Burke Mountain Academy Math Reference Algebra...

For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins

Algebra Cheat Sheet

Basic Properties & Facts Arithmetic Operations

( )

, 0

b abab ac a b c a

c c

a

a a acb

bc bc b

c

a c ad bc a c ad bc

b d bd b d bd

a b b a a b a b

c d d c c c c

a

ab ac adbb c a

ca bc

d

! "+ = + =# $

% &

! "# $% & = =

! "# $% &

+ '+ = ' =

' ' += = +

' '

! "# $+ % &= + ( =! "# $% &

Exponent Properties

( )

( )

( ) ( )1

1

0

1

1, 0

1 1

nm

m m

nn m n m n m

m m n

mn nm

n nn n n

n

n n

n n

n n n nn

n

aa a a a

a a

a a a a

a aab a b

b b

a aa a

a b ba a a

b a a

+ '

'

'

'

'

= = =

= = (

! "= =# $

% &

= =

! " ! "= = = =# $ # $

% & % &

Properties of Radicals

1

, if is odd

, if is even

nn n n n

nm n nm n

n

n n

n n

a a ab a b

a aa a

b b

a a n

a a n

= =

= =

=

=

Properties of Inequalities If then and

If and 0 then and

If and 0 then and

a b a c b c a c b c

a ba b c ac bc

c c

a ba b c ac bc

c c

< + < + ' < '

< > < <

< < > >

Properties of Absolute Value

if 0

if 0

a aa

a a

)*= +

' <,

0

Triangle Inequality

a a a

aaab a b

b b

a b a b

) ' =

= =

+ - +

Distance Formula

If ( )1 1 1,P x y= and ( )2 2 2,P x y= are two

points the distance between them is

( ) ( ) ( )2 2

1 2 2 1 2 1,d P P x x y y= ' + '

Complex Numbers

( ) ( ) ( )

( ) ( ) ( )

( )( ) ( )

( )( )

( )

( )( )

2

2 2

2 2

2

1 1 , 0

Complex Modulus

Complex Conjugate

i i a i a a

a bi c di a c b d i

a bi c di a c b d i

a bi c di ac bd ad bc i

a bi a bi a b

a bi a b

a bi a bi

a bi a bi a bi

= ' = ' ' = )

+ + + = + + +

+ ' + = ' + '

+ + = ' + +

+ ' = +

+ = +

+ = '

+ + = +

Burke Mountain AcademyMath Reference


Logarithms and Log Properties Definition

log is equivalent to y

by x x b= =

Example 3

5log 125 3 because 5 125= =

Special Logarithms

10

ln log natural log

log log common log

ex x

x x

=

=

where 2.718281828e = K

Logarithm Properties

( )( )

log

log 1 log 1 0

log

log log

log log log

log log log

b

b b

xx

b

r

b b

b b b

b b b

b

b x b x

x r x

xy x y

xx y

y

= =

= =

=

= +

! "= #$ %

& '

The domain of logb

x is 0x >

Factoring and Solving Factoring Formulas

( ) ( )

( )

( )

( ) ( ) ( )

2 2

22 2

22 2

2

2

2

x a x a x a

x ax a x a

x ax a x a

x a b x ab x a x b

# = + #

+ + = +

# + = #

+ + + = + +

( )

( )

( )( )

( ) ( )

33 2 2 3

33 2 2 3

3 3 2 2

3 3 2 2

3 3

3 3

x ax a x a x a

x ax a x a x a

x a x a x ax a

x a x a x ax a

+ + + = +

# + # = #

+ = + # +

# = # + +

( )( )2 2n n n n n nx a x a x a# = # +

If n is odd then,

( ) ( )

( )( )

1 2 1

1 2 2 3 1

n n n n n

n n

n n n n

x a x a x ax a

x a

x a x ax a x a

# # #

# # # #

# = # + + +

+

= + # + # +

L

L

Quadratic Formula

Solve 2 0ax bx c+ + = , 0a ( 2 4

2

b b acx

a

# ± #=

If 2 4 0b ac# > - Two real unequal solns.

If 2 4 0b ac# = - Repeated real solution.

If 2 4 0b ac# < - Two complex solutions.

