8 cm 11 cm 43°
a
a
h
θ
θ b A c b
B a C
A
O
x
B 5 cm
30°
A 10 cm
C
4 cm
B 50° 12 cm D 100°
30° 5 cm
A
C
D
13
E
5
x
F
H
E
θ
β G
F
N
N P
34°
O
O P
68°
N
P
N
62° O
O 23°
P N
N P P 28° O
45° O
N
N A
33° O
O
47° A
N
N A 48° O
O
38° A
N
N A
31° O
O 22° A
x
x
h
θ
θ y
f mm 36°
10 cm
f cm
15 mm 39°
7 cm 16 mm
20° g cm
63° f mm
bm 40° 19 m hm 44° 13 m
41° hm 75 m
40 m
8 30° 30°
a 56°
10
a 66°
12
31°
26°
94°
5.5
a 40°
26°
300
15° a
a
20 48°
a
29°
31°
19.2
50°
18 40°
60°
x 22
10 12.2
x
64°
x
9
16
32°
x
5.6
44°
x
x 120°
260
3.7
A 70° 22 m
B 30 m x D
60
95
C
300
y 15
30
z x 35
C
C
15
23.5
a
24°
20°
A
B
21
A C
a A 18
a
120
22
46° B
B
17.6
C
A
a
55° 270
B
30 11
24
9 w 40
w 15.1
3
9
7
10.6
A A
5 7.8
4
36 26
10.23 6.7
B 16
A
55 5 9 66 50 B
C 7
N
w 225° 120 km
N
N b
Start
4.5 km
165° 8.75 km
y km
End
20 cm
42° 25 cm
52°
33° 30 cm 18.5 cm 68°
A B
3m C
4m
900 m
65° A
Fence
110° 500 m
540 m
P (x)
A
P (x)
x
x P (x)
C
x
B
P (x)
D
x
50
y
Relative frequency (%)
45 40 35 30 25 20 15 10 5 0
x 145
150
155
160 Height (cm)
165
170
175
25
y
Relative frequency (%)
20
15
10
5
x
0 90
95
100
105
110
115
120
Upper limit of score
y 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 x 0 1100 1500 1900 2300 2700 3100
125
130
y
Relative frequency (%)
20
15
10
5
x
0 0
100 200 300 400 500 600 700 800 900 1000 Upper limit of digits (not inclusive)
25
y
Relative frequency (%)
20
15
10
5
x
0 105
115
125
135
145
155
165
175
Upper limit of time spent (minutes)
185
195
P ( X = x)
P ( X = x)
1
1
0.75
0.75
0.5
0.5
0.25 0
x 0
1
2
0.25 0
x 0
1
2
f(w)
f(w)
w
w 70 80 90 100 Weight (g)
70 80 90 100 Weight (g)
Frequency histogram
Frequency histogram with probability density function
f(x)
Shaded area represents probability 1
y = f ( x)
x a
b
f(X)
Probability density
This area is equal to the probability of a < X < b.
X 0
a
b
p (x) 0.40 0.32 0.24 0.16 0.08 x 1 2 3 4 5 6 7 8
p (x) 0.40 0.32 0.24 0.16 0.08 x 1 2 3 4 5 6 7 8
p (x) 0.40 0.32 0.24 0.16 0.08 x 1 2 3 4 5 6 7 8
p (x) 0.40 0.32 0.24 0.16 0.08 x 1 2 3 4 5 6 7 8
p (x) 0.40 0.32 0.24 0.16 0.08 x 1 2 3 4 5 6 7 8
p (x) 3 16 1 8 1 16
x 2 4 6 8 10 12 14 16 18
p (x) 3 16 1 8 1 16
x 2 4 6 8 10 12 14 16 18
