SAT Math2 Formula

SAT Math II Formula 1 Real Numbers Rational Numbers

Integers Whole Numbers Irrational Numbers Natural Numbers

Absolute Value

n-th root (n even)

 a a  0 a a  0

a  

n

Ex) 3  3 , 3  3

n-th root (n odd)

 a a  0 a a  0

an   

2 3 3,

2

3  3

n

3

a

3 3 3,

n

 a 3

3 3  3

Invented Operations - Follow the directions and plug in. If x  y  4 x 

Percent change :

2 y 3

, then 3  6  4  3 

Amount Change Original

2 6 3

 12  4  16 .

100%

Percent-increase / Percent Decrease Final Amount = Original × (1 + Rate )

number of change

; Rate가 + 면 increase / – 면 decrease

The compound interest

 r  A = P  1 +   n A = Pe rt

nt

A : amount of principal and interest, P : principal, r : annual interest rate t : # of years, n : # of compounding in a year( k=4 quarterly, k=12 monthly, etc. )

Continuously compounding ( as k → ∞ )

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SAT Math II Formula 2

Imaginary numbers : i  1 Ex) i 2  1 , i 3  i , i 4  1 , i 5  i ,

Complex Numbers : a  bi where

a



and b are real numbers.

Conjugate : a  bi  a  bi c  di

Rationalization : Reciprocal :



a  bi

1 a  bi bi



c  di a  bi

1 a  bi bi





a  bi bi a  bi

a  bi bi



a  bi bi



ac  bd    ad  bc i a 2  b2

a  bi bi a 2  b2

Complex Plane

Im z = a + bi

bi z

θ a

O

Absolute value (or modulus) of complex number nu mber

Re

z  a  bi  a 2  b 2

Vectors  





A vector in a plane : V  a, b  ai  b j  







A vector in a space : V  a, b, c  ai  b j  c k Magnitude (or Norm) of vector

y    V = a, b = ai + b j  V =

 j = 1, 0  O i = 1, 0

a +b 2

x

2

   2 2 V  a, b , V  a  b    2 2 2 V  a , b, c , V  a  b  c  V is unit vector , then V  1 .  



Resultant vector 

 

V  a, b , U  c, d  

kV  ka , kb  



V  U  a, b  c, d  a  c, b  d  



V  U  a, b  c, d  a  c, b  d

Counting Factorial : n!  n  n  1n  2 3 21 , 0!  1 

n!

Permutation : n Pr 

Combination :

n

n  r !

 n n 1 n  r  2 n  r  1, n Pn  n!

 n P n! Cr     n r  , C  C r!  r  n  r ! r ! n r n nr

Matrices Adding, Subtracting, and scalar multiplication a b   e    c d   g

f 

 a  e b  f     , h  c  g d  h

a b   e      c d   g

f 

 a  e b  f  a b ka kb      , k          h   c  g d  h  c d  kc kd 

Matrix multiplication  p q  a b c     ap  br  ct aq  bs  cu   r s       dp  er  ft f t dq  es  fu fu  d e f    t u 

2  3  3 2   2 2 1

Inverse matrix : XX

Determinant :

1

a

b

c

d


X

1

X  I ,

a b    1   d b   ad  bc c a   c d  a b c

 ad  bc ,

d e f  aei  bfg  cdh  bdi  afh  ceg g h i


Arithmetic Sequences and Series a1

a2

a3

a4



a1

a1  d

a1  2d

a1  3d



an a1  n  1d

n th term : an  a1  n  1d

Sum of the first n terms : S n 

a1  an  n 2

 2a1  n 1 d   n   2

Geometric Sequences and Series a1

a2

a3

a4



an

a1

a1 r

a1r 2

a1r 3



a1r n1

n1 n th term : an  a1r

a1  1  r

n

Sum of the first n terms : S n 



1  r

Sum of infinite geometric series : S 

a1 1  r

n  1 1  r  1

Direct and Indirect Variation Direct variation

Indirect variation y 

y  kx

k x

x varies directly as y

x varies indirectly as y

x and y change proportionally

x and y are inversely proportional

x and y are in proportion

x and y are in inverse proportion

Quantity

x y

will always have the same value

Quantity xy will always have the same value

Travel, Work, and Average

Distance time

speed

Work time

rate of work

Total # of average things



Basic Exponential Properties x ⋅ x = x a

b

( x ) a

b

x a

a +b

x

= x ab

b

= x a −b

x − a =

x = 1 0

x a ⋅ y a = ( xy )

