CLEP College Algebra

1^XX

1

1 to any power is ALWAYS 1

Negative Exponents

indicate how many times to DIVIDE one by that number

EX: 8^-1 -- indicates how many times to divide 1 by. 1/8 = 0.125

EX: 8^-3 -- 1 / 8 / 8 / 8 ---> or 1/8^3

if a number has a negative exponent then put it in the denominator and make the exponent positive

Perpendicular Lines

two lines with slopes that are negative reciprocals:

-1

m2 = --- , or m1m2 = -1

m1

Difference of Squares

a^2 - b^2 = (a + b) (a - b)

EX: 4x^2 - 25y^2

= (2x + 5y) (2x - 5y)

Intercept Form

x y a = x-intercept

--- + --- = 1 b = y-intercept

a b

Y-Axis Symmetry

"even functions"

if the curve is the same on either side of the y-axis

Q1 = Q2 or Q3 = Q4

Conjugate

change the addition to subtraction and vice versa

i.e. (c + di) --> (c - di)

I r I >= 1

the sum is NOT convergent -- it is DIVERGENT and we CANNOT determine a sum

Counting Principle

m x n

one event can occur m ways

and a second event can occur in n ways

the total way both events can occur is:

= m x n

Inverse of a function

switch x and y then solve for the new y

EX: y = 3/2x - 8

Inverse: x = 3/2y - 8

**Distributive**

a(b+c) = ab + ac

anything to the ZERO power = 1

8^0 = 1

Rational

***HELP*

- a number that can be expressed as a/b (and is in

its LOWEST terms)

- the ratio of 2 integers, b =/= 0

- All fractions and terminating or repeating decimals are rational

Irrational

Any number that is NOT rational

Natural Numbers

"counting numbers"

does NOT include negatives or zero

{1, 2, 3, 4 ...}

Whole Numbers

Includes zero

{0, 1, 2, 3, 4, 5 ...}

Integers

natural whole numbers:

negative & positive whole numbers INCLUDING zero

{...-3, -2, -1, 0, 1, 2, 3 ...}

Set

a collection of objects

the objects can be related or completely unrelated

usually denoted by capital letters and using {}

EX: A = {2, 4, 6, 7}

Subset

Given two sets A and B, A is said to be a subset of B if every element of set A is also a member of set B

Element

each individual item belonging to a set is an element or member of that set

A = {2, 4, 6, 7}

set A has 4 elements

Union

the union of sets A and B, denoted A u B, is the set of ALL elements that are EITHER in A or in B or in BOTH

Intersection

the set of all elements that belong to BOTH A and B

Disjoint

no elements in common

Complement of a set

the complement of set A, denoted as A', is all elements of the universal set NOT contained in A

Absolute Value

the distance of a number from zero on a number line; NEVER negative

Base

a^b

a is the base

0^XX

0

Zero to any power is ALWAYS zero

Radical

in the form of n root a

Order of Operations

P = ( )

E = exponents

M = multiplication

D = division {if both M&D, go left to right}

A = addition

S = subtraction {same as M and D}

Nested Parentheses

work from innermost to the outermost parentheses

Factors

one or more expressions that when multiplied together, produce a given result

- the way to reduce a fraction is to write the numerator and denominator as the product of 2 (or more integers) called factors, the cancel all the factors that appear in both

Reciprocal

1 over the fraction

used when dividing fractions

dividing by a fraction is the same as multiplying by a fractions reciprocal

a/b // c/d = a/b x d/c = ad/bc

Scientific Notation

Standard Notation

Polynomials

exponents must be POSITIVE integers

Constant

a quantity that does NOT change

Coefficient

the number that is infront (multiplied) of a variable or a term

6a + 1b --> 6 and 1 are the coeffs

FOIL Method

used when multiplying two BINOMIALS

Common Factor

ab + ac = a(b + c)

Grouping

a(c + d) + b(c + d) = (c + d)(a + b)

Rational Expression

an algebraic expression that can be written as a quotient of two polynomials: p/q

Domain

the x coordinates; if repeated only list once

EX: The domain of the Relation

R = { (-2,-2), (-1,0), (0,-1), (0,-2) }

Domain: {-2, -1, 0}

Range

the y coordinates; if repeated only list once

EX: The range of the Relation

R = { (-2,-2), (-1,0), (0,-1), (0,-2) }

Range: {-2, 0, -1}

Relation

a set of points

EX: Relation R = {(2,2), (1,1), (1,0)}

Ordered Pair

a "point" in the form (x,y)

Function

a relation, or set of points (x,y), such that for every x, there is one and ONLY one y

The x values CANNOT repeat but the y values CAN repeat

Vertical Line Test

used to determine whether a graph is a function or not

vertical line cannot intersect at more than one point

Linear Function

a straight line

Slope

the change in y over the change in x

measures the steepness of a line

RISE Y2-Y1

m = ------ = -----------

RUN X2-X1

Slope Intercept Form

y = mx + b

m = slope

b = y-intercept

(only vertical lines will not have a y-intercept)

Y-Intercept

(0 , b)

the point at which the line crosses the Y axis

y = mx + b

When to use a RECIPROCAL:

