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Algebra Cheat Sheet

Basic Properties & Facts

Arithmetic Operations

Exponent Properties

Properties of Radicals


,if n is odd
,if n is even

Properties of Inequalities

If then and
If and then and
If and then and

Properties of Absolute Value

Triangle Inequality

Distance Formula

If and are two
points the distance between them is

Complex Numbers


Complex Modulus
Complex Conjugate

 

   
Logarithms and Log Properties  
Definition
is equivalent to

Example
because

Special Logarithms
natural log
common log
where e = 2.718281828K
 
Logarithm Properties

The domain of

   
Factoring and Solving
Factoring Formulas

If n is odd then


 

Quadratic Formula

Solve ax2 + bx + c = 0, a ≠ 0

If b2 - 4ac > 0 - Two real unequal solns.
If b2 - 4ac = 0 - Repeated real solution.
If b2 - 4ac < 0 - Two complex solutions.

Square Root Property

If x2 = p then

Absolute Value Equations/Inequalities
If b is a positive number

   
Completing the Square
Solve 2x2 - 6x -10 = 0

(1) Divide by the coefficient of the x2

x2 - 3x - 5 = 0

(2) Move the constant to the other side.

x2 - 3x = 5

(3) Take half the coefficient of x, square
it and add it to both sides

(4) Factor the left side

(5) Use Square Root Property

(6) Solve for x

   
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 and is

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 is



Parabola/Quadratic Function


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 = ax2 + bx + c f (x ) = ax2 + bx + c

The graph is a parabola that opens up if
a > 0 or down if a < 0 and has a vertex
at

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
at .
 

Circle
( x - h)2 + ( y - k )2 = r2
Graph is a circle with radius r and center
(h, k ) .

Ellipse

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

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 .

Hyperbola

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 .

Common Algebraic Errors

Error Reason/Correct/Justification/Example
and Division by zero is undefined !
Watch parenthesis !
A more complex version of the previous
error

Beware of incorrect canceling!

Make sure you distribute the “-“!
See previous error .
and More general versions of previous three
errors.

Square first then distribute!
See the previous example. You can not
factor out a constant if there is a power on
the parethesis!

Now see the previous error.
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