1 x a a

1

a

b

x  x 2

x a  x b

Basic Logarithmic Properties  p   = log a p − loga q q

log a ( p ⋅ q ) = log a p + log a q

log a y p x =

x

log a p =

log a p

y

log a 1 = 0 , log a a = 1

log a 

logb p

a

logb a

loga p

=p

Factoring 2

2

x 2  2 xy  y 2   x  y 

x 2  2 xy  y 2   x  y 

x 2  y 2   x  y  x  y 

Quadratic Formula b  b 2  4ac

 a  0 2a b c where α and β are zeros α + β = − , αβ = a a

2

For ax  bx  c  0

Discriminant :

x 

b2  4ac  0 two real, unequal roots b2  4ac  0 one real root (a double root) b2  4ac  0 no real roots (two complex, unequal roots a  bi )

Binomial Theorem n

n

 x  y    n Cnk x nk y k k 0

= nC0 x n y 0 n C1 x n1 y n n Cn2 x n2 y 2  n C 2 x 2 y n2 n C1 x1 y n1 n C 0 x 0 y n



Odd and Even Functions Odd function : f x    f  x  , symmetric with respect to the origin Even function : f x  f  x , symmetry with respect to the y -axis

Compound Functions

( f + g ) ( x ) = f ( x) + g ( x)

( f − g ) ( x ) = f ( x ) − g ( x)

( f ⋅ g ) ( x ) = f ( x ) × g ( x)

f ( x )  f    ( x ) = g ( x)  g 

( f

( g ( x ) ≠ 0)

 g ) ( x ) = f ( g ( x) )

Inverse Functions f − has property that 1

( f



f −1

)( x) = ( f −

1



f

)(x) = x

Horizontal line test : the inverse of the one-to-one function is also a function. 1 Graph of f and f − are reflections of each other with respect to the line y = x .

f −1 ( x ) = y ⇔ x = f ( y )

Graphs of Basic Functions

y

O

y

y

y

x

x

O

1

x

O

x

O

y = x

Domain Range

( −∞, ∞ ) ( −∞, ∞ )

y = x Domain Range

2

( −∞, ∞ ) [0, ∞ )

y = x Domain Range

3

( −∞, ∞ ) ( −∞, ∞ )

y = a

x

Domain Range

( a > 1) ( −∞, ∞ ) ( 0, ∞ )



Graphs of Basic Functions - continued

y

y

y

y =

x

=x

1

1

−1

y =

Domain

( −∞, 0 )  ( 0, ∞) Range ( −∞, 0 )  ( 0, ∞)

y =

x = x2

Domain Range

y

O

x

1

x

O

1

x

O

x

O

y

[0, ∞ ) [0, ∞ )

3

Domain Range

y = log a x ( a > 1)

x = x3

( −∞, ∞ ) ( −∞, ∞ )

Range

y

y

( 0, ∞ ) ( −∞, ∞ )

Domain

y

c O

x

O

x O

x

x

O

y = [ x ] y = 0

( −∞, ∞ ) Range {0}

Domain

y = x

y = c

( −∞, ∞ ) Range {a}

Domain

Domain Range

greatest integer function

( −∞, ∞ ) [0, ∞ )

Domain

( −∞, ∞ )

Range n ( n is a integer)

Linear Functions (Lines) Slope-intercept Slope-intercept form : f ( x ) = mx + b , slope  m 

rise run

=

changes in y changes in x



y2  y1 x2  x1

A horizontal line : slope of 0 A vertical lines : undefined slope Parallel lines : m1  m2 Perpendicular Perpendicular lines : m1m2  1


m

θ

rise run

 tan 


Quadratic Functions 2

Standard form : y = a ( x − h ) + k vertex ( h, k ) , axis of symmetry x = h , minimum(or maximum) k 2 General form : y = ax + bx + c where a ≠ 0 , b and c are constant 2

2 b  b − 4 ac  y = ax + bx + c = a  x +  − 2a  4a  2

Higher-Degree Higher-Degree Polynomial Functions A polynomial function of x with degree n is given by P ( x ) = an x n + an −1 x n −1 +  a2 x 2 + a1 x1 + a0

n odd, an > 0

n odd, an < 0

n even, an > 0

n even, an < 0

Remainder theorem : P ( x ) is divided by ( x − r ) , then the remainder, R = P ( r ) Factor theorem : P ( k ) = 0 ⇔ P ( x ) has a factor of ( x − k ) Rational root theorem : all possible pos sible rational roots are derived using x = ±

p q

p ; factor of the constant term a0 , q ; factor of the leading coefficient an

Complex zeros occurs in conjugate pairs : a + bi ( b ≠ 0 ) is a zero ⇔ a − bi is a zero.

