Algebra Cheat Sheet Reduced PDF

Preview

Summary

cheat sheet...

Description

Logarithms and Log Pr operties Defini tion y y = logb x is equivalent to x = b

Algebra Cheat Sheet Basic Properties & Facts Properties of Inequalities If a < b then a + c < b+ c and a- c< b- c

Arithmetic Ope rations æ b ö ab aç ÷= èc ø c

ab +ac = a (b + c ) æa ö ç ÷ a èb ø = c bc

a b < c c a b If a < b and c < 0 then ac > bc and > c c If a < b and c > 0 then ac < bc and

a ac = æ bö b ç c÷ è ø

a c ad + bc + = b d bd

a c ad - bc - = b d bd

a -b b - a = c- d d - c

a+b a b = + c c c æ aö ç b ÷ ad è ø = æ c ö bc çd÷ è ø

ab + ac =b +c, a ¹ 0 a

a+ b £ a + b

(a )

m

= anm

( ab)

n

= ab

n

Triangle Inequality

a 1 = a n -m = m -n am a

points the distance betwe en them is

a0 = 1, a ¹ 0

d ( P1, P2 ) =

a -n = æ aö ç ÷ è bø

1 n

a

-n

n

bn æ bö =ç ÷ = n a è aø

n

Properties of Radicals n

1

a =a n

mn

a = nm a

Complex Numbers

( ) =( a ) 1

am = am

n

n

1 m

n

ab = n a n b

a + bi =

n

a na = b nb

( a + bi) = a - bi ( a + bi)( a + bi) =

n

a n = a, if n is odd

n

a n = a , if n is even

- a = i a, a ³ 0

( 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) ( a + bi)( a - bi) = a 2 + b2

Fo r a complete set of o nline Algebra no tes visit http://tutorial.math.lamar.edu.

a 2 + b2

Factoring Formulas x 2 - a 2 = ( x + a)( x- a)

Quadratic Formula Solve ax2 + bx + c = 0 , a¹ 0

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

2

x 2 - 2 ax+ a 2 = ( x- a)

2

- b ± b2 - 4 ac 2a If b 2 - 4 ac > 0 - Two real unequal solns. 2 If b - 4 ac = 0 - Repeated real solution. If b2 - 4 ac < 0 - Two complex solutions. x=

x 3 + 3 ax 2 + 3a 2 x + a3 = ( x+ a)

3

x 3 - 3 ax 2 + 3 a 2 x - a3 = ( x- a )

3

3

3

3

3

2

Square Root Property If x 2 = p then x = ± p

2

2

2n

n

2

n

n

n

If n is odd then, x n - a n = ( x- a) ( xn-1 + axn-2 + L + an-1 ) xn + an = ( x + a) ( x

i

Complex Modulus Complex C onjugate a + bi

The domain of logb x is x > 0

) x - a = ( x - a) ( x + ax+ a ) x - a = ( x - a )( x + a )

( x2 - x1) 2 + ( y2 - y1 ) 2

i 2 = -1

æx ö logb ç ÷ = logb x - logb y èy ø

log x = log 10 x common log where e = 2.7 182818 28K

2n

i = -1

logb ( xy ) = logb x + logb y

Special Logarithms ln x = log e x natural log

x + a = ( x + a) ( x - ax+ a

n

n æa ö a ç ÷ = n b èbø 1 =a n a n

n n

blog b x = x

logb ( xr ) = r logb x

x 2 + ( a+ b) x+ ab= ( x+ a)( x+ b)

Distance Formula If P1 = (x 1, y 1 ) and P2 = ( x2 , y2 ) are two

n

logb bx = x Example log 5 125 = 3 because 53 = 125

Factoring and Solving

a a = b b

ab = a b

Exponent Properties + a na m = a n m

Properties of Absolute Value if a ³ 0 ìa a =í if a < 0 î- a a ³0 -a = a

Logarithm Properties logb b = 1 logb 1= 0

2

n -1

n -2

- ax

n -3

+ a2 x

p b

Þ

p < -b or

p >b

n -L + a -1 )

Completing the Square (4) Factor the left side

Solve 2x 2 - 6x - 10 = 0

2

(1) Divide by the coefficient of the x 2 x2 - 3 x - 5 = 0 (2) Move the constant to the other side. x2 - 3 x = 5 (3) Take half the coefficient of x, square it and add it to both sides 2

2

9 29 æ 3ö æ 3ö x 2 - 3 x + ç- ÷ = 5 + ç- ÷ = 5 + = 4 4 è 2ø è 2ø

© 2005 Paul Dawkins

Absolute Value Equations/Inequalities If b is a positive number p =b p = -b or p = b Þ

29 æx 3ö ç - 2÷ = 4 è ø (5) Use Square Root Property 3 29 29 =± =± 2 4 2 (6) Solve for x 3 29 x= ± 2 2 x-

Fo r a complete set of o nline Algebra no tes visit http://tutorial.math.lamar.edu.