Reciprocal = 1/x

or flip the denominator and multiply by

numerator

dividing fractions is the SAME as multiplying by the fractions reciprocal

Use a reciprocal when DIVIDING fractions

i.e.

a/b a c a d

------- or --- / --- ---> --- x ---

c/d b d b c

When to multiply each term by the Denominator:

to get rid of a fraction

How to get rid of a fraction

multiply each term by the denominator

(NOT the reciprocal)

EX: y = (-2/3)x - 2

*multiply each term by the denominator*

--> y(3) = (-2/3)(3)x - 2(3)

--> y(3) = -2x - 6

Point Slope Form

Y - Y1 = M(X - X1)

easiest way to find the equation of a line if given 2 points

Parallel Lines

2 lines with the SAME slope

m1 = m2

Quadrants on a Graph

Q2 I Q1

------I--------

Q3 I Q4

Symmetry

a curve that has the same exact shape on either side of a particular line

Origin Symmetry

"odd functions"

if the curve is the same on either side of the origin (0,0)

Q1 = Q3 and Q2 = Q4

Characteristics of a Function:

1) a set of points (x,y)

2) for every x there is one and ONLY one y

x values CANNOT repeat but

y values CAN repeat

use the Vertical Line Test to check

Second Degree Equations

ax^2 + bx + c = 0

called "Quadratic Equations"

Quadratic Functions

ax^2 + bx + c = 0

Roots: can have either 2,1, or 0 roots -- these indicate where the function equals zero

To solve - set the function = 0 by moving all terms to one side -- then simplify, factor, set each factor = 0 and solve for x

OR you can use the Quadratic Formula to solve for x

Roots

where it crosses/touches the x-axis

(NOT the y-axis)

# of roots = # of solutions

Characteristics of a Polynomial

1. consist of 1 or more terms

2. terms are either constants or the product

of constants

3. exponents must be POSITIVE integers

1 term = monomial

2 terms = binomial

3 terms = trinomial

Degree

the HIGHEST degree of all of the terms

(the degree of a constant is 0)

Prime

cannot be factored

Point Slope Form

y - y1 = m(x - x1)

find the equation of a line passing through point (x1, y1) with slope m

Intercept Form

x y

--- + --- = 1

a b

find equation of a line with x-intercept a and y-intercept b ---

System of Linear Equations

a set of two or more linear equations with solutions that are TRUE at the same time

"Simultaneous Equations"

Form: ax + by = c

dx + ey = f

Consistent

A system of linear equations with ONE solution (independent)

Inconsistent

A system of linear equations with NO solutions

same slope but different y-intercepts

AKA - parallel lines

Powers raised to powers

indicate MULTIPLICATION of the powers

Logarithms

the opposite of exponential functions

Common Logarithm

base of 10

if base not specified -- it's assumed to be 10

Natural Logs (ln)

base of e

e = 2.718...

Natural Numbers

1, 2, 3, 4, ...

Whole Numbers

0, 1, 2, 3, 4

T/F Every natural number is a WHOLE number, but NOT every whole number is a natural number.

Every natural number is a whole number, but NOT every whole number is a natural number.

True

Integers

... -2, -1, 0, 1, 2, 3...

Rational Number

written as a fraction in its simplest terms

a/b when b =/= 0

either a fraction or decimal

that has an end

or repeats

Mixed Number

a number with a WHOLE NUMBER part and a decimal part

i.e.: 1.7979

if the repeating decimal occurs ---

Irrational Numbers

all numbers that are not rational

- they CANNOT be written as fractions

- include decimals that never end

- never repeat

EX: pi

Real Numbers

include all rational numbers and all irrational numbers

Imaginary Numbers

invented to include square roots of negative numbers

any number in the form of bi i.e. 3i

b = a real number

i = imaginary

Imaginary Unit

the square root of negative -1

"i"

root -1 = i and i^2 = -1

Complex Numbers

form: a + bi

ALL numbers are complex

EX: 4, 5i, -2, -3i, 1/2, 4.7i, root3, -iroot21

Series

the sum of a sequence

Arithmetic Series

The sum of the terms of an arithmetic sequence

Sn

d = common difference

a1 = the first term

Geometric Series

The sum of the terms of an geometric sequence

constant = r

Geometric Sequence

r = common ratio

r is multiplied by EACH term to get to the next

If an infinite geometric series does have a sum, it is convergent

I r I < 1

use the formula to find the sum of the infinite geometric series:

a1

S = -----

1 - r

S is to infiniti

As n gets LARGER, r^n gets SMALLER

since r^n is a fraction of a fraction of a fraction... getting close to, but NOT reaching 0

T

n factorial

n!

Permutation

order makes a difference

Combination

order does NOT make a difference

0! = 1

true

Multiplicative Inverse****

the RECIPROCAL of a number a which is 1/a

NOT to be confused with the inverse of a function

Additive Inverse****

the NEGATIVE of a given number

ie. a --> -a

Domain of a Function

The set of allowable x-values -->

The domain of a function f is (-infiniti, +infiniti) EXCEPT for:

1. values of x that create a ZERO in the denominator

2. an EVEN ROOT of a negative number

3. a logarithm of a negative number

if there is NO radical and NO denominator -- then there are NO restrictions and thus the domain is (-infiniti, +infiniti)