Vertical asymptotes y =

After simplify

f ( x ) g ( x )

f ( x ) g ( x )

, vertical asymptote occurs at x = a that satisfy g ( a ) = 0



f ( x )

Horizontal asymptotes y =

g ( x )

Degree of f ( x ) > degree of g ( x ) : no horizontal asymptotes Degree of f ( x ) = degree of g ( x ) : y = the ratio of leading coefficients Degree of f ( x ) < degree of g ( x ) : y = 0

Triangle inequality theorem b ~ c
a

a ~c
c

Three Special Triangle Types C

C 60°

Hypotenuse

a

b

(leg)

a=b A

c

Opposite

60°

60°

A

s

∠ A = ∠B

Isosceles triangle

Equilateral triangle

The area of triangle

1 2

B

Adjacent Adjacent (leg) (leg)

Right triangle

× (base) × (height ) or

1 2

ab sin θ

Special Right Triangles

13

5

60°

2

1

5

3

30° 4

3 − 4 − 5 Triangle


12

5 − 12 − 13 Triangle

45°

2

1 45°

3

30° − 60° − 90° Triangle

1

45° − 45° − 90° Triangle


Special Quadrilaterals b1

b1

b1 y°

Trapezoids Area 

b1  b2 2

h

h

h

h

x°

y°

b2

x°

a

h

Area  b  h

x°

 ab sin x  ab sin y

Rectangle

x°

b2

b2

Parallelogram

y°

y° b

h

Area  b h

b

y °

Rhombus

x°

Area  b  h

y°

x°

 s 2 sin x  s 2 sin y 

Square 2

Area  s 

1 2

d 2

d

s

Circles and Sector A r r

O

r

x° l = θ (rad)

x° l = θ (rad)

d B

Circumference  2r  d

  l  x  2 r  r  AB 360

Area 

x 360

1

1

2

2

 r 2  rl  r 2

2

Area  r

2 (r (rad)  360 1 

 180

(rad)

1(rad) 

180







Solids Rectangular Rectangular prism h

S  2lw  2lh  2hw

d h

V  lwh

l 2 + w2

w

d  l 2  w2  h 2

l

w

l

Cube s

S  6s 2 V  s3

d f

s

d s 3

s

r

Cylinder r

2π r

2

S  2 r  2 rh 2

V  Bh  r h 2  2r   h 2

d

d

h

h

h

r

Pyramids 1 V  Bh 3

Cones S

1 2

l

2 2 cl   r   rl   r

V

1

l

h

1

2

Bh   r h 3 c 2r  (rad )   l l 3

θ

c

r

r

중심 (0,0,0) , 반지름 r 인 구의 방정식

Spheres

x 2 + y 2 + z 2 = r 2

S  4r 2

V

4 3

3

 r


r

P ( x1 , y1 , z 1 ) , Q ( x2 , y2 , z 2 ) 일 때,

PQ =

2

2

( x2 − x1 ) + ( y2 − y1 ) + ( z2 − z 1 )

2


Inscribed Solids

c a

s

b

2 2 2 d  a  b  c  2R

d  3s  2 R

( R ; radius of a sphere) r H

h

r

r

R s

d

R  r , H  2r

2

2r   h2  2R

s  2r

( R ; radius of a sphere)

( R , H ; radius and height of

( r ; radius of a sphere)

cylinder,

r

; radius of a sphere,)

Changing Dimensions Ratio of sides m : n Ratio of area m 2 : n 2 . Ratio of volumes m3 : n 3

Coordinates in Three Dimensions z

 x2 , y2 , z 2 

( 3,5,4) d

z 2  z 1

 x1 , y1 , z 1  y

x2  x1

y 2  y1

x

The distance between origin and  x1, y1 , z 1  2 2 2 d  x1  y1  z1

The distance between  x1, y1 , z 1  and  x2 , y2 , z 2  d

2

2

2

 x2  x1    y2  y1    z2  z1 


SAT Math II Formula 13 y II

I

( −, + )

