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Algebra Cheat Sheet

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Algebra Cheat Sheet
Basic Properties & Facts
Arithmetic Operations
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
æ b ö ab
aç ÷ =
ècø c
ab + ac = a ( b + c )
æaö
ç ÷ a
èbø =
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 ÷ ad
èbø =
æ c ö bc
ç ÷
èdø
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
a n a m = a n+m
an
1
= a n-m = m-n
m
a
a
(a )
a 0 = 1, a ¹ 0
( ab )
n
a -n =
æaö
ç ÷
èbø
-n
= a nm
n
1
an
n
bn
æbö
=ç ÷ = n
a
èaø
n
m
1
a = an
m n
a = nm a
( x2 - x1 ) + ( y2 - y1 )
2
2
n
Complex Numbers
i = -1
( ) = (a )
a = a
Properties of Radicals
n
d ( P1 , P2 ) =
a
æaö
ç ÷ = n
b
èbø
1
= an
-n
a
= a nb n
Triangle Inequality
Distance Formula
If P1 = ( x1 , y1 ) and P2 = ( x2 , y2 ) are two
points the distance between them is
Exponent Properties
n m
a
a
=
b
b
ab = a b
1
m
n
n
1
m
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
n
ab = n a n b
a + bi = a 2 + b 2
n
a na
=
b nb
( a + bi ) = a - bi Complex Conjugate
2
( a + bi )( a + bi ) = a + bi
n
a n = a, if n is odd
n
a n = a , if n is even
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
Complex Modulus
© 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
log 5 125 = 3 because 53 = 125
b logb x = x
log b ( x r ) = r log b x
log b ( xy ) = log b x + log b y
Special Logarithms
ln x = log e x
natural log
æxö
log b ç ÷ = log b x - log b y
è yø
log x = log10 x common log
where e = 2.718281828K
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 + 2ax + 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 )
x3 + 3ax 2 + 3a 2 x + a 3 = ( x + a )
x3 - 3ax 2 + 3a 2 x - a 3 = ( x - a )
3
3
Square Root Property
If x 2 = p then x = ± p
x3 + a3 = ( x + a ) ( x 2 - ax + a 2 )
x3 - a 3 = ( x - a ) ( x 2 + ax + a 2 )
x -a
2n
2n
= (x -a
n
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 + a n
Absolute Value Equations/Inequalities
If b is a positive number
p =b
Þ
p = -b or p = b
p <b
Þ
-b < p < b
p >b
Þ
p < -b or
p>b
= ( x + a ) ( x n -1 - ax n - 2 + a 2 x n -3 - L + a n -1 )
Completing the Square
(4) Factor the left side
Solve 2 x - 6 x - 10 = 0
2
2
(1) Divide by the coefficient of the x 2
x 2 - 3x - 5 = 0
(2) Move the constant to the other side.
x 2 - 3x = 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 - 3x + ç - ÷ = 5 + ç - ÷ = 5 + =
4 4
è 2ø
è 2ø
3ö
29
æ
çx- ÷ =
2ø
4
è
(5) Use Square Root Property
3
29
29
x- = ±
=±
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.
© 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 ( x2 , y2 ) is
y2 - y1 rise
=
x2 - 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 = y1 + m ( x - x1 )
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 = ax 2 + bx + 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 2a è 2 a ø ø
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
2
2
( x - h) + ( y - k ) = r 2
Graph is a circle with radius r and center
( h, k ) .
Ellipse
( x - h)
2
( 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
( x - h)
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
=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 ,
(x )
(x )
2 3
2 3
¹ x5
a
a a
¹ +
b+c b c
1
¹ x -2 + x -3
2
3
x +x
- a ( x - 1) ¹ - ax - a
2
x+a ¹ x + a
n
= 9 Watch parenthesis!
= x2 x2 x2 = x6
( x + a)
¹ x2 + a2
x2 + a2 ¹ x + a
( x + a)
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 canceling!
- a ( x - 1) = - ax + a
Make sure you distribute the “-“!
a + bx
¹ 1 + bx
a
( x + a)
( -3 )
¹ x n + a n and
n
x+a ¹ n x + n a
= ( x + a )( x + a ) = x 2 + 2ax + a 2
2
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 x + 2)
2
2
2
¹ 2 ( x + 1)
( 2 x + 2)
2
= 4 x2 + 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!
1
- x2 + a2 ¹ - x2 + a2
a
ab
¹
æbö c
ç ÷
ècø
æaö
ç ÷ ac
èbø ¹
c
b
- x2 + a2 = ( - x2 + a 2 ) 2
Now see the previous error.
æaö
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 ÷
ècø ècø
æaö æaö
ç ÷ ç ÷
è b ø = è b ø = æ a öæ 1 ö = a
ç ÷ç ÷
c
æ c ö è b øè c ø bc
ç ÷
è1ø
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
© 2005 Paul Dawkins
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