p (x) 3 16 1 8 1 16
x 2 4 6 8 10 12 14 16 18
p (x)
1 70
x 10 20 30 40 50 60 70 80 90
p (x)
5 18
x 1
2
3
4
5
6
7
p (x) 2 5 8 25 6 25 4 25 2 25
x 1
2 3 4 5 6 7
8
p (x)
1 10
x 2
4
6
8 10 12 14
p (x)
1 3
x 1
2
3
4
5
6
7
8
p (x) 3 32 1 16 1 32
x 2 4 6 8 10 12 14 16
p (x)
4
2
x 1
2
3
p (x) 4 3 2 1 x 1π 6
1π 3
1π 2
p (x) 0.2
0.1 x 2
4
6
8
10 12 14
y 2 19 1 19 k
x
p (x) 1 4 1 8 x 2
4
6
8
10 12 14
p (x)
x 0
10
20
30
40
50
5 13 4 13 3 13 2 13 1 13
p (x)
p (x) 3 5 2 5 1 5 x 2 4 6 8 10 12 14 16 18 20
1 3 4 15 1 5 2 15 1 15
p (x)
x 5
10
15
20
x 5
10
15
20
p (x) 3 10 1 5 1 10 x
0
5 10 15 20 25 30 35 40 45 p (x) 1 5
3 20 1 10 1 20 x 2
4
6
8 10 12 14
p (x) 3 5 2 5 1 5 k
x
p (x)
1 2
x 4
8
12
16
p (x)
0.5
x 0
1π 2
1π
f (t)
k
a
t
2 1
−2
µ=0 σ = 0.2
1
−1
2 z−score
2 1
−2
−1
µ=0 σ = 0.9
1
2 z−score
µ = 0, σ = 0.3 µ = 0, σ = 0.9
z−score
68% −4 −3 −2 −1
0
1
2
3
4
z−score
2
3
4
z−score
2
3
4
z−score
2
3
4
z−score
95% −4 −3 −2 −1
0
1
99.7% −4 −3 −2 −1
0
1
47.5% −4 −3 −2 −1
0
1
46 53 60 67 74 81 88
Score
46 53 60 67 74 81 88
Score
99.7% 95% 68%
Mean −3sd
Mean −2sd
Mean −1sd
Mean
Mean +1sd
Mean +2sd
Mean +3sd
35
30
30 Percentage
Percentage
25 20 15 10 5 0
Frequency
Percentage
20 15 10 5 80 85 90 95 100 105 110 Weight
Frequency
Frequency
15 10 0
80 85 90 95 100 105 110 Weight
25
10 9 8 7 6 5 4 3 2 1 0
20
5
30
0
25
8 9 10 11 12 13 14 15 16 17 Score
10 9 8 7 6 5 4 3 2 1 0
10 9 8 7 6 5 4 3 2 1 0
80 85 90 95 100 105 110 Weight
8 9 10 11 12 13 14 15 16 17 Score
8 9 10 11 12 13 14 15 16 17 Score
185
190
195
200
205
210
215
145
151
157
163 Height (cm)
169
175
181
Score
38
x
56
65
y
83
92 Exam score
−4 −3 −2 −1 0
1
2
3
4 z−score
−4 −3 −2 −1 0
1
2
3
4 z−score
−4 −3 −2 −1 0
1
2
3
4 z−score
−4 −3 −2 −1 0
1
2
3
4 z−score
−4 −3 −2 −1 0
1
2
3
4 z−score
−4 −3 −2 −1 0
1
2
3
4 z−score
34%
34%
13.5%
13.5%
2.35%
2.35%
0.15% µ−3σ
0.15% µ−2σ
µ−1σ
µ
µ+1σ
µ+2σ
µ+3σ
34%
34%
13.5%
13.5%
2.35%
2.35%
0.15% µ−3σ
0.15% µ−2σ
µ−1σ
µ
µ+1σ
µ+2σ
µ+3σ
1% of students scored IN the 78th percentile 78% of students scored BELOW the 78th percentile
0
10
20
30
21% of students scored ABOVE the 78th percentile
40
50
60
70
80
Some students may score AT the 78th percentile