A Plane Rectangular Coordinate System (or Cartesian Plane)

( +, + ) x

O IV ( +, − )

III ( −, − )

Distance 2

2

The distance between any two points  x1 , y1  and  x2 , y2  is d   x2  x1    y2  y1  Midpoint

 x  x2 y1  y2  The midpoint of a line segment with endpoints  x1 , y1  and  x2 , y2  is M  1 ,   2 2 

Distance between a Point and a Line y

Ax + By + C = 0

Ax + By + C = 0

d

d

d 

( x1 , y1 )

x

O

C

d 

2 2 A  B

Ax1  By1  C 2 2 A  B

Transformation Reflection

x -axis

y -axis

origin

y = x

Points & graph

y → − y

x → − x

x → − x , y → − y

x → y , y → x

Translation

→ h unit

↑ y unit

→ h unit, ↑ y unit

Points ( x, y )

x → x + h

y → y + k

x → x + h

y → y + k

Graph y = f ( x )

x → x − h

y → y − k

x → x − h

y → y − k

Stretch / Shrink

horizontally horizonta lly a

vertically b

Points ( x, y )

x → ax

y → by

Graph y = f ( x )


x →

x a

y →

y b

horizontally a , vertically b x → ax x →

x a

y → by y →

y b


Conics Sections

Circle

Parabola x 2  4 py

x 2  y 2  r 2

 p  0

y

y

Equal Distances C ( 0,0 0, 0 )

F ( 0, p )

P ( x, y )

r

x

O

P ( x, y ) x

O V ( 0, 0 ) y = − p

0,0 

Center

0,0  Directrix y 0, p Axis of symmetry

Vertex Focus

 

a2



y2 b2

1 y

x  0

Hyperbola x 2

a  b  0

a2

y2 b2

( c, 0 )

( − a, 0 )

x

c, 0 where c 2  a 2  b2

Major axis 2a , Minor axis 2b y intercepts 0,b ,

O

( c, 0 )

x

( 0, −b )

0,0  , Vertex a, 0 , a, 0

Foci c, 0 ,

a  0, b  0

( a, 0 )

( −c, 0 )

( 0, −b )

Center

1

0, b ) ( 0,b

P ( x, y )

( a, 0 ) O



y 0, b ) ( 0,b

( − a, 0 ) ( −c, 0 )

r

p

Ellipse x 2

Radius

0, b 0,b

Center

0,0  , Vertex a, 0 , a, 0

Foci c, 0 ,

c, 0

where c 2  a 2  b2 b

Asymptotes y   x a



Polar Coordinates Relationship Relationship between rectangular and polar coordinates:

( x, y )

y

( r ,θ )

y

r θ

θ x

O

x

O

 x, y   r ,  

 r ,     x, y 

r  x 2  y 2      y  1    t a n      Verify the quadrant   x   

 x  r cos    y  r sin 

Six Trigonometric Functions B

sin  

csc  

O

cos  

H

1 sin 



H

sec  

O

A

tan  

H

1 cos 



H A

cot  

O Hypotenuse

A

1 tan 



Opposite

A O

A

θ Adjacent

C

Trigonometric Identities tan  

sin  cos 

cot  

cos 2   sin 2   1

cos  sin 

1  tan 2   sec2 

cot 2   1  csc 2 

Cofunction Identities

 A  B  90 (complementary angle) ⇒ sin A  cos B , cos A  sin B , tan A 


1 tan B

 cot B


Special Right Triangles

2

30° =

45° =

6

3



0 0

6

Trig Trig. ft . sin 

π

2

1

 4

 45

 3

3

0

 90

1

1

0

2

2

Undefined

3

1

3

2

2

1 tan 



3

1

2

1

 60

2

3

cos 

1

π

1

1

0

60° =

4

 30

3

1

1

π



2

2

All Students Take Calculus.

