Pre calc chapter 1
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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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Pre Calc—Chapter 1

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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


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 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 Numbers

  • Commutative laws: For all real numbers a,b,a + b = b + a and a . b = b . a.


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 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 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

    • Collection of objects or elements

    • Elements

      • Objects in a set

  • Set Builder Notation


Set Notation

  • Let S and T be sets:

  • Union

  • Intersection

  • Empty


Intervals

  • Open Intervals

  • Closed Intervals

    • Can Infinity be a closed interval?


Absolute Value


Properties of Absolute Value


p.12 #1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49


Exponents and Radicals—1.2


Exponential Notation


Properties of Exponents


Properties of Exponents


Properties of Exponents


Properties of Exponents


Radicals


Properties of Radicals


Properties of Radicals


Rational Exponents

=????


Rational Exponents

  • Examples:


Examples


p.23 #13, 16, 19-22, 37-40, 48,49


Simplifying radicals by rewriting as rational expressions


Simplifying radicals by rewriting as rational expressions


Rationalizing the Denominator

  • Multiply the entire fraction (top and bottom) by the denominator…or by 1


Rationalizing the Denominator

  • Examples:


p.24 #53-67, 70, 71, 86


Algebraic Expressions—1.3


Scientific Notation


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


Definitions

  • Degree

    • Highest power of the variable

  • Monomials

  • Binomial

  • Trinomial

  • Polynomials


Polynomials

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


Adding/Subtracting Polynomials


Product of Polynomials

  • FOIL


Product of Polynomials

  • Examples:


Special Product Formulas


Special Factoring Formulas


Factoring


Factoring


Factoring


Factoring


p.33 #6-16, 30-33


Factoring Completely


Factoring


Factoring


Factoring


Factoring


Factoring By Grouping


Factoring By Grouping


p.33 # 35-38, 47-54, 65-68, 78-80


Fractional Expressions—1.4


Simplifying Fractional Expressions


Simplifying Fractional Expressions


Simplifying Fractional Expressions


Simplifying Fractional Expressions


Simplifying Fractional Expressions


Simplifying Fractional Expressions


Compound Fractions

  • Is a fraction in which the numerator, the denominator, or both, are themselves fractional expressions


Compound Fractions


Compound Fractions


Compound Fractions


Rationalizing the Denominator

  • Multiply by the conjugate:

    • Denominator:

    • Conjugate:


Rationalizing the Denominator


Rationalizing the Denominator


Rationalizing the Numerator

  • Used often in calculus type settings

  • Same rules apply as rationalizing the denominator


Rationalizing the Numerator


Rationalizing the Numerator


p.43 #25. 27, 29, 41, 42, 51, 59-68


Equations—1.5


Properties of Equality


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 Equations


Quadratic Equations

  • Zero Product Property


Quadratic Equations


Completing the Square

  • Rewrite equation so it is in the form:

  • Example:


Completing the Square


The Quadratic Formula


The Discriminant


Using the Discriminant

  • Determine how many solutions (roots) an equation has without actually solving


p.57 #1-41 Every Other Odd


Equations—1.5 (Day 2)


Solving Equations


Solving Equations


Solving Equations


Solving Equations


Solving Equations


Solving Equations


p.57-58 #45, 49, 53, 56-62


Modeling Equations—1.6


Consecutive Integers

  • What does it mean to be consecutive?

    • In terms of addition?

  • How can you write consecutive integers algebraically?


Consecutive Integers

  • The sum of the squares of two consecutive integers is 1252. Find the integers:


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


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?


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?


p.70-71 #1-3, 9-11, 16-18, 30-32


Inequalities—1.7


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 Inequalities

1.) Move all terms to one side

2.) Factor

3.) Find the Intervals

4.) Make a table

5.) Solve

6.) Smile 


Non-Linear Inequalities

  • Example:


Non-Linear Inequalities

  • Example:


Absolute Value


p.87 #22-27, 47-49


Coordinate Geometry—1.8


Coordinate Geometry

  • X-axis

  • Y-axis

  • Origin

  • Ordered Pair

  • Coordinates


Coordinate Geometry


Coordinate Geometry


Coordinate Geometry


Distance Formula

  • Finds the linear distance between any two points, a and b


Distance Formula

  • Is P(-1, 3) or Q(7, 10) closer to point A(5, -1)


Midpoint Formula

  • The midpoint between point A and B is:


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

  • A circle with a center (h, k) and radius r is:

  • Is a circle a function?

  • How can it become a function?


Circles

  • Graph:


Circles

  • Graph:


Circles

  • Find the equation of a circle with radius 3 and center at (1, -2)


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


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


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

  • Example:

    • x-axis?

    • y-axis?

    • Origin?


p.100#3-5, 10-11, 13-15, 22-24, 27, 34, 73,74


Lines—1.10


Slope

  • The slope, m, of a non-vertical line that passes through the points


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

  • 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 line, thru the point (a, b)

    • x = a

    • Function??

  • Horizontal line, thru the point (a, b)

    • y = b

    • Function?


Standard Form

  • The Graph of every linear equation

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


Standard Form--Graphing


Parallel lines

  • Two non-vertical lines are parallel iff they have the same slope


Perpendicular Lines

  • Two lines with slopes are perpendicular iff , that is their slopes are negative reciprocals:

  • Also, horizontal lines are perpendicular to vertical lines


p.124#1-31 Odd, 54-56, 58-59


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