PRE-CALCULUS
.ALGEBRA.
LOGARITHM PROPERTIES OF EXPONENT
𝑦𝑦 = 𝑒𝑒 𝑥𝑥 𝑥𝑥 𝑎𝑎 ∙ 𝑥𝑥 𝑏𝑏 = 𝑥𝑥 𝑎𝑎+𝑏𝑏
ln(𝑦𝑦) = 𝑥𝑥 ln(𝑒𝑒) = 𝑥𝑥 𝑥𝑥 𝑎𝑎
= 𝑥𝑥 𝑎𝑎−𝑏𝑏
𝑥𝑥 𝑏𝑏
ln(𝑒𝑒) = 1
(𝑥𝑥𝑥𝑥)𝑎𝑎 = 𝑥𝑥 𝑎𝑎 𝑦𝑦 𝑎𝑎
(𝑥𝑥 𝑎𝑎 )𝑏𝑏 = 𝑥𝑥 𝑎𝑎𝑎𝑎
𝑦𝑦 = 𝑏𝑏 𝑥𝑥
𝑎𝑎 𝑏𝑏 𝑎𝑎
𝑏𝑏
𝑥𝑥 𝑏𝑏 = √𝑥𝑥 𝑎𝑎 = � √𝑥𝑥 �
𝑙𝑙𝑙𝑙𝑙𝑙𝑏𝑏 (𝑦𝑦) = 𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑏𝑏 (𝑏𝑏) = 𝑥𝑥
𝑎𝑎 𝑎𝑎 𝑎𝑎
√𝑥𝑥 ∙ �𝑦𝑦 = �𝑥𝑥𝑥𝑥
𝒍𝒍𝒍𝒍( ) 𝒍𝒍𝒍𝒍𝒍𝒍( ) 𝑎𝑎
√𝑥𝑥 𝑎𝑎 𝑥𝑥
= �
𝑎𝑎
�𝑦𝑦 𝑦𝑦
ln (𝑥𝑥) log (𝑥𝑥)
ln(x) = = = 𝑙𝑙𝑙𝑙𝑙𝑙𝑒𝑒 (𝑥𝑥)
ln (𝑒𝑒) log (𝑒𝑒)
𝑙𝑙𝑙𝑙𝑙𝑙𝑒𝑒 (𝑥𝑥) = ln(𝑥𝑥) → 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛/𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑙𝑙𝑙𝑙𝑙𝑙 REMAINDER THEOREM
𝐹𝐹(𝑥𝑥)
log (𝑥𝑥) ln (𝑥𝑥)
𝑙𝑙𝑙𝑙𝑙𝑙𝑏𝑏 (𝑥𝑥) = = 𝑥𝑥 − 𝑘𝑘
log (𝑏𝑏) ln (𝑏𝑏)
𝐹𝐹(𝑘𝑘) ℎ𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
Then: (𝑥𝑥 − 𝑘𝑘) 𝑖𝑖𝑖𝑖 𝑛𝑛𝑛𝑛𝑛𝑛 𝑎𝑎 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑜𝑜𝑜𝑜 𝐹𝐹(𝑥𝑥)
𝒍𝒍𝒍𝒍𝒍𝒍() & 𝒍𝒍𝒍𝒍() proper�es
𝑙𝑙𝑙𝑙(𝑥𝑥𝑥𝑥) = 𝑙𝑙𝑙𝑙(𝑥𝑥) + 𝑙𝑙𝑙𝑙(𝑦𝑦)
FACTOR THEOREM
𝑥𝑥
𝑙𝑙𝑙𝑙 � � = 𝑙𝑙𝑙𝑙(𝑥𝑥) − 𝑙𝑙𝑙𝑙(𝑦𝑦) 𝐹𝐹(𝑥𝑥)
𝑦𝑦
𝑥𝑥 − 𝑘𝑘
𝑙𝑙𝑙𝑙(𝑥𝑥 𝑛𝑛 ) = 𝑛𝑛𝑛𝑛𝑛𝑛(𝑥𝑥)
𝐹𝐹(𝑘𝑘) = 0
Then: (𝑥𝑥 − 𝑘𝑘)𝑖𝑖𝑖𝑖 𝑎𝑎 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑜𝑜𝑜𝑜 𝐹𝐹(𝑥𝑥)
An�-𝒍𝒍𝒍𝒍() and an�log
ln (𝑥𝑥)
𝑒𝑒 ln (𝑥𝑥) = 𝑥𝑥
𝑙𝑙𝑙𝑙𝑙𝑙𝑏𝑏 (𝑥𝑥)
𝑏𝑏 𝑙𝑙𝑙𝑙𝑙𝑙𝑏𝑏(𝑥𝑥) = 𝑥𝑥
PRE-CALCULUS
QUADRATIC EQUATION BINOMIAL PROPERTY
𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 = 0
(𝑥𝑥 + 𝑦𝑦)𝑛𝑛
−𝑏𝑏 ± √𝑏𝑏 2 − 4𝑎𝑎𝑎𝑎 # 𝑜𝑜𝑜𝑜 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = (𝑛𝑛 + 1)
𝑥𝑥 =
2𝑎𝑎
𝑁𝑁 𝑡𝑡ℎ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = [ 𝑛𝑛 𝐶𝐶(𝑟𝑟−1) ][𝑥𝑥 𝑛𝑛−(𝑟𝑟−1) 𝑦𝑦 𝑟𝑟−1 ]
𝑏𝑏 2 − 4𝑎𝑎𝑎𝑎 > 0 two real dis�nct roots