Square Root Property

If 2x p= then x p= ±

Absolute Value Equations/Inequalities

If b is a positive number

or

or

p b p b p b

p b b p b

p b p b p b

= ) = # =

< ) # < <

> ) < # >

Completing the Square

Solve 22 6 10 0x x# # =

(1) Divide by the coefficient of the 2x

2 3 5 0x x# # =

(2) Move the constant to the other side. 2 3 5x x# =

(3) Take half the coefficient of x, square

it and add it to both sides 2 2

2 3 3 9 293 5 5

2 2 4 4x x

! " ! "# + # = + # = + =$ % $ %

& ' & '

(4) Factor the left side 2

3 29

2 4x! "# =$ %

& '

(5) Use Square Root Property

3 29 29

2 4 2x # = ± = ±

(6) Solve for x

3 29

2 2x = ±


Functions and Graphs Constant Function

( )or y a f x a= =

Graph is a horizontal line passing

through the point ( )0,a .

Line/Linear Function

( )or y mx b f x mx b= + = +

Graph is a line with point ( )0,b and

slope m.

Slope

Slope of the line containing the two

points ( )1 1,x y and ( )2 2,x y is

2 1

2 1

rise

run

y ym

x x

!= =

!

Slope – intercept form

The equation of the line with slope m

and y-intercept ( )0,b is

y mx b= +

Point – Slope form

The equation of the line with slope m

and passing through the point ( )1 1,x y is

( )1 1y y m x x= + !

Parabola/Quadratic Function

( ) ( ) ( )2 2

y a x h k f x a x h k= ! + = ! +

The graph is a parabola that opens up if

0a > or down if 0a < and has a vertex

at ( ),h k .


( )2 2y ax bx c f x ax bx c= + + = + +

The graph is a parabola that opens up if

0a > or down if 0a < and has a vertex

at ,2 2

b bf

a a

" #" #! !$ %$ %

& '& '.


( )2 2x ay by c g y ay by c= + + = + +

The graph is a parabola that opens right

if 0a > or left if 0a < and has a vertex

at ,2 2

b bg

a a

" #" #! !$ %$ %& '& '

.

Circle

( ) ( )2 2 2

x h y k r! + ! =

Graph is a circle with radius r and center

( ),h k .

Ellipse

( ) ( )2 2

2 21

x h y k

a b

! !+ =

Graph is an ellipse with center ( ),h k

with vertices a units right/left from the

center and vertices b units up/down from

the center.

Hyperbola

( ) ( )2 2

2 21

x h y k

a b

! !! =

Graph is a hyperbola that opens left and

right, has a center at ( ),h k , vertices a

units left/right of center and asymptotes

that pass through center with slope b

a± .

Hyperbola

( ) ( )2 2

2 21

y k x h

b a

! !! =

Graph is a hyperbola that opens up and

down, has a center at ( ),h k , vertices b

units up/down from the center and

asymptotes that pass through center with

slope b

a± .


Common Algebraic Errors Error Reason/Correct/Justification/Example

20

0! and

22

0! Division by zero is undefined!

23 9" ! 23 9" = " , ( )

23 9" = Watch parenthesis!

( )3

2 5x x! ( )

32 2 2 2 6x x x x x= =

a a a

b c b c! +

+

1 1 1 12

2 1 1 1 1= ! +=

+

2 3

2 3

1x x

x x

" "! ++

A more complex version of the previous

error.

a bx

a

+1 bx! +

1a bx a bx bx

a a a a

+= + = +

Beware of incorrect canceling!

( )1a x ax a" " ! " " ( )1a x ax a" " = " +

Make sure you distribute the “-“!

( )2 2 2

x a x a+ ! + ( ) ( ) ( )2 2 22x a x a x a x ax a+ = + + = + +

2 2x a x a+ ! + 2 2 2 25 25 3 4 3 4 3 4 7= = + ! + = + =

x a x a+ ! + See previous error.

( )n n n

x a x a+ ! + and n n nx a x a+ ! +

More general versions of previous three

errors.

( ) ( )2 2

2 1 2 2x x+ ! +

( ) ( )2 2 22 1 2 2 1 2 4 2x x x x x+ = + + = + +

( )2 22 2 4 8 4x x x+ = + +

Square first then distribute!

