Complex Numbers
a+bi , where "a" is the real part and "bi" is
A complex number is any number that can be written as
the imaginary part.
Examples:
32 i , 4 +8 i , 7 i , -4i
Complex
Numbers
Real
Numbers
Rational
Numbers
3
The imaginary number
Imaginary
Numb
Simplifying Square Roots
To simplify a square root, make the number under the radical as small as possible (but still a whole
number). For example, simplify
12
Start by factoring the number under the radical, using perfect squares as
factors.
The Product
Solving Quadratic Equations
In Algebra 1, you learned to solve quadratic equations several different ways: by graphing, by factoring,
by taking square roots, and using the quadratic formula. We are going to review these methods and
expand on them in Algeb
A linear equation in three variables
where
y ,
x ,
y , and
z , is an equation of the form ax +by +cz=d ,
a , b , and c are not all zero. A solution to such a system is an ordered triple ( x ,
z ) whose coordinates are a solution to each equation and a sol
Tell whether the ordered pair is a solution to the system
(4, 1)
cfw_
x +2 y=6
x y =3
Substitute 4 for
x and 1 for
y
in both
equations. If the resulting statements are true,
the ordered pair is a solution to that system.
x+ 2 y =6
x y=3
41=3
4 +2(1)=6
3=3
Solving Systems by Elimination
Solving
cfw_x52x +2y =19
y=1
by elimination
1. Add the two equations
together to eliminate one of the
variables.
x2 y=19
+
5 x+2 y=1
6 x=18
x=3
x2 y=19
2. Substitute the value of that
variable into one of the original
equati
A parent function is the most basic function in a family.
Function
Constant
In the equation, look for:
y = a number with no variable
There is no x-term.
Parent:
Linear
Absolute Value
f (x)= x
The x-term has an exponent
of 2.
Parent:
Square Root
f (x)=x
Th
A relation is a set of inputs and outputs, often written as ordered pairs (input, output). We can also
represent a relation as a mapping diagram, xy table or a graph. For example, the relation can be
represented as:
Mapping Diagram
0
1
2
3
-2
1
2
4
Ordere
In mathematics, a collection of objects is called a set. Each one of these objects is an element of the set.
If all the members of one set are also members of another set, they are called a subset. For example:
Elements of this set are A, B, and 1
Subsets
Some of the basic transformations that you learned in Algebra I and Geometry are translations and
reflections. A translation is a slide and a reflection is a flip.
Horizontal translations change the domain of a function and appear inside parentheses, abso
Solving Rational Inequalities
Graphically and Algebraically
EQ:
Graphically:
1)
On the graphing calculator enter the left side of the inequality into y1
and enter the right side of the inequality into y2. Be sure to use parentheses
around
y and
y.
1
2
qua
Solving Log Equations
EQ:
Methods for solving logarithmic equations:
1) If logarithms with the same base on both sides of the equation ,
simplify to a single logarithm on each side. If the bases are the same,
then the values must be equal, so drop the log
Synthetic Division
EQ:
Synthetic division is a shorthand method of dividing a polynomial by
a first-degree or linear binomial (ex. x+3; 4x 1) using only the
coefficients and constant.
Synthetic Division method:
1. Arrange the terms of the polynomial in de
Solving Rational Equations
EQ:
An equation with rational expressions can be difficult to solve with
the fractions in it. You can make these equations easier to solve by
using properties of equations to eliminate the rational expressions.
Steps:
1. Find va
VARIATION FUNCTIONS
EQ:
Direct variation is a relationship between two variables x and y that can be
written in the form
is k, so the ratio of
. The graph of a direct variation is a line passing
k
y where k 0.
x
The product of
x y k
. The
graph of an inve
Solving Radical Equations I
EQ:
A radical equation contains a variable within the radical.
(Ex.
;
)
3 x2 9
3
x 3 2 x 1
If equation has just one radical expression:
1. Isolate the radical on one side of the equation and move
everything else to the other si
Solving Exponential Equations
EQ:
Methods for solving exponential equations:
1) Change both sides of the equation to the same base. If the bases in an
equation are equal, then the exponents must also be equal. Drop the
bases and solve for the variable by
Characteristics of Functions
EQ:
Remember that calculator functions listed under CALC (2nd Trace) can be
used to find zeros, maximum and minimum values of a function. Use the AP
standard to give values to 3 decimal places.
The Location Principle: If is a
Polynomial Functions
EQ:
A polynomial function of degree n is a function in the form
P ( x) an x n an 1 x n 1 L a1 x a0
an 0
. The numbers
The number
a0
where n is a nonnegative integer and
a0 , a1 , a2 ,K , an
are coefficients of the polynomial.
is the c
The Quadratic Formula and the Discriminant
EQ:
In
f ( x) ax bx c
2
the quadratic formula will yield the real or non-
real solutions. For non-real (complex/imaginary) solutions, the
parabola will not cross the x-axis and for the real solutions it crosses
t
Solving Polynomial Equations
EQ:
Some polynomial equations can be solved by factoring. Zeros of
the function sometimes can be found by factoring.
Rational Root Theorem: If a polynomial has integer coefficients,
then every rational root has the following f
Complex Numbers
EQ:
A complex number is any number that can be written as a + bi, where "a" is
the real part and "bi" is the imaginary part. Real numbers are complex
numbers where
b 0.
Examples:
3 2i; 7i; 4i 3 (The two last ones are called pure imaginary
Radical Inequalities
EQ:
A radical inequality is an inequality that contains a variable within a
radical. You can solve by graphing or using algebra.
Solving Square Root Inequalities Graphically:
1) Using the graphing calculator, enter the left side of t
Completing the Square
EQ:
Square root property:
If x a and a is a non-negative real number, then x a .
2
In other words,
to solve a quadratic equation, you can take the square root of both
sides of the equation. Be sure to consider the positive and negati
Operations with Radicals and Rational Expressions
EQ:
Exponent Properties from Algebra 1 to remember:
a m a n a
a 0 1 (a 0)
1
a
a
1
a
b
1
b
a
mn
am
mn
a
n
a
a b
m
a
m n
n
a b
m
m
a
b
a mn
an
n
b
To perform operations with radicals or rational
Operations with Polynomials
EQ:
To add polynomials, add like terms. To subtract polynomials, add the opposite of the
second polynomial. Write polynomial in standard form.
To multiply a polynomial use the distributive property:
then
a( x b) ax ab
combine l
Simplifying Radicals and Operations
EQ:
Radical: an indicated root of a quantity (Ex.
;
)
3
24
35
Radicand: the expression under the radical symbol
Index or Root: in the radical
which represents the nth root of x, n is
n
x
the index; when no index appears
Multiplying and Dividing Rational Expressions
EQ:
A rational expression is a quotient of two polynomials.
(Ex. of rational expressions:
).
x 2 4 10 x 3
; ;
x2 x x7
Simplifying, multiplying or dividing rational expressions works the same way as
multiplying
Applications of Radical Equations
EQ:
Radical equations are used to represent relationships between various objects.
Substitute known values into the formula given and then use inverse
operations to isolate the variable remaining. Usually a calculator wil
Solving Radical Equations II
EQ:
Things to remember about solving radical equations:
1) A radical equation contains a variable in the radicand.
2) Two types of radical equations: those with just one radical expression
where you isolate the radical express