𝑏𝑏 2 − 4𝑎𝑎𝑎𝑎 = 0 two real equal roots
𝑏𝑏 2 − 4𝑎𝑎𝑎𝑎 < 0 complex conjugate roots
.TRIGONOMETRY.
Sine, Cosine, and Tangent
Suppose 𝑟𝑟1 𝑎𝑎𝑎𝑎𝑎𝑎 𝑟𝑟2 are the roots:
𝑏𝑏
Sum of Roots: 𝑟𝑟1 + 𝑟𝑟2 = −
𝑎𝑎
𝑐𝑐
Product of Roots : 𝑟𝑟1 ∙ 𝑟𝑟2 =
𝑎𝑎
DEGREE OF AN EQUATION
3𝑥𝑥 4 𝑦𝑦 − 2𝑥𝑥 3 𝑧𝑧 4 + 7𝑦𝑦𝑧𝑧 5 → 7𝑡𝑡ℎ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
𝑐𝑐𝑐𝑐𝑐𝑐(𝑥𝑥) = 𝑠𝑠𝑠𝑠𝑠𝑠(𝑥𝑥 + 90)
si n(𝑥𝑥)
FACTORING USING SYNTHETIC DIVISION 𝑡𝑡𝑡𝑡𝑡𝑡(𝑥𝑥) =
co s(𝑥𝑥)
𝑥𝑥 3 + 𝑥𝑥 2 − 2
𝑥𝑥 3 + 𝑥𝑥 2 + 0𝑥𝑥 − 2
Cosine is an even fxn because its graph is symmetric about the y-
1 1 0 −2 axis.
(𝑥𝑥 − 1) Even Func�on: 𝐹𝐹(−𝑥𝑥) = 𝐹𝐹(𝑥𝑥)
2 1 � 𝑜𝑜𝑜𝑜 � 𝑖𝑖𝑖𝑖 𝑡𝑡ℎ𝑒𝑒 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
(𝑥𝑥 − 2)
Odd Func�on: 𝐹𝐹(−𝑥𝑥) = −𝐹𝐹(𝑥𝑥)
2 1 1 0 −2
2 6 12
1 3 6 10 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 ≠ 0
RIGHT TRIANGLE
𝑥𝑥 − 2 𝑖𝑖𝑖𝑖 𝑛𝑛𝑛𝑛𝑛𝑛 𝑎𝑎 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
𝑺𝑺𝑺𝑺𝑺𝑺 − 𝑪𝑪𝑪𝑪𝑪𝑪 − 𝑻𝑻𝑻𝑻𝑻𝑻
1 1 1 0 −2
𝒄𝒄𝟐𝟐 = 𝒂𝒂𝟐𝟐 + 𝒃𝒃𝟐𝟐
1 2 2
1 2 2 0 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 = 0
𝑥𝑥 2 + 2𝑥𝑥 + 2 𝑥𝑥 − 1 𝑖𝑖𝑖𝑖 𝑎𝑎 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
Therefore:
𝑥𝑥 3 + 𝑥𝑥 2 − 2 = (𝑥𝑥 − 1)(𝑥𝑥 2 + 2𝑥𝑥 + 2)
PRE-CALCULUS
OBLIQUE TRIANGLE
Square of a Trigo Fxn
Sine Law 𝑠𝑠𝑠𝑠𝑛𝑛2 (𝜃𝜃) =
1
[1 − cos(2𝜃𝜃)]
2
𝑎𝑎 𝑏𝑏 𝑐𝑐
= = 1
𝑆𝑆𝑆𝑆𝑆𝑆(𝐴𝐴) 𝑆𝑆𝑆𝑆𝑆𝑆(𝐵𝐵) 𝑆𝑆𝑆𝑆𝑆𝑆(𝐶𝐶) cos2 (𝜃𝜃) = [1 + cos(2𝜃𝜃)]
2
1 − co s(2𝜃𝜃)
𝑡𝑡𝑡𝑡𝑛𝑛2 (𝜃𝜃) =
1 + co s(2𝜃𝜃)
Cosine Law
𝑎𝑎2 = 𝑏𝑏2 + 𝑐𝑐 2 − 2𝑏𝑏𝑏𝑏 ∙ 𝑐𝑐𝑐𝑐𝑐𝑐(𝐴𝐴)
Half Angles Trigo Fxn
𝑏𝑏2 = 𝑎𝑎2 + 𝑐𝑐 2 − 2𝑎𝑎𝑎𝑎 ∙ 𝑐𝑐𝑐𝑐𝑐𝑐(𝐵𝐵)
𝜃𝜃 1
𝑐𝑐 2 = 𝑎𝑎2 + 𝑏𝑏2 − 2𝑎𝑎𝑎𝑎 ∙ 𝑐𝑐𝑐𝑐𝑐𝑐(𝐶𝐶) sin � � = � [1 − cos(𝜃𝜃)]