( ) ( )2 2

2 2 2 1x x+ ! +

See the previous example. You can not

factor out a constant if there is a power on

the parethesis!

2 2 2 2x a x a" + ! " + ( )

12 2 2 2 2x a x a" + = " +

Now see the previous error.

a ab

b c

c

!# $% &' (

1

1

a

a a c ac

b b b b

c c

# $% &

# $# $' (= = =% &% &# $ # $ ' (' (% & % &' ( ' (

a

acb

c b

# $% &' ( !

1

1

a a

a ab b

cc b c bc

# $ # $% & % &

# $# $' ( ' (= = =% &% &# $ ' (' (% &' (

2D GEOMETRY FORMULAS

SQUARE

s = sideArea: A = s2

Perimeter: P = 4s

s

s

RECTANGLE

l = length, w = widthArea: A = lw

Perimeter: P = 2l + 2w

w

l

TRIANGLE

b = base, h = heightArea: A = 1

2bh

Perimeter: P = a + b + c

h

b

a c

EQUILATERAL TRIANGLE

s = sideHeight: h =

!

3

2s

Area: A =!

3

4s2

h

s

s s

PARALLELOGRAM

b = base, h = height, a = sideArea: A = bh

Perimeter: P = 2a + 2b

b

h a

TRAPEZOID

a, b = bases; h = height;c, d = sidesArea: A = 1

2(a + b)h

Perimeter:P = a + b + c + d b

h

a

c d

CIRCLE

r = radius, d = diameterDiameter: d = 2rArea: A = !r2

Circumference: C = 2!r = !d r

d

SECTOR OF CIRCLE

r = radius, " = angle in radiansArea: A = 1

2"r2

Arc Length: s = "rr

s

"

ELLIPSE

a = semimajor axisb = semiminor axisArea: A = !ab

a

b

Circumference:C ! !

!

3(a + b) ""

(a + 3b)(b + 3a)#

ANNULUS

r = inner radius,R = outer radiusAverage Radius: # = 1

2(r + R)

Width: w = R " r

Area: A = !(R2 " r2)or A = 2!#w

r

R

REGULAR POLYGON

s = side length,n = number of sidesCircumradius: R = 1

2s csc(!

n)

Area: A = 1

4ns2 cot(!

n)

or A = 1

2nR2 sin(2!

n)

R

s

3D GEOMETRY FORMULAS

CUBE

s = sideVolume: V = s3

Surface Area: S = 6s2

ss

s

RECTANGULAR SOLID

l = length, w = width,h = heightVolume: V = lwh

Surface Area:S = 2lw + 2lh + 2wh

l

w

h

SPHERE

r = radiusVolume: V = 4

3!r3

Surface Area: S = 4!r2

r

RIGHT CIRCULAR

CYLINDER

r = radius, h = heightVolume: V = !r2h

Surface Area: S = 2!rh + 2!r2

r

h

TORUS

r = tube radius,R = torus radiusVolume: V = 2!2r2R

Surface Area: S = 4!2rR

R r

GENERAL CONE

OR PYRAMID

A = area of base, h = heightVolume: V = 1

3Ah

h

A

RIGHT CIRCULAR CONE

r = radius, h = heightVolume: V = 1

3!r2h

Surface Area:S = !r

!r2 + h2 + !r2

h

r

FRUSTUM OF A CONE

r = top radius, R = base radius,h = height, s = slant heightVolume: V = !

3(r2 + rR + R2)h

Surface Area:S = !s(R + r) + !r2 + !R2

s

h R

r

SQUARE PYRAMID

s = side, h = heightVolume: V = 1

3s2h

Surface Area:S = s(s +

!s2 + 4h2)

h

s

s

REGULAR TETRAHEDRON

s = sideVolume: V = 1

12

!2s3

Surface Area: S =!

