Title | Algebra Cheat Sheet - Law and Formulas |
---|---|
Author | Wee Man |
Course | Modern mathematics |
Institution | Ryerson University |
Pages | 4 |
File Size | 193 KB |
File Type | |
Total Downloads | 65 |
Total Views | 205 |
Algebra skills cheat sheet, assumes no prior knowledge
- includes exponent, log, ln laws
- trig identities, radical properties among other things...
Algebra Cheat Sheet Basic Properties & Facts Properties of Inequalities If a < b then a + c < b + c and a - c < b - c a b If a < b and c > 0 then ac < bc and < c c a b If a < b and c < 0 then ac > bc and > c c
Arithmetic Operations b ab a æç ö÷ = ècø c
ab + ac = a ( b + c ) æ aö ç ÷ a è bø = c bc
a ac = æbö b ç ÷ ècø
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 æaö ç b ÷ ad è ø= æ c ö bc çd ÷ è ø
ab + ac = b + c, a ¹ 0 a
Properties of Absolute Value if a ³ 0 ìa a =í if a < 0 î- a a ³0 -a = a
a+b £ a + b
an 1 = a n- m = m -n m a a
(a )
a 0 = 1, a ¹ 0
n m
=a
nm
n
n æaö = æbö = b çb÷ ça÷ an è ø è ø
n m
1
a = an
m n
a=
nm
a
n
ab = n a n b
n
a = b
n
an = a , if n is odd
n
an = a , if n is even
( ) 1
a = am
Properties of Radicals n
( x 2 - x1 ) + ( y 2 - y1 ) 2
2
n
1 an
-n
d ( P1, P2 ) =
n æa ö = a ç ÷ n èb ø b 1 = an a- n
(ab )n = an bn a -n =
Triangle Inequality
Distance Formula If P1 = ( x1 , y1 ) and P2 = ( x 2 , y 2 ) are two points the distance between them is
Exponent Properties a n a m = a n+ m
a a = b b
ab = a b
n n
a b
n
Complex Numbers i = -1 = ( an ) m 1
i 2 = -1
-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 )i (a + bi )(a - bi ) = a 2 + b 2 a + bi = a 2 + b 2
Complex Modulus
(a + bi ) = a - bi Complex Conjugate (a + bi )(a + bi ) = a + bi 2
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
© 2005 Paul Dawkins
Logarithms and Log Properties Definition y = log b x is equivalent to x = b y
Logarithm Properties log b b = 1 log b 1 = 0 log b b x = x
Example log5 125 = 3 because 53 = 125
b logb x = x
log b ( xr ) = r log b x log b ( xy ) = log b x + log b y
Special Logarithms ln x = loge x natural log
æ log b ç è
log x = log10 x common log where e = 2.718281828K
xö = log b x - log b y y ÷ø
The domain of log b x is x > 0
Factoring and Solving Factoring Formulas x 2 - a 2 = ( x + a ) (x - a )
Quadratic Formula Solve ax 2 + bx + c = 0 , a ¹ 0
x 2 + 2 ax + a 2 = ( x + a )
2
x 2 - 2ax + a 2 = ( x - a )
2
-b ± b 2 - 4ac 2a 2 If b - 4ac > 0 - Two real unequal solns. If b 2 - 4ac = 0 - Repeated real solution. If b 2 - 4ac < 0 - Two complex solutions. x=
x 2 + ( a + b ) x + ab = ( x + a )( x + b ) x 3 + 3ax 2 + 3a 2x + a 3 = ( x + a )
3
x 3 - 3ax 2 + 3a 2x - a 3 = ( x - a )
3
Square Root Property If x 2 = p then x = ± p
x 3 + a 3 = ( x + a ) (x 2 - ax + a 2 ) x 3 - a 3 = ( x - a ) ( x 2 + ax + a 2 ) x 2n - a 2n = (x n - a n )( x n + a n ) If n is odd then, x n - a n = ( x - a ) ( x n-1 + ax n-2 + L + a n-1 ) xn +an
Absolute Value Equations/Inequalities If b is a positive number p=b p = - b or p = b Þ pb
Þ
p < - b or
p> b
= ( x + a ) ( x n -1 - ax n -2 + a 2 x n-3 -L + a n-1 ) Solve 2 x 2 - 6 x -10 = 0