2 2
𝜃𝜃 1
cos � � = � [1 + cos(𝜃𝜃)]
TRIGONOMETRIC IDENTITIES 2 2
𝜃𝜃 1 − co s(𝜃𝜃) sin (𝜃𝜃)
Pythagorean Rela�ons tan � � =
2 sin (𝜃𝜃)
=
1 + co s(𝜃𝜃)
𝑠𝑠𝑠𝑠𝑛𝑛2 (𝜃𝜃) + 𝑐𝑐𝑐𝑐𝑠𝑠 2 (𝜃𝜃) = 1
𝑠𝑠𝑠𝑠𝑐𝑐 2 (𝜃𝜃) − 𝑡𝑡𝑡𝑡𝑛𝑛2 (𝜃𝜃) = 1
Product of Two Func�on
𝑐𝑐𝑐𝑐𝑐𝑐 2 (𝜃𝜃) − 𝑐𝑐𝑐𝑐𝑡𝑡 2 (𝜃𝜃) = 1
2 sin(𝐴𝐴) sin(𝐵𝐵) = cos(𝐴𝐴 − 𝐵𝐵) − cos(𝐴𝐴 + 𝐵𝐵)
2 cos(𝐴𝐴) cos(𝐵𝐵) = cos(𝐴𝐴 + 𝐵𝐵) + cos(𝐴𝐴 − 𝐵𝐵)
Sum and Difference of Angles
2 sin(𝐴𝐴) cos(𝐵𝐵) = sin(𝐴𝐴 + 𝐵𝐵) + sin(𝐴𝐴 − 𝐵𝐵)
𝑠𝑠𝑠𝑠𝑠𝑠(𝐴𝐴 ± 𝐵𝐵) = 𝑠𝑠𝑠𝑠𝑠𝑠(𝐴𝐴)𝑐𝑐𝑐𝑐𝑐𝑐(𝐵𝐵) ± co s(𝐴𝐴) si n(𝐵𝐵)
𝑐𝑐𝑐𝑐𝑐𝑐(𝐴𝐴 ± 𝐵𝐵) = 𝑐𝑐𝑐𝑐𝑐𝑐(𝐴𝐴)𝑐𝑐𝑐𝑐𝑐𝑐(𝐵𝐵) ∓ si n(𝐴𝐴) si n(𝐵𝐵)
Sum of Two Func�on
− → 2 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 (𝑠𝑠𝑠𝑠𝑠𝑠 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠)
1 1
𝑡𝑡𝑡𝑡𝑡𝑡(𝐴𝐴) ± 𝑡𝑡𝑡𝑡𝑡𝑡(𝐵𝐵) sin(𝐴𝐴) + sin(𝐵𝐵) = 2 sin � (𝐴𝐴 + 𝐵𝐵)� cos � (𝐴𝐴 − 𝐵𝐵)�
𝑡𝑡𝑡𝑡𝑡𝑡(𝐴𝐴 ± 𝐵𝐵) = 2 2
1 ∓ 𝑡𝑡𝑡𝑡𝑡𝑡(𝐴𝐴)𝑡𝑡𝑡𝑡𝑡𝑡(𝐵𝐵)
𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 1 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 1 1
cos(𝐴𝐴) + cos(𝐵𝐵) = 2 cos � (𝐴𝐴 + 𝐵𝐵)� cos � (𝐴𝐴 − 𝐵𝐵)�
2 2
sin (𝐴𝐴 + 𝐵𝐵)
tan(𝐴𝐴) + tan(𝐵𝐵) =
Double of Angles cos (𝐴𝐴)cos (𝐵𝐵)
Derive from sum of angles (𝐴𝐴 = 𝐵𝐵)
sin(2𝜃𝜃) = 2 sin(𝜃𝜃) cos (𝜃𝜃)
Difference of Two Func�on
cos (2𝜃𝜃) = cos2 (𝜃𝜃) − sin2 (𝜃𝜃)
1 1
sin(𝐴𝐴) − sin(𝐵𝐵) = 2 cos � (𝐴𝐴 + 𝐵𝐵)� sin � (𝐴𝐴 − 𝐵𝐵)�
2tan (𝜃𝜃) 2 2
tan(2𝜃𝜃) =
1 − tan2 (𝜃𝜃)
1 1
cos(𝐴𝐴) − cos(𝐵𝐵) = 2 sin � (𝐴𝐴 + 𝐵𝐵)� sin � (𝐴𝐴 − 𝐵𝐵)�
2 2
sin (𝐴𝐴 − 𝐵𝐵)
tan(𝐴𝐴) − tan(𝐵𝐵) =
cos (𝐴𝐴)cos (𝐵𝐵)
PRE-CALCULUS
.Plane and Solid Geometry. .ANALYTIC GEOMETRY.
RECTANGULAR COORDINATE SYSTEM
𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨
AKA Cartesian Coordinates System
𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 1
𝐴𝐴 = 𝑏𝑏ℎ
2
𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 1
𝐴𝐴 = (𝑏𝑏1 + 𝑏𝑏2 ) × ℎ
2
𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 1 1
𝐴𝐴 = 𝑟𝑟𝑟𝑟 = 𝑟𝑟 2 𝜃𝜃
2 2
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝐴𝐴 = 𝜋𝜋𝑟𝑟 2 y ordinate range (YOR)
x abscissa domain (XAD)
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐴𝐴 = 𝜋𝜋(𝑟𝑟𝑜𝑜 2 − 𝑟𝑟𝑖𝑖2 )
Lines
𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐴𝐴 = 𝜋𝜋𝜋𝜋𝜋𝜋 1. Distance between Two Points
𝑑𝑑 = �(𝑥𝑥2 − 𝑥𝑥1 )2 + (𝑦𝑦2 − 𝑦𝑦1 )2
𝑽𝑽𝑽𝑽𝑽𝑽𝑽𝑽𝑽𝑽𝑽𝑽 (𝑨𝑨𝒃𝒃 × 𝒉𝒉)
2. Slope of a line
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑉𝑉 = 𝜋𝜋𝑟𝑟 2 ℎ
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏) 1
𝑉𝑉 = 𝜋𝜋𝑟𝑟 2 ℎ
3
𝑆𝑆𝑆𝑆ℎ𝑒𝑒𝑒𝑒𝑒𝑒 4
𝑉𝑉 = 𝜋𝜋𝑟𝑟 3
3
𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 4 ∆𝑦𝑦 𝑦𝑦2 − 𝑦𝑦1
𝑉𝑉 = 𝜋𝜋𝜋𝜋𝜋𝜋𝜋𝜋 𝑚𝑚 = tan(𝜃𝜃) = =
3 ∆𝑥𝑥 𝑥𝑥2 − 𝑥𝑥1
𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒: 𝑐𝑐 = 𝑎𝑎 𝑜𝑜𝑜𝑜 𝑏𝑏
PRE-CALCULUS
3. Division of Line Segment 7. Area by Coordinates
1 𝑥𝑥1 𝑦𝑦1 1
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 = �𝑥𝑥2 𝑦𝑦2 1�
2 𝑥𝑥 𝑦𝑦3 1
3
8. Equa�on of Lines
∆𝑦𝑦 𝑦𝑦2 − 𝑦𝑦1
𝑚𝑚 = =
∆𝑥𝑥 𝑥𝑥2 − 𝑥𝑥1
i. Point-Slope Form
𝑦𝑦 − 𝑦𝑦1 = 𝑚𝑚(𝑥𝑥 − 𝑥𝑥1 )
4. Distance between a Line and a Point
ii. Two-Point Form
𝐴𝐴𝐴𝐴 + 𝐵𝐵𝐵𝐵 + 𝐶𝐶
𝑑𝑑 =
±√𝐴𝐴2 + 𝐵𝐵2 𝑦𝑦2 − 𝑦𝑦1
𝑦𝑦 − 𝑦𝑦1 = � � (𝑥𝑥 − 𝑥𝑥1 )
𝑥𝑥2 − 𝑥𝑥1
5. Distance between Two Parallel Lines
𝐶𝐶2 − 𝐶𝐶1 iii. Slope – y-Intercept Form (𝒚𝒚 = 𝒃𝒃, 𝒙𝒙 = 𝟎𝟎)
𝑑𝑑 =