3s2

s

© 2005 Paul Dawkins

Trig Cheat Sheet

Definition of the Trig Functions

Right triangle definition

For this definition we assume that

02

!"# # or 0 90"$ # # $ .

opposite

sinhypotenuse

" % hypotenuse

cscopposite

" %

adjacentcos

hypotenuse" %

hypotenusesec

adjacent" %

oppositetan

adjacent" %

adjacentcot

opposite" %

Unit circle definition

For this definition " is any angle.

sin1

yy" % %

1csc

y" %

cos1

xx" % %

1sec

x" %

tany

x" % cot

x

y" %

Facts and Properties Domain

The domain is all the values of " that

can be plugged into the function.

sin" , " can be any angle

cos" , " can be any angle

tan" , 1

, 0, 1, 2,2

n n" !& '( ) % * *+ ,- .

!

csc" , , 0, 1, 2,n n" !( % * * !

sec" , 1

, 0, 1, 2,2

n n" !& '( ) % * *+ ,- .

!

cot" , , 0, 1, 2,n n" !( % * * !

Range

The range is all possible values to get

out of the function.

1 sin 1"/ 0 0 csc 1 and csc 1" "1 0 /

1 cos 1"/ 0 0 sec 1 andsec 1" "1 0 /

tan"/2 0 0 2 cot"/2 0 0 2

Period

The period of a function is the number,

T, such that 3 4 3 4f T f" ") % . So, if 5

is a fixed number and " is any angle we

have the following periods.

3 4sin 5" 6 2

T!5

%

3 4cos 5" 6 2

T!5

%

3 4tan 5" 6 T!5

%

3 4csc 5" 6 2

T!5

%

3 4sec 5" 6 2

T!5

%

3 4cot 5" 6 T!5

%

"

adjacent

opposite

hypotenuse

x

y

3 4,x y

"

x

y 1


Formulas and Identities Tangent and Cotangent Identities

sin costan cot

cos sin

! !! !

! !" "

Reciprocal Identities

1 1csc sin

sin csc

1 1sec cos

cos sec

1 1cot tan

tan cot

! !! !

! !! !

! !! !

" "

" "

" "

Pythagorean Identities 2 2

2 2

2 2

sin cos 1

tan 1 sec

1 cot csc

! !

! !

! !

# "

# "

# "

Even/Odd Formulas

$ % $ %$ % $ %$ % $ %

sin sin csc csc

cos cos sec sec

tan tan cot cot

! ! ! !

! ! ! !

! ! ! !

& " & & " &

& " & "

& " & & " &

Periodic Formulas

If n is an integer.

$ % $ %$ % $ %$ % $ %

sin 2 sin csc 2 csc

cos 2 cos sec 2 sec

tan tan cot cot

n n

n n

n n

! ' ! ! ' !

! ' ! ! ' !

! ' ! ! ' !

# " # "

# " # "

# " # "

Double Angle Formulas

$ %$ %

$ %

2 2

2

2

2

sin 2 2sin cos

cos 2 cos sin

2cos 1

1 2sin

2 tantan 2

1 tan

! ! !

! ! !

!

!!

!!

"

" &

" &

" &

"&

Degrees to Radians Formulas

If x is an angle in degrees and t is an

angle in radians then

180and

180 180

t x tt x

x

' ''

" ( " "

Half Angle Formulas

$ %$ %

$ %$ %$ %$ %

2

2

2

1sin 1 cos 2

2

1cos 1 cos 2

2

1 cos 2tan

1 cos 2

! !

! !

!!

!

" &

" #

&"

#

Sum and Difference Formulas

$ %$ %

$ %

sin sin cos cos sin

cos cos cos sin sin

tan tantan

1 tan tan

) * ) * ) *

) * ) * ) *

) *) *

) *

+ " +

+ "

++ "

!

!

Product to Sum Formulas

$ % $ %

$ % $ %

$ % $ %

$ % $ %

1sin sin cos cos

2

1cos cos cos cos

2

1sin cos sin sin

2

1cos sin sin sin

2

) * ) * ) *

) * ) * ) *

) * ) * ) *

) * ) * ) *

" & & #, -. /

" & # #, -. /

" # # &, -. /

" # & &, -. /

Sum to Product Formulas

sin sin 2sin cos2 2

sin sin 2cos sin2 2

cos cos 2cos cos2 2

cos cos 2sin sin2 2

) * ) *) *

) * ) *) *

) * ) *) *

) * ) *) *

# &0 1 0 1# " 2 3 2 34 5 4 5

# &0 1 0 1& " 2 3 2 34 5 4 5

# &0 1 0 1# " 2 3 2 34 5 4 5

# &0 1 0 1& " & 2 3 2 34 5 4 5

Cofunction Formulas

sin cos cos sin2 2

csc sec sec csc2 2

tan cot cot tan2 2

' '! ! ! !