Completing the Square (4) Factor the left side 2
2
(1) Divide by the coefficient of the x x2 - 3 x - 5 = 0 (2) Move the constant to the other side. x 2 -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 2 x - 3x + ç - ÷ = 5 + ç - ÷ = 5 + = 4 4 è 2ø è 2ø
æ x - 3 ö = 29 ç ÷ 2ø 4 è (5) Use Square Root Property 3 29 29 =± =± 2 4 2 (6) Solve for x 3 29 x= ± 2 2
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
x-
© 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. Slope Slope of the line containing the two points ( x1, y1 ) and ( x 2 , y 2 ) is y 2 - y1 rise = x 2 - x1 run 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 (x1 , y1 ) is m=
y = y 1 +m (x -x 1 ) 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 a > 0 or down if a< 0 and has a vertex at ( h, k ) . Parabola/Quadratic Function y = a x 2 + b x + c f ( x ) = ax 2 + bx + c The graph is a parabola that opens up if a > 0 or down if a< 0 and has a vertex æ b æ b öö at ç - , f ç ÷÷ . è 2 a è 2 a øø
Parabola/Quadratic Function x = ay 2 + by + c g ( y ) = ay 2 + by + c The graph is a parabola that opens right if a > 0 or left if a < 0 and has a vertex æ æ b ö b ö at ç g ç ÷, - ÷ . è è 2a ø 2 a ø Circle (x -h )2 + ( y - k )2 = r 2 Graph is a circle with radius r and center ( h, k ) . Ellipse 2 ( x - h)
( y - k) +
2
=1 a2 b2 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 ( x - h) 2 ( y - k ) 2 =1 a2 b2 Graph is a hyperbola that opens left and right, has a center at (h, k ) , vertices a units left/right of center and asymptotes b that pass through center with slope ± . a Hyperbola ( y - k )2 - ( x - h )2 = 1 b2 a2 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 b slope ± . a
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
© 2005 Paul Dawkins
Common Algebraic Errors Error
Reason/Correct/Justification/Example
2 2 ¹ 0 and ¹ 2 0 0
Division by zero is undefined!
- 32 ¹ 9
- 32 = - 9 , (- 3 ) = 9 Watch parenthesis!
(x )
(x )
2
3
2
¹ x5
2
a + bx ¹ 1 + bx a - a ( x -1 ) ¹ - ax - a
(x + a )
¹ x +a 2
Make sure you distribute the “-“! ( x + a )2 = ( x + a )( x + a ) = x 2 + 2ax + a 2
2
x2 +a2 ¹ x +a x +a ¹ x + a
(x + a ) n ¹ x n + a n
= x2 x2x 2 = x 6
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 canceling! - a ( x -1) = - ax + a
a a a ¹ + b+c b c 1 ¹ x -2 + x -3 2 x + x3
2
3
and n x + a ¹ n x + n a
5 = 25 = 32 + 42 ¹ 32 + 42 = 3 + 4 = 7 See previous error. More general versions of previous three errors. 2 ( x +1) = 2 ( x 2 + 2 x +1 ) = 2 x 2 + 4 x + 2 2
2 ( x + 1) ¹ ( 2 x + 2 ) 2
2
(2 x + 2 ) 2 ¹ 2 ( x + 1 ) 2 -x 2 + a 2 ¹ - x 2 + a 2 a ab ¹ æbö c ç c÷ è ø æ aö ç b ÷ ac è ø¹ c b
( 2x + 2 )2 = 4x 2 + 8x + 4 Square first then distribute! See the previous example. You can not factor out a constant if there is a power on the parenthesis! -x + a = ( - x + a 2
2
2
1 2 2
)
Now see the previous error. æ aö ç ÷ a 1 æ a öæ c ö ac = è ø = ç ÷ç ÷ = æ b ö æ b ö è 1 øè b ø b ç ÷ ç ÷ è cø è cø æ aö æ aö ç b÷ ç b÷ è ø = è ø = æ a öæ 1 ö = a ÷ç c ÷ bc c æ c ö èç b øè ø ç 1÷ è ø
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
© 2005 Paul Dawkins...