90
100
Score
99.7% 95% 68%
−3
−2
−1
0
1
2
3
y 3 19 2 19 1 19
k
x
p ( x) 2 5 3 10 1 5 1 10 x 5
10
15 20 25 30 35
p ( x)
1 2
x 2
4
6
8
10
12
166
168
170
172 Height (cm)
174
176
178
Beach
Bushwalking
Beach
Frequency 0
0.2
0.3
p
Frequency
0.1
p 0.06
0.12
0.18
0.24
0.3
Frequency
0
0
p 0.04
0.08
0.12
0.16
Frequency 0
p 0.04
0.08
0.12
0.16
Frequency
3
2
1 0
0.0
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Sample proportions of home team supporters
1.0
p
p
p
p
p
p
p
p
p
y 5 x y=5 4 y
3
5
y=3
x
x
2
( 1, 4)
4 3
1
2 1
y=2
y=0
( 0, 1) 1
−5 −4 −3 −2 −1 −1
y=0
x
−3
−2
2
1
−1
x
3
−1
2 3 4 5
−2 −3 −4 −5
y 5 ( −1, 4)
( 1, 4)
4 3 2 1
−5 −4 −3 −2 −1 −1
(0, 1 ) 1
y=0 x 2 3 4 5 y
−2
12
−3
10
−4
x
y=5 + 3
8
−5
6 4
y=3
2 −3
−2
x 1
−1 −2
2
3
y 18 16
8 6
x
14
y=3 −5
4 2 −2
2
3
8
4
6 4
−2 −4
y = −5 −3
−8
−1
−2
x 1
2
3
1
2 3
y
−7 −6 −5 −4 −3 −2 −1 −4
3
x
−8
2 y=2
−12
(x − 2)
−16 1
−1
−2
y=0
y
−2
y=4
2
−6
1
y = 2x+ 4
12 10
x 1
−1
y
2
y=0 3
x
−20
4
−24
y = −3 −x
−1
−28 −32
−2
−36
−3
y=2
(x + 4)
y=0
18 16 14 12 10 8 6 4 2
−8 −7 −6 −5 −4 −3 −2 −1 −2
y 18 y = 3x + 2 16
y
14 12 10 8 6 4
y=2
2 x 1
−3
−2
−1
−2
x 1
2
3
9 8 7 6 5 4 3 2 1 −3
−2
−1
−3
−2
−1
18 16 x
y = 2 −2
14
−x
y= 3 − 1
12 10 8 6 4
x 1
−1 −2 −3
2
y
y
y
2 3 y = −2
2
−2 −4 −6 −8 −10 −12 −14 −16 −18
2
1
2
18
x 2
1
y
y =2 1
x
−5 −4 −3 −2 −1 −2 y = −1
16
3 y=3
−x
14
+1
12 10 8 6 4
y=1
x
y = −3 + 2
2
−5 −4 −3 −2 −1
−2
x
y 3 −3
−2
−1
−3
x 1
y
2 3 y = −2
18 16
−6 −9
14
−12
12
−15
10
−18
8
−21 −24
y=4
(x − 2)
2
3
6
x
y = −3 − 2
4
−27
2 −1
1
y=0 4 5
x
2 −1
y y=0 1
−2
2
3
−4 −6
4
y 25 y = 5 x
x
20
5
y=5
15
(x − 3)
10
(x − 4)
y = −5
5
−8
x 1
−5 −4 −3 −2 −1 −5
−10 −12
2 3 4 5 y=0
−10
−14 −16
−15 −20
−18
−25
y 5 4 3 2
y=4
(x − 3)
−2
1 −2 −1
−1
20 18 16 14 12 10 8 6 4 2
x 1
2
3
4
5
6
y = −2
−2 −3
−2 −1
y
y
(x − 3)
y=5
−5
x 1
−2 −4 −6
2
3
4
5
y = −5
y=5
5 4 3
1 −5 −4 −3 −2 −1 −1 −2 −3
y
3x y = −2 + 5
2
15 12 9 6 3
x 1
2 3 4 5 −2
−1
−3 −6 −9 −12 −15
y=3×2
x
y=0 1
2
3
4
x
y 15 x
y=3×2 +2
10 5 −2
y=2 1
−1
2
3
4
y=5×2
(x − 3)
x
−5 −10 −15
y 9 8 7 6 5 4 3 2 1 −3
−2
−1
1
−1
y=0 2 3
x
y −3
−2
y = −4
1
−1
(x + 2)
−4 −8 −12
y=0 2 3
x
22 20 18 16 14 12 10 8 6 4 2 −5 −4 −3 −2 −1