' '! ! ! !

' '! ! ! !

0 1 0 1& " & "2 3 2 34 5 4 5

0 1 0 1& " & "2 3 2 34 5 4 5

0 1 0 1& " & "2 3 2 34 5 4 5


Unit Circle

For any ordered pair on the unit circle ! ",x y : cos x# $ and sin y# $

Example

5 1 5 3cos sin

3 2 3 2

% %& ' & '$ $ () * ) *+ , + ,

3

%

4

%

6

%

2 2,

2 2

& ') *) *+ ,

3 1,

2 2

& ') *) *+ ,

1 3,

2 2

& ') *) *+ ,

60-

45-

30-

2

3

%

3

4

%

5

6

%

7

6

%

5

4

%

4

3

%

11

6

%

7

4

%

5

3

%

2

%

%

3

2

%

0

2%

1 3,

2 2

& '() *+ ,

2 2,

2 2

& '() *+ ,

3 1,

2 2

& '() *+ ,

3 1,

2 2

& '( () *+ ,

2 2,

2 2

& '( () *+ ,

1 3,

2 2

& '( () *+ ,

3 1,

2 2

& '() *

+ ,

2 2,

2 2

& '() *

+ ,

1 3,

2 2

& '() *

+ ,

! "0,1

! "0, 1(

! "1,0(

90-

120-

135-

150-

180-

210-

225-

240-270- 300-

315-

330-

360-

0-

x

! "1,0

y


Inverse Trig Functions Definition

1

1

1

sin is equivalent to sin

cos is equivalent to cos

tan is equivalent to tan

y x x y

y x x y

y x x y

!

!

!

" "

" "

" "

Domain and Range

Function Domain Range

1siny x!" 1 1x! # # 2 2

y$ $

! # #

1cosy x!" 1 1x! # # 0 y $# #

1tany x!" x!% & & % 2 2

y$ $

! & &

Inverse Properties

' (' ( ' (' (

' (' ( ' (' (

' (' ( ' (' (

1 1

1 1

1 1

cos cos cos cos

sin sin sin sin

tan tan tan tan

x x

x x

x x

) )

) )

) )

! !

! !

! !

" "

" "

" "

Alternate Notation 1

1

1

sin arcsin

cos arccos

tan arctan

x x

x x

x x

!

!

!

"

"

"

Law of Sines, Cosines and Tangents

Law of Sines

sin sin sin

a b c

* + ," "

Law of Cosines 2 2 2

2 2 2

2 2 2

2 cos

2 cos

2 cos

a b c bc

b a c ac

c a b ab

*

+

,

" - !

" - !

" - !

Mollweide’s Formula

' (12

12

cos

sin

a b

c

* +

,

!-"

Law of Tangents

' (

' (

' (

' (

' (

' (

12

12

12

12

12

12

tan

tan

tan

tan

tan

tan

a b

a b

b c

b c

a c

a c

* +

* +

+ ,

+ ,

* ,

* ,

!!"

- -

!!"

- -

!!"

- -

c a

b

*.

+.

,.

!

f x( ) = x0

= 1

!

f x( ) = x1

= x

!

f x( ) = x2

!

f x( ) = x1

2 = x

!

f x( ) = x3

!

f x( ) = x3" x

!

f x( ) = x4

!

f x( ) = x4" x

2

y xsin=

y xcos=

y xtan=

!

f x( ) = 2x

x2 y2+ 1=

x2 y2" 1=

x2

9

y2

4+ 1=

!

f x( ) = log x

Di�erentiation Formulas

d

dxk = 0 (1)

d

dx[f(x) ± g(x)] = f ⇥(x) ± g⇥(x) (2)

d

dx[k · f(x)] = k · f ⇥(x) (3)

d

dx[f(x)g(x)] = f(x)g⇥(x) + g(x)f ⇥(x) (4)

d

dx

�f(x)g(x)

