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Pre Calc—Chapter 1. Fundamentals. Real Numbers. Whole (Natural) Numbers Counting (kindergarten) numbers Integers Natural numbers along with their negatives and 0 Rational Numbers Ratios of integers Irrational Numbers Numbers that cannot be expressed as ratios of integers.

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real numbers
Real Numbers
  • Whole (Natural) Numbers
    • Counting (kindergarten) numbers
  • Integers
    • Natural numbers along with their negatives and 0
  • Rational Numbers
    • Ratios of integers
  • Irrational Numbers
    • Numbers that cannot be expressed as ratios of integers
properties of real numbers
Properties of Real Numbers
  • Closure: For all real numbers a,b, the sum a + b and the product a . b are real numbers.
properties of real numbers1
Properties of Real Numbers
  • Associative laws: For all real numbers a,b,c, a + (b + c) = (a + b) + c and a . (b . c) = (a . b) . c.
properties of real numbers2
Properties of Real Numbers
  • Commutative laws: For all real numbers a,b,a + b = b + a and a . b = b . a.
properties of real numbers3
Properties of Real Numbers
  • Distributive laws: For all real numbers a,b,c, a . (b + c) = a . b + a . c and (a + b) . c = a . c + b . c.
properties of real numbers4
Properties of Real Numbers
  • Identity elements: There are real numbers 0 and 1 such that for all real numbers a,a + 0 = a and 0 + a = a (addition)

a . 1 = a and 1 . a = a (multiplication)

properties of real numbers5
Properties of Real Numbers
  • Inverse elements: For each real number a, the equations a + x = 0 and x + a = 0 have a solution x in the set of real numbers, called the additive inverseof a, denoted by -a.
  • For each nonzero real number a, the equations a . x = 1 and x . a = 1 have a solution x in the set of real numbers, called the multiplicative inverse of a, denoted by a-1.
set notation
Set Notation
  • Set
    • Collection of objects or elements
    • Elements
      • Objects in a set
  • Set Builder Notation
set notation1
Set Notation
  • Let S and T be sets:
  • Union
  • Intersection
  • Empty
intervals
Intervals
  • Open Intervals
  • Closed Intervals
    • Can Infinity be a closed interval?
rationalizing the denominator
Rationalizing the Denominator
  • Multiply the entire fraction (top and bottom) by the denominator…or by 1
definitions
Definitions
  • Variable
    • Letter or symbol representing a number
  • Constant
    • Fixed or specific number
  • Domain
    • The set of numbers a variable is permitted to have
    • Input
definitions1
Definitions
  • Degree
    • Highest power of the variable
  • Monomials
  • Binomial
  • Trinomial
  • Polynomials
polynomials
Polynomials

A polynomial of degree n, where n is a non-negative integer, and

compound fractions
Compound Fractions
  • Is a fraction in which the numerator, the denominator, or both, are themselves fractional expressions
rationalizing the denominator2
Rationalizing the Denominator
  • Multiply by the conjugate:
    • Denominator:
    • Conjugate:
rationalizing the numerator
Rationalizing the Numerator
  • Used often in calculus type settings
  • Same rules apply as rationalizing the denominator
linear equations
Linear Equations
  • 1st degree equation
  • Each term is either a constant or a nonzero multiple of the variable
  • Equivalent to an equation of the form
quadratic equations1
Quadratic Equations
  • Zero Product Property
completing the square
Completing the Square
  • Rewrite equation so it is in the form:
  • Example:
using the discriminant
Using the Discriminant
  • Determine how many solutions (roots) an equation has without actually solving
consecutive integers
Consecutive Integers
  • What does it mean to be consecutive?
    • In terms of addition?
  • How can you write consecutive integers algebraically?
consecutive integers1
Consecutive Integers
  • The sum of the squares of two consecutive integers is 1252. Find the integers:
guidelines
Guidelines
  • 1.)Identify the variable.
  • 2.) Express all unknown quantities in terms of 1 variable
    • Draw a diagram or make a table to help
  • 3.) Set up the model
  • 4.) Solve and check answer
examples1
Examples
  • Mary has $3.00 in nickels, dimes, and quarters. If she has twice as many dimes as quarters and five more nickels than dimes, how many coins of each type does she have?
examples2
Examples
  • Craig is saving to buy a vacation home. He inherits some money from a wealthy uncle, then combines this with the $22,000 he has already saved and doubles the total in a lucky investment. He ends up with $134,000, just enough to buy the cabin. How much did he inherit?
non linear inequalities
Non-Linear Inequalities
  • If a product or a quotient has an even number of negative factors, then its value is positive
  • If a product or a quotient has an odd number of negative factors, then its value is negative
non linear inequalities1
Non-Linear Inequalities

1.) Move all terms to one side

2.) Factor

3.) Find the Intervals

4.) Make a table

5.) Solve

6.) Smile 

coordinate geometry
Coordinate Geometry
  • X-axis
  • Y-axis
  • Origin
  • Ordered Pair
  • Coordinates
distance formula
Distance Formula
  • Finds the linear distance between any two points, a and b
distance formula1
Distance Formula
  • Is P(-1, 3) or Q(7, 10) closer to point A(5, -1)
midpoint formula
Midpoint Formula
  • The midpoint between point A and B is:
intercepts
Intercepts
  • X-intercept
    • Where a graph crosses or touches the x-axis
    • Set y=0 and solve for x
  • Y-intercept
    • Where a graph crosses or touches the y-axis
    • Set x=0 and solve for y
circles
Circles
  • A circle with a center (h, k) and radius r is:
  • Is a circle a function?
  • How can it become a function?
circles1
Circles
  • Graph:
circles2
Circles
  • Graph:
circles3
Circles
  • Find the equation of a circle with radius 3 and center at (1, -2)
symmetry
Symmetry
  • Symmetry with respect to x-axis
    • Test:
      • The equation is unchanged when y is replaced by –y
    • Graph looks like:
      • Graph is unchanged when reflected in the x-axis
symmetry1
Symmetry
  • Symmetry with respect to y-axis
    • Test:
      • The equation is unchanged when x is replaced by –x
    • Graph looks like:
      • Graph is unchanged when reflected in the y-axis
symmetry2
Symmetry
  • Symmetry with respect to the origin
    • Test:
      • The equation is unchanged when x is replaced by -x and y is replaced by –y
    • Graph looks like:
      • Graph is unchanged when rotated 180 degrees
using symmetry to sketch a graph
Using Symmetry to sketch a graph
  • Example:
    • x-axis?
    • y-axis?
    • Origin?
slope
Slope
  • The slope, m, of a non-vertical line that passes through the points
point slope form
Point-Slope Form
  • An equation of the line that passes through
  • Example: Find the equation of the line through (2, -4) with slope of 3
slope intercept form
Slope-Intercept Form
  • An equation of the line that has slope m and y-intercept b is:
  • Example: Find the equation of the line passing through point (3,2) with b=9
vertical and horizontal lines
Vertical and Horizontal Lines
  • Vertical line, thru the point (a, b)
    • x = a
    • Function??
  • Horizontal line, thru the point (a, b)
    • y = b
    • Function?
standard form
Standard Form
  • The Graph of every linear equation

is a line. Conversely, every line is the graph of a linear equation

parallel lines
Parallel lines
  • Two non-vertical lines are parallel iff they have the same slope
perpendicular lines
Perpendicular Lines
  • Two lines with slopes are perpendicular iff , that is their slopes are negative reciprocals:
  • Also, horizontal lines are perpendicular to vertical lines
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