−16
y
x 1
2 3 4 5
y 8 6 4 y = e−x −6
−4
2
y = ex 2
−2
x 4
6
−2 −4
y 8
−6 −8
6
g(x)=4–ex
4
h(x)=2+e2x f(x)=–e x −6 −4
2
x 2
−2
4
6
−2 −4 −6 −8 y 6
4
g (x) = 2e x + 1
2
f (x) = e x
h (x) = e 2 x − 1
−4 −3 −2 −1
1
2
3
x 4
y − f ( x + 2) + 3 =−e2( x+2) 8 +7
y 14 12
6
10
4
8
2
6 y = ex + 3
−2
4
f ( x) = e 2x − 4
2
y = ex −6 −5 −4 −3 −2 −1 −2 y = ex − 4 −4
x
−8
4 2 x 4
6
8
10
−2 −4 −6
y 4 2 x −2
1
−1
f ( x) = e2x − 4
2
−4
−10
6
2
−2
x 1
−6
1 2 3 4 5 6
y
−4 −2
−1
−2 −4 −6
−8 2f ( x − 1 ) = 2e2(x − 1) − 8
2
f ( x) = 2e2( x − 1) − 8
y 4 3 2 1 −4 −3 −2 −1
x 1
−1
2
3
4
y = log3 (x)
−2 −3 −4
2
x = −3 (Asymptote)
1 0, −
−3
−2
1 2
(6, 0) 1
−1 −1
f (x) = log9 (x + 3) − 1 −2 −3 −4
2
3
4
5
6
7
8
y 2 1 x 1
−1
2
3
4
5
6
7
8
−1 −2 −3 −4
y 4 3 2 1 f (x) −5 −4 −3 −2 −1 −1 −2 −3 −4
y=x x 1
2 3 4 5
f –1 (x)
x = −1
y
x=1
4
y = f (x)
3 2 1 (− 2, 0) −4 −3 −2 −1
1 −1 −2 −3 −4
2
x ( 2, 0) 3 4
y 3 g (x)
2 1
f (x) x 1
−2 −1 −1 −2
2
3
4
5
6
y 2 x = −2 (Asymptote)
1
f (x) = log4 ( x + 2) − 1 ( 2, 0)
−2
1
−1 −1 −2 −3 −4
1 0, − 2
2
x 3
4
5
6
7
y 2 1 x 1
−1
2
3
4
5
6
7
8
−1 −2 −3 −4
x = −2
y
x=2
4
y = f (x)
3 2
(− 5, 0)
1
−5 −4 −3 −2 −1 −1 −2 −3 −4
( 5, 0) 1
2 3 4 5
x
y 6 5 y = 6x 4 3
y = 3x
y = 4x
2 1 −2
−1
1
y=0 2
1
2
1
2
x
−1
8
y
6 4 2 −2
−1
f ( x) = e3x − 5
−2
x
−4 −6 −8 −10
−12 3f ( x − 1) = 3e3 ( x − 1 ) − 15 y −f ( x + 3) + 4 = −e3(x+3) + 9 8 6 4 2 −2 f ( x) = e 3x − 5
−1
−2 −4 −6 −8 −10
−12 f ( x) = 3e3( x− 1) − 15
x
y
1
0
−1
y = g (x) x π 2
π
y = f (x)
3π 2
y
2
1 x y = sin 4 0
π 6
π 3
π 2
2π 3
5π 6
π
x 7π 6
4π 3
−1 y = 2 cos x − π 3 −2
3π 2
5π 3
11π 6
y 1.5 −1 0.5 x −11π −10π−3π −8π−7π −2π −5π −4π −π −2π − π 3 3 3 3 3 3 3 3
π 3
2π 3
4π 5π 7π 8π 10π 11π π 2π 3π 3 3 3 3 3 3
π 3
2π 3
4π 5π 7π 8π 10π 11π π 2π 3π 3 3 3 3 3 3
−0.5 −1 −1.5
y 1.5 −1 0.5 x −11π −10π−3π −8π−7π −2π −5π −4π −π −2π − π 3 3 3 3 3 3 3 3 −0.5
y = −0.5 y= cos
x 4
−1 −1.5
y 4 3 2 1 x − 5π 6
− 2π 3
− π 2
− π 3
− π 6
−1
π 6
π 3
π 2
2π 3
5π 6
π 6
π 3
π 2
2π 3
5π 6
−2
y 4 y= 5 2
3 2 1 x
− 5π 6
− 2π 3
− π 2
− π 3
− π 6
y = 3 cos(2x) + 1
−1 −2
y 6
5 y = sin x −
π +5 3
4
3
2
1 x − 5π 3
− 4π 3
−π
− 2π 3
−π 3
π 3
2π 3
π
4π 3
5π 3
y y=
11 2
6
5 y = sin x −
π +5 3