© 2005 Paul Dawkins

Functions and Graphs Constant Function y = a or f ( x ) = a Graph is a horizontal line passing through the point ( 0, a) . Line/Linear Function y = mx + b or f ( x) = mx + b Graph is a line with point( 0, b ) and slope m. Slo pe Slope of the line containing the two points ( x1, y1) and ( x 2, y 2) is y2 - y1 rise = x2 - x1 run Slo pe – intercept form The equation of the line with slope m a nd y-interce pt ( 0,b) is m=

y = mx + b Poi nt – Slope form The equation of the line with slope m and passing through the point( x1 , y1 ) is y =y 1 +m (x -x 1 ) Parab ola/Quadratic Function 2 2 y = a ( x -h ) + k f ( x) = a ( x - h) + k The graph is a parabola that opens up if a> 0 or down if a< 0 and has a vertex at ( h, k ).

Common Algebraic Errors

Parabola/Quadratic Function x = ay2 + by + c g( y ) = ay2 + by+ c The graph is a parabola that opens right if a > 0 or left if a < 0 and has a vertex b bö æ at ç g æç - ö÷ , - ÷ . è è 2 a ø 2 aø

Er ror

Division by zero is undefined!

- 32 ¹ 9

-3 = -9 , ( -3 ) = 9 Watch parent hesi s!

( x)

2 3

Graph is a circle with radius r and center

a + bx a

2

+

2

¹ 1 + bx

-a ( x -1) ¹ -ax - a

Ellipse

( x -h ) 2 ( y - k ) 2

(x)

2 3

¹x

( x - h) 2 +( y - k) 2 = r2 ( h, k ) .

=1

a b Graph is an ellipse with center( h, k ) wi th verti ces a unit s ri ght/left from the center and vertic es b units up/down from the center.

(

x + a)

2

x +a ¹ x +a 2

¹ xn + an and n x + a ¹ n x + n a

n

2

1 1 1 1 = ¹ + =2 2 1 +1 1 1 A more complex version of the previous error. a +bx a bx bx = + =1 + a a a a Beware of incorrect c anc eling! -a ( x -1) = -ax + a Make sure you distribute the “-“! x + a) = x2 +2 ax + a2

5 = 25 = 3 + 4 ¹ 3 2 + 4 2 = 3 + 4 = 7 See previous error. More gene ral ve rsions of previous three errors. 2

x +a ¹ x + a x + a)

= x x x = x6 2 2

( x +a) 2 =( x + a)(

¹ x2 + a2

2

(

2

2

5

a a a ¹ + b+ c b c 1 ¹ x-2 + x-3 x2 + x3

Circle

Reason/Corre ct/Justifi cation/Example

2 2 ¹ 0 and ¹ 2 0 0

2

2 ( x +1) = 2 ( x2 + 2 x +1) = 2 x2 + 4 x + 2 2

Hyperbola 2 2 ( x -h ) ( y - k ) 1 = a2 b2 Graph is a hype rbola that opens left and right, has a center at( h, k ) , vertices a units left/right of center a nd asymptotes b that pass through center with slope ± . a Hyperbola

2 ( x +1 ) ¹ (2 x + 2 ) 2

(2 x + 2 )

2

2

¹ 2 ( x + 1)

Parabola/Quadratic Function y = ax 2 + bx + c f ( x) = ax 2 + bx + c

=1 b2 a2 Graph is a hype rbola that opens up and down, has a center at( h, k ) , vertices b

a ab ¹ æb ö c çc ÷ è ø

The graph is a parabola that opens up if a> 0 or down if a< 0 and has a vertex b ö æ b at ç - , f æç - ö÷ ÷ . è 2a è 2a ø ø

units up /d own from the center and asymptotes that pass through c enter with b slope ± . a

æa ö ç b ÷ ac è ø ¹ c b

© 2005 Paul Dawkins

= 4 x2 + 8 x + 4

1

- x 2 + a 2 ¹ - x 2 + a2

( y - k ) 2 ( x - h) 2

Fo r a complete set of o nline Algebra no tes visit http://tutorial.math.lamar.edu.

2

( 2 x + 2)2

Square first then distribute! See the previous example. You can not factor out a constant if there is a power on the parethesis! -x 2 + a 2 = ( - x 2 + a 2 ) 2

Now see the previous error. æa ö ç ÷ a è 1 ø æ a ö æ c ö ac = =ç ÷ç ÷ = æ b ö æ b ö è 1 ø è bø b ç c÷ ç c÷ è ø è ø æ aö æ aö ç b÷ ç b÷ a è ø è ø æ aö æ 1 ö = =ç ÷ ç ÷ = c æ c ö è b ø è c ø bc ç1 ÷ è ø

Fo r a complete set of o nline Algebra no tes visit http://tutorial.math.lamar.edu.

© 2005 Paul Dawkins...