±√𝐴𝐴2 + 𝐵𝐵2
𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑏𝑏
6. Angle between Two Lines
iv. Intercept Form (𝒚𝒚 = 𝒃𝒃, 𝒙𝒙 = 𝟎𝟎 | 𝒙𝒙 = 𝒂𝒂, 𝒚𝒚 = 𝟎𝟎)
𝑚𝑚2 − 𝑚𝑚1
tan(𝜃𝜃) = 𝑥𝑥 𝑦𝑦
1 + 𝑚𝑚2 𝑚𝑚1 + =1
𝑎𝑎 𝑏𝑏
v. General Equa�on
𝐴𝐴𝐴𝐴 + 𝐵𝐵𝐵𝐵 + 𝐶𝐶 = 0
PRE-CALCULUS
CONIC SECTION PARABOLA
If this term exist, the principal axis (symetry) is not
perpendicular or parallel to x or y axis.
ELLIPSE
STANDARD EQUATIONS
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃: (𝑥𝑥 − ℎ)2 = ±4𝑎𝑎(𝑦𝑦 − 𝑘𝑘)
HYPERBOLA
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶: (𝑥𝑥 − ℎ)2 + (𝑦𝑦 − 𝑘𝑘)2 = 𝑟𝑟 2
(𝑥𝑥 − ℎ)2 (𝑦𝑦 − 𝑘𝑘)2
𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸: + = 1, 𝐴𝐴 > 𝐶𝐶
𝑎𝑎2 𝑏𝑏 2
(𝑦𝑦 − 𝑘𝑘)2 (𝑥𝑥 − ℎ)2
+ = 1, 𝐴𝐴 > 𝐶𝐶
𝑎𝑎2 𝑏𝑏 2
(𝑥𝑥 − ℎ)2 (𝑦𝑦 − 𝑘𝑘)2
ℎ𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦: − = 1, 𝐴𝐴 > 𝐶𝐶
𝑎𝑎2 𝑏𝑏 2
(𝑦𝑦 − ℎ)2 (𝑥𝑥 − 𝑘𝑘)2
− = 1, 𝐴𝐴 > 𝐶𝐶
𝑎𝑎2 𝑏𝑏 2
Shortcuts:
𝐷𝐷 𝐸𝐸
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 (ℎ, 𝑘𝑘) = �− ,− �
2𝐴𝐴 2𝐶𝐶
PRE-CALCULUS
𝑓𝑓 Discrimant Latus Rectum (LR) Major Axis (a)
𝑒𝑒 =
𝑑𝑑 𝐵𝐵2 − 4𝐴𝐴𝐴𝐴
Parabola 𝑒𝑒 = 1 𝐵𝐵2 − 4𝐴𝐴𝐴𝐴 = 0 4𝑎𝑎 −𝐷𝐷
𝑦𝑦 2 → 𝑎𝑎 =
4𝐶𝐶
−𝐸𝐸
𝑥𝑥 2 → 𝑎𝑎 =
4𝐴𝐴
Circle 𝑒𝑒 = 0 𝐵𝐵2 − 4𝐴𝐴𝐴𝐴 < 0,
𝐹𝐹
𝐴𝐴 = 𝐶𝐶 𝑟𝑟 = �ℎ2 + 𝑘𝑘 2 −
𝐴𝐴
Ellipse 𝑒𝑒 < 1 𝐵𝐵2 − 4𝐴𝐴𝐴𝐴 < 0, 2𝑏𝑏 2 𝑐𝑐
𝑎𝑎 = 𝑑𝑑𝑑𝑑 =
𝑎𝑎 𝑒𝑒
±𝐴𝐴 ≠ ±𝐶𝐶
Hyperbola 𝑒𝑒 > 1 𝐵𝐵2 − 4𝐴𝐴𝐴𝐴 > 0 2𝑏𝑏 2 𝑐𝑐
𝑎𝑎 = 𝑑𝑑𝑑𝑑 =
𝑎𝑎 𝑒𝑒
±𝐴𝐴 ≠ ∓𝐶𝐶
PRE-CALCULUS
.INVERSE FUNCTIONS. .EVEN and ODD FUNCTIONS.
Inverse of a Func�on a func�on → 𝑭𝑭−𝟏𝟏 (𝒙𝒙) CALTECH:
𝐹𝐹(𝑥𝑥) +1 → 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
i. Interchange y and x. � −1 → 𝑜𝑜𝑜𝑜𝑜𝑜