4
3
2
1 x − 5π 3
− 4π 3
−π
− 2π 3
−π 3
π 3
2π 3
π
4π 3
5π 3
y
1
x −
3π π −π − 2 2
π 2
π
3π 2
π
3π 2
−1
y 1.5 1 y
0.5 x π − 2
−0.5
π 2
1
−1 −1.5
x − 3π 2
π −π − 2
π 2 −1
y 3 2 1 0 −1 −2
y = 2 cos
x 3
7π 6
4π 3
x π 6
π 3
π 2
2π 3
5π 6
π
y = 3 sin x +
3π 2
5π 3
11π 6
2π
π 4
−3
y
1
x − 3π −π 2
−π 2
π 2 −1
π
3π 2
y 4 3 2 1 x − 5π 6
−π
− 2π 3
−π 2
−π 3
−π 6
−1
π 6
π 3
π 2
2π 3
5π 6
π 6
π 3
π 2
2π 3
5π 6
π
−2
y 4 3 2
y=1
1 x
− 5π 6
−π
− 2π 3
−π 2
−π 3
−π 6
y = 3sin (3x) +1
−2
y
1.5
1
0.5
x −π 2
0
π 2
−1
π
y 1 x −1 −1 −2 −3 −4 −5
1
2 3 4 5 6 7 8 9
y 4 3 2 1 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4
x 1
2
26 24 22 20 18 16 14 12 10 8 6 4 2
y
18 f ( x 16 14 12 10 8 6 4 2
x
−1 −2
1
2
3
4
5
6
7
−1 −2 −4 −6 −8 −10
x
1
9
2 3 4 5 6 7
8 9
f(x
8 7 90
f(x
6 5
80
4
70
3
60
2
50
1
40
−2 −1 −1
30
x 1
2 3 4 5 6 7 8 9
20 f(x
10 −4 −3 −2 −1
x 1
2
3
10
4
5 x −10
−5
5 −5
6 4 2 −1 −2 −4 −6 −8 −10 −12 −14 −16 −18
f(x
−10 −15 x
1 2 3 4 5 6 7 8 9 10 11
−20 −25
10
f(x 8 6 4 2 −4 −3 −2 −1 −2
x 1
2
3
f(x
4
20
−4
15
−6
10
−8
5 −4 −3 −2 −1
x 1
2
3
4
−5 −10
f(x
9 8 7 6 5 4 3 2 1 −6 −5 −4 −3 −2 −1
−15 −20
x 1
−1 −2
2
6 3
f(x 15
−2 − 7 − 3 − 5 −1 − 3 − 1 − 1 4 2 4 4 2 4−3
10 5 x −4 −3 −2 −1
1 −5
−10 −15
2
3
4
−6 −9 −12 −15 −18 −21 −24 −27
f(x x 1 1 3 1 4 2 4
5 4
f(x 40 30 20
f(x
10 1 2 3
−20
x − 31 − 61
x
−9 −8 −7 −6−5−4 −3 −2 −1 −10
1
−30
1 1 1 2 5 1 7 4 3 5 11 6 6 6 6 6 6 3 2 3 6
−40
−1
8 f(x 6 4 2 8 7 6 5 4 3 2 1 −4 −3 −2 −1
−1 −2 −3 −4
−3
f(x
x 1
2
3
−2
−1 −2 −4 −6 −8 −10 −12 −14 −16 −18 −20 −22
x 1
2
3
4
f(x 6 3 −3 −2 −1 −3 −6 −9 −12 −15 −18 −21 −24 −27
x 1
2
3
4
5
12 9 6 3 −4 −3 −2 −1−3
9 8 7 6 5 4 3 2 1
y
x 1
2 3 4 5 6 7
−6 −9 −12 −15 −18 −21 −24 −27 −30 −33
f(x
x 0.5 1
−0.5 −1 −2 −3
1.5 2 2.5 3 3.5
f(x 16 12 8 4
8
−4 3
f(x
−8
6 4
−12
2 −2 −1
−2 −4 −6 −8
−10 −12
−2 3 −4
x 1
2
3
4
5
6
−16
x 2 3
4 3
f(x
f(x
5
8
4
6 4
3
2
2
x
1 x −4 −3 −2 −1
1
2
3
4
−6
−2
−8
y 9 8 7 6 5 4 3 2 1
x 1
−1
2
y 4 3 2 1 x −4 −3 −2 −1 −1
1
−4
−1
10
−2
−8 −6 −4 −2 −2
2 3 4 5 6 7
2
4
6
8
=
0.8 x2
x 25 20 15 10 5 t 0
3
6
9
a (t) (m/s2) 2 1 t (s)
0 π −1 −2
2π
y y 6 4 2 −2 −4 −6 −8 −10
x 4
8 12 16 20 24 28 32 36
18 16 14 12 10 8 6 4 2 −1 −2
