SPRING SEMESTER S2

Conic Sections

Chapter 8 Rationals

Rational Expressions
Find common denominators, combine like terms, factor is possible and reduce.

Adding and Subtracting Compound Rationals (nested fractions) and negative exponents:

Chapter 7 Polynomials

Graphing Polynomials in Factored Form: (Polynomial Graphs KEY)

FALL SEMESTER S1

Changes to FER#2:
Change the right hand side of the equation to
ln(62-25x)
and it should work

Problem XVII should have a starting amount of 100 bacteria instead of just 1.

Chapter 6 Logarithms

Chapter 3 Systems

October 15, 2013
Linear Inequalities Quiz Review and More Mixture Problems

October 14, 2013

Chapter 2 Functions

Chapter 1 Foundations Review

September 9, 2013

Nine zulu queens rule China but the king awakes and surveys his
Domain (horizon from left to right, x-values) and imagines the
Range (vertical up and down y-values) of possibilities.  "Sky's the limit."  y goes with sky.

Exponents are defined as repeated multiplication.
5^2 means multiply 5 by itself twice, 5*5 = 25.
2^3 means multiply 2 by itself three times, 2*2*2 = 8.

This also means add exponents when multiplying identical bases:
x^2 * x^3 = x*x*x*x*x = x^5 = x^(2+3).
Likewise, subtract exponents when dividing identical bases:
a^4 / a^3 = a*a*a*a / (a*a*a) = a (a*a*a) / (a*a*a) = a ^ 1 = a ^ (4-3) = a.

Since a negative is the opposite of a positive, a negative exponent means
do the opposite of multiplying.  So divide.  For example,
5^(-1) means do the opposite of multiplying by 5 once.  So instead of 1*5,
do 1/5.  So 5^(-1) = 0.2.
5^(-2) means divide by 5 twice.  1/5/5 = 1/5^2 = 1/25.

If 2*2*2 = 8, then
8^(1/3) = 2 (one third of the product) and
8^(2/3) = 2*2 = 4 (two thirds of the product.)
Therefore, 8^(1/3) is the cube root of 8, or 2.
Square that to get 8^(2/3), or 2^2 = 4.

Unit 2

Nine Zulu Queens Rule China:
N = Natural "counting" numbers
Z = Integers, 0 with positive and negative natural numbers
Q = Rational numbers, ratios of two integers, Q meaning quotient.
(Proper and improper fractions, mixed numbers, repeating and terminating decimals.)
R = Real numbers, which is the set of Q along with irrational numbers
(non-repeating non-terminating decimals, e.g. Φ = 1.618033..., √2, 0.12345678910111213..., 0.101001000100001...)

Closure is the property by which the resulting answer is a member of the same set as the two operands.
For example, adding or multiplying two natural numbers results in another natural number.
So the Natural number set is closed under addition and multiplication.  But not subtraction because 2-5 = -3.
However, the set Z of integers is closed under addition, multiplication, AND subtraction as sums, products, and differences of integers are still integers.  But not quotients.  Sometimes dividing two integers yields a fraction or a decimal, in other words, a rational number.
Therefore, the set of rationals Q is closed under all four operations (+, -, *, /) and integral powers.  However, inverting powers, i.e. taking roots, often results in an irrational number.  √2, √3, √5, etc. all result in non-terminating non-repeating decimals.  This development was once so controversial it cost a mathematician his life as Hipparchus showed his Pythagoreans colleagues that √2 was not a fraction, and therefore all was not number as they had previously believed every number could be written as a fraction.  They were none too happy with him about this discovery.

Quite often, a new number set was discovered and developed by inverting an operation on a simpler set
(subtracting Naturals, dividing Integers, rooting Rationals.)  Indeed, square rooting a negative number is not closed in the set of Reals, which leads to another larger number set denoted by C.  We will learn these imaginary complex numbers in a later chapter.

Review Unit 1

Unit 1 Lesson 1 Notesheet 1.1 (Slope Triangles and x- and y-intercepts)

Notesheet 1.2 (Linear Equations and Solving Systems of Equations)

Unit 1 Lesson 8/22/2013 (Factoring Trinomials, Parallel Lines)

Perpendicular Lines, Graphing Parabolas

Solving absolute value EQuations and Quadratic EQuations