⇥=

g(x)f ⇥(x)� f(x)g⇥(x)[g(x)]2

(5)

d

dxf(g(x)) = f ⇥(g(x)) · g⇥(x) (6)

d

dxxn = nxn�1 (7)

d

dxsinx = cos x (8)

d

dxcos x = � sinx (9)

d

dxtanx = sec2 x (10)

d

dxcot x = � csc2 x (11)

d

dxsec x = sec x tanx (12)

d

dxcsc x = � csc x cot x (13)

d

dxex = ex (14)

d

dxax = ax ln a (15)

d

dxln |x| =

1x

(16)

d

dxsin�1 x =

1⇥1� x2

(17)

d

dxcos�1 x =

�1⇥1� x2

(18)

d

dxtan�1 x =

1x2 + 1

(19)

d

dxcot�1 x =

�1x2 + 1

(20)

d

dxsec�1 x =

1|x|⇥

x2 � 1(21)

d

dxcsc�1 x =

�1|x|⇥

x2 � 1(22)

Integration Formulas

⇤dx = x + C (1)

⇤xn dx =

xn+1

n + 1+ C (2)

⇤dx

x= ln |x| + C (3)

⇤ex dx = ex + C (4)

⇤ax dx =

1ln a

ax + C (5)

⇤lnx dx = x lnx� x + C (6)

⇤sinx dx = � cos x + C (7)

⇤cos x dx = sinx + C (8)

⇤tanx dx = � ln | cos x| + C (9)

⇤cot x dx = ln | sinx| + C (10)

⇤sec x dx = ln | sec x + tanx| + C (11)

⇤csc x dx = � ln | csc x + cot x| + C (12)

⇤sec2 x dx = tan x + C (13)

⇤csc2 x dx = � cot x + C (14)

⇤sec x tanx dx = sec x + C (15)

⇤csc x cot x dx = � csc x + C (16)

⇤dx⇥

a2 � x2= sin�1 x

a+ C (17)

⇤dx

a2 + x2=

1a

tan�1 x

a+ C (18)

⇤dx

x⇥

x2 � a2=

1a

sec�1 |x|a

+ C (19)

Standard Normal Distribution Table

0 z

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359

0.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753

0.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141

0.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517

0.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879

0.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224

0.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549

0.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852

0.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133

0.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389

1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621

1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3830

1.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015

1.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177

1.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4319

1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441

1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545

1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633

1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706

1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767

2.0 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817

2.1 .4821 .4826 .4830 .4834 .4838 .4842 .4846 .4850 .4854 .4857

2.2 .4861 .4864 .4868 .4871 .4875 .4878 .4881 .4884 .4887 .4890

2.3 .4893 .4896 .4898 .4901 .4904 .4906 .4909 .4911 .4913 .4916

2.4 .4918 .4920 .4922 .4925 .4927 .4929 .4931 .4932 .4934 .4936

2.5 .4938 .4940 .4941 .4943 .4945 .4946 .4948 .4949 .4951 .4952

2.6 .4953 .4955 .4956 .4957 .4959 .4960 .4961 .4962 .4963 .4964

2.7 .4965 .4966 .4967 .4968 .4969 .4970 .4971 .4972 .4973 .4974

2.8 .4974 .4975 .4976 .4977 .4977 .4978 .4979 .4979 .4980 .4981

2.9 .4981 .4982 .4982 .4983 .4984 .4984 .4985 .4985 .4986 .4986

3.0 .4987 .4987 .4987 .4988 .4988 .4989 .4989 .4989 .4990 .4990

3.1 .4990 .4991 .4991 .4991 .4992 .4992 .4992 .4992 .4993 .4993

3.2 .4993 .4993 .4994 .4994 .4994 .4994 .4994 .4995 .4995 .4995

3.3 .4995 .4995 .4995 .4996 .4996 .4996 .4996 .4996 .4996 .4997

3.4 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4998

3.5 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998

Gilles Cazelais. Typeset with LATEX on April 20, 2006.

Burke Mountain Academy Math Reference Algebra...

Documents

Transcript of Burke Mountain Academy Math Reference Algebra...