x 1
2
3
4
5
6
7
f ( x) 6 f ( x)
4
20 15
2
10
x
5 x 1
−4 −3 −2 −1
2
3
4
− 41
1 4
1 2
3 4
5 4
1
3 2
7 4
−2
−5 −10 −15 −20
f ( x) 25 20 15 10 5 5 − 2 − 23
− 1 − 21 −5 −10 −15 −20 −25 −30 −35 −40 −45
f ( x)
x x
1 2
1
3 2
2
−5 −4 −3 −2 −1 −5
1
2
f ( x)
f ( x)
40
4
30 3
20 10
2 x
−9 −8 − 7 −6 −5 −4 −3 −2 −1 −10
1
1
x
−20
−3
−40
f ( x) 3 2 1 x
−1
0.5
1
1.5
2
−1
1 −1
−30
− 0.5
−2
2.5
2
3
25
f ( x)
20 15 10 5 −6 −4 −2 −5 − 10 − 15
x 2
4
6
8
y 4 3 2 1 x −4 −3 −2 −1
1
2
3
4
1
2
3
4
−1 −2 −3
y 7 6 5 4 3 2 1 −4 −3 −2 −1
x
−1 y 8 7 6 5
y 1
4 3
−4
−2
−2
1
x 1
2
3
4
x 1
−1 −1
2
−4 −3 −2 −1
−3
−3 −4 −5 −6 −7
2
y 7 6 5 4 3 2 1 −9 −8 −7 −6 −5 −4 −3 −2 −1 −1
x 1
y 6 4 2 x 2
−2 −2 −4 −6
4
6
8
10
v 3 2
v(t
1 0
t 2
4
6
8
10
12
6
8
10
12
−1 −2 −3
7
x and v
6 5 x(t
4 3 2
v(t
1 0 −1 −2
t 2
4
y 5 4 3 2 1 x −7 −6 −5 −4 −3 −2 −1 −1 −2
1
2 3
Flower
Flower Vegetable
Flower
Flower Vegetable Flower
Vegetable Flower Vegetable
Vegetable
Flower Vegetable
Vegetable
Male Male
Male Female Male
Female
Female
Male
Male
Female
Female
Female
Male Female
Female
Heads 0.3 0.3
0.7
0.7
0.3 Tails
0.7
Heads
Tails Heads
Tails
Right Right 1 4
3 4 Wrong
1 4
1 4
3 4
3 4
1 4 Wrong
3 4
Right
1 4
1 4
3 4
3 4
1 4 Wrong
3 4
Right
Wrong Right
Wrong Right
Wrong Right
Wrong
0.5
P( X = x)
Distribution B
0.4 0.3 0.2 0.1 0
x 1
2
3
4
5
6 Jo Jo
0.7 0.3
0.6
0.4
0.2 Ky
0.8
Jo
0.7 0.3
0.2 Ky
0.8 Ky
Jo Ky Jo Ky
3 7 4 7
Car
3 7 Dinosaur 4 7
Car
3 7 4 7
Car
3 7 4 Dinosaur 7
Car
Car Car 3 7
4 7
3 7 4 7
Car 3 7 4 Dinosaur 7
Dinosaur
Dinosaur
Dinosaur
Dinosaur
Probability distribution 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
P X= x
x 0
1
Probability distribution P X= x
x 0 1 Probability distribution P X= x
x 0
1
f(x 7 6 5 4 3 2 1 x 0
13 12 11 10 9 8 7 6 5 4 3 2 1 0
1
2
f(x
x 1
2
3
4
f(x 4 3
f(x 30
2
25 20
1 x 0
1
2
3
4
5
6
7
8
15 10 5
f(x
x
7
0
6
1
2
3
4
5
6
5 4 f (x)
3 12
2
10
1 x 0
1
2
3
4
5
6
7
8
9
8 6 4 2 x 0
1
2
3
y 10 8 6 4 2 −5 −4 −3 −2 −1 −2
1
2 3 4 5
x
−4 −6 −8 −10
y 10 8 6 4 2 x −1
1
2
3
4
5
6
1
2
3
4
5
6
7
−2
y 4 3 2 1 −1 −1 −2 −3 −4
x 7
y
y 14
7
12
6
10
5
8
4
6
3
4
2
2 −5 −4 −3 −2 −1 −2
1
2 3 4 5
x
1
x
−5 −4 −3 −2 −1 −1
−4
−2
−6
−3
1
2 3 4 5
1