Unit Circle Approach π 2 y

Quadrant

P ( x, y )

π

1 β β α α α α

sin  x

II

III

IV

 sin 

 sin 

sin 

sin 

 cos 

 cos 

 cos 

 cos 

 cos 

 cos 

 cos 

 cos 

 sin 

sin 

sin 

 sin 

 tan 

 tan 

 tan 

 tan 

 cot 

 cot 

 cot 

 cot 

0

β β

cos 

3π

I

tan 

2

Reference angle  (formed by the x axis and the terminal side ) sin    sin  , cos    cos  , tan    tan  (sign is determined by terminal side, ASTC)

Reference angle  (formed by the y axis and the terminal side ) sin    cos  , cos    sin  , tan    cot  (sign is determined by terminal side, ASTC) http:// 고강사 고강사.com


Graphs of Six Trigonometric Functions y T

1

y  sin x (Odd function) A

−2π

−π

π

O

2π

Domain ,  , Range 1,1

x

Period 2 , Amplitude 1 −1 y T

1

y  cos x (Even function) A

−π

−2π

2π

π

O

Domain ,  , Range 1,1

x

Period 2 , Amplitude 1 −1

y T

y  tan x (Odd function)

1

−

3π 2

−

π 2

 2

Domain ,  except 2n  1 , π

3π

2

2

−1

Range , 

x

Period



Vertical asymptotes at x   2n 1

Sinusoidal Graphs    

y  A sin  x    B  A sin   x 

   

 

  B  

y  A tan  x    B  A tan    x 


 

  B  

Period 

Period 

2

  

, Amplitude  A , Phase shift 

, Amplitude  A , Phase shift 

   

 2


Addition and Subtraction Formula sin( A ± B) = si sin A co cos B ± cos A sin B tan( A ± B) =

cos( A ± B) = co cos A cos B  sin A si sin B

tan A ± tan B 1  tan A tan B

Double-Angle Formula cos 2 A = cos 2 A − sin 2 A = 2 cos 2 A − 1 = 1 − 2 sin 2 A

sin 2 A = 2 si sin A co cos A tan2 A =

2tan A 1 − tan 2 A

The Law of Sines B

sin A a

c

a

=

si n B b

=

sin C c

=

1 2R

( R ; the radius of the circumscribed circle) A

C

b

The Law of Cosines B

a 2 = b2 + c 2 − 2bc cos A

a

c

b 2 = c 2 + a2 − 2ca cos B 2 2 2 c = a + b − 2ab cos A

A

b

C



Probability Definition: P  A 

# of possible outcomes of A total # of possible outcomes

Properties of probability probability - 0  P  A  1 - P  A  1  P  Ac  - P  A  B   P  A  P  B  P  A  B  Probability Probability of independent events - P  A  B   P( A) P( B)  Event A and B are independent Mutually exclusive events - P  A  B  0  Event A and B are mutually exclusive Binomial trial P  x  nC x p 1 p x

n x

n

C x 

P x

n

x !



n!

n  x! x!

where n ; total # of trials, x ; # of successes, p ; probability of success

Statistics N

 X

i

Mean :  

i 1

N

where X i ; data values, N ; total # of data

Median : the middle value when the values are ordered from smallest to largest Mode : the value that occurs most often. N

Standard deviation (SD):  

 X ; mean of X  X ; SD of X 2

  X i   i 1

N

If Y  aX  b , then

Y  a X  b , Y  a  X

Range : The difference between the maximum and the minimum Interquartile range (IQR) : IQR  Q3  Q1

Q2  median

Quartile - Three values which divide the sorted data set into four equal parts.



Regression 1) Linear regression regression y  ax  b

Speed

Fuel used

(Km/h)

(liters/100km)

60

5.90

70

6.30

80

6.95

90

7.57

100

8.27

110

9.03

2) Quadratic regression y  ax 2  bx  c

3) Exponential regression regression y  a b x

Limit 2 x + 2 x − 3 2

lim

x →∞

2 x 2 − 1

2 x 2

= lim x x →∞

2

+

2 x 2 x 2

2x x

2

−

− 1

3 2

x = lim

2+

x→ ∞

x2

2

−

3

x2 = 2 + 0 − 0 = 1 1 2−0 2− 2 x x

 2 x 2 + 2 x − 3  ∞ TI-89 : limit  , x ,  =1 2  2 x − 1 

lim x →1

x 2 + 2 x − 3 x 2 − 1

( x − 1) ( x + 3) 4 = =2 x →1 ( x + 1) ( x − 1) 2

= lim

 x 2 + 2 x − 3  , x , 1 TI-89 : limit  =2 2  x − 1  x

 1 lim 1 +  = e ; natural constant e x →∞  x 

  1  x  TI-89 : limit   1 +  , x, ∞  = e   x     lim+ x x = ?????

x → 0

TI-89 : limit ( x x , x, 0, 1) = 1


SAT Math2 Formula

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