ii. Isolate y. 𝐹𝐹(−𝑥𝑥)
≠ ±1 → 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛ℎ𝑒𝑒𝑒𝑒
SUM and DIFFERENCE:
𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ± 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸
𝑂𝑂𝑂𝑂𝑂𝑂 ± 𝑂𝑂𝑂𝑂𝑂𝑂 = 𝑂𝑂𝑂𝑂𝑂𝑂
𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ± 𝑂𝑂𝑂𝑂𝑂𝑂 = 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 (𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑜𝑜𝑜𝑜𝑜𝑜 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑖𝑖𝑖𝑖 𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧)
Product and Quo�ent:
𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸
𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 × 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸
Inverse of Trigonometric func�ons 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸
𝑂𝑂𝑂𝑂𝑂𝑂
𝑂𝑂𝑂𝑂𝑂𝑂 × 𝑂𝑂𝑂𝑂𝑂𝑂 = 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸
1 𝑂𝑂𝑂𝑂𝑂𝑂
csc −1 (𝑥𝑥) = sin−1 � �
𝑥𝑥 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸
𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 × 𝑂𝑂𝑂𝑂𝑂𝑂 = 𝑂𝑂𝑂𝑂𝑂𝑂 = 𝑂𝑂𝑂𝑂𝑂𝑂
𝑂𝑂𝑂𝑂𝑂𝑂
1
sec −1 (𝑥𝑥) = cos −1 � �
𝑥𝑥
1 Deriva�ves:
cot −1 (𝑥𝑥) = tan−1 � �
𝑥𝑥 𝑑𝑑
𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 𝑂𝑂𝑂𝑂𝑂𝑂
𝑑𝑑𝑑𝑑
𝑑𝑑
𝑂𝑂𝑂𝑂𝑂𝑂 = 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸
Deriva�ve of Inverse func�on 𝑑𝑑𝑑𝑑
𝑑𝑑 −1 1 Composi�on:
𝐹𝐹 (𝑥𝑥) =
𝑑𝑑𝑑𝑑 𝑑𝑑 • The composi�on of two even func�ons is even, and the
𝐹𝐹(𝑥𝑥)
𝑑𝑑𝑑𝑑 composi�on of two odd func�ons is odd.
• The composi�on of an even func�on and an odd func�on is
even.