2 3 4 5
y
y
3
12
2
10
1
8 6
−5 −4 −3 −2 −1 −1
4
−2
x
−3
2
−4
x
−5 −4 −3 −2 −1 −2
1
2 3 4 5
−5 −6
−4
−7
y
y 3
4
2 3
1 x
2
−3
1
−1 −1
1 x −3 −2 −1
−2
1
2
3
4
5
6
−2 −3
2
3
y
1 x
0
π 2
π
3π 2
−1
y
3
2
1
0
x
π 2
π
3π 2
y 3 2 1 0 −1
x π 2
π
3π 2
−2 −3
5 4 3 2 1 −4 −3 −2 −1 −1 −2 −3 −4 y=x −5
y
x 1 2 3 4 5 6 x+y=2
y 7 6 5 4 3 x+y=6
2 1 −1 −1 −2
y = 2x 1
2
x 3 4
5
6
7
8
y 5 18 16 14 12 10 8 6 4 2
4
y
3 2 y = x2
1
1 2 3 4 5
−5 −4 −3 −2 −1 −1
y = 2x y = ( x − 4)2
y=x+2
x
x
−2
1 2 3 4 5 6 7 8 9
−1 −2
y 12 10 8
y 10
6 y = x2
4 2
8 6
−6
−4
x 2
−2 −2
−4 y = x −6
4
4
6
y = x ( x − 4)
2 y = ( x − 2)2 −3 −2 −1
x
1 2 3 4 5 6 7 8 9
y y = −x + 3
9
y = − x2 + 9
6 3 x −5 −4 −3 −2 −1 −3 −6
1 2 3 4 5
y
y 10
2
8
y=x
6 4
0
2
y = x ( x − 4)2 1
−1 −2
x
2 3 4 5 6 7 8 9
−1 −2
−4 −6
3 2 1 −2 −1−1 −2 −3 −4 −5 −6 −7
y
x 1 2 3 4 5 6 7 8 y = −x ( x − 3)2
y = −x
g (x)
1
x π 8
π 4
3π 8
f (x)
v( t ) 3 2 1 0
−1 −2 −3
v( t ) 100 80 60 40 20 t 0
1
2
3
4
5
6
7
8
9
t 2
4
6
8
10
v (t) 15 7.5 t 4
5
10
45 42 39 36 33 30 27 24 21 18 15 12 9 6 3
g(x) 20 15
y
y = 3x
y = (x−6)2 x 3
10 5 x 0
1
2
3
4
6
9
12
15
9 8 7 6 5 4 3 2 1
y = (x+4)2
−7 −6 −5 −4 −3 −2 −1 −1
y y = x2
x 1 2 3
v (t) 100 80 60 40 20 0 1
2
3
4
5
6
7
8
9
t
18 16 14 12 10 8 6 4 2 0 −2 −4 −6 −8
v
t 1
2
3
4
5
6
7
8
P (t) 2 3
1 3 t 1
2
3
y
y
x
x 0
30
y
P (x)
1
x x 0
10 20 30 40 50 60 70
y
x 0
5
10
15
20
Relative frequency (%)
20
y
15
10
5
x
0
0
10
20
30
40
50
60
70
80
90
100
110
120
Relative frequency (%)
Upper limit of time interval (seconds)
40 35 30 25 20 15 10 5 0
y
x 90
100
110
120 130 Time (minutes)
140
150
y 0.1 x 140
150
f (t)
1 15 t 0
1
CRF
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
Hours 750 825 900 975 1050 1125 1200 1275
5
10
15
20
25
p (x)
p (x) 1 20
15 34
3 80
5 17
1 40
5 34
x 1
2
3
4
5
1 80
x
6 20 40 60 80 100 120 140
p (x) 3 16 1 8 1 16
x 2
4
6
8 10 12 14
p (x) 1 5 3 20 1 10 1 20
x 2 4 6 8 10 12 14 16 18
y
1 34
x 0
10 20 30 40 50 60 70
y
1 7
x 0
5
10
15 20 25 30 35
P (x)
1
x 0
30 40 50 60 70 80 90
p ( x)
3 16 1 8 1 16 x 2 4 6 8 10 12 14 16 18
Frequency
3
2
1
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Sample proportions of home team supporters
0.9
1.0