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

Pre Calc—Chapter 1

Fundamentals


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?


Absolute value

Absolute Value


Properties of absolute value

Properties of Absolute Value


P 12 1 5 9 13 17 21 25 29 33 37 41 45 49

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


Exponents and radicals 1 2

Exponents and Radicals—1.2


Exponential notation

Exponential Notation


Properties of exponents

Properties of Exponents


Properties of exponents1

Properties of Exponents


Properties of exponents2

Properties of Exponents


Properties of exponents3

Properties of Exponents


Radicals

Radicals


Properties of radicals

Properties of Radicals


Properties of radicals1

Properties of Radicals


Rational exponents

Rational Exponents

=????


Rational exponents1

Rational Exponents

  • Examples:


Examples

Examples


P 23 13 16 19 22 37 40 48 49

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


Simplifying radicals by rewriting as rational expressions

Simplifying radicals by rewriting as rational expressions


Simplifying radicals by rewriting as rational expressions1

Simplifying radicals by rewriting as rational expressions


Rationalizing the denominator

Rationalizing the Denominator

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


Rationalizing the denominator1

Rationalizing the Denominator

  • Examples:


P 24 53 67 70 71 86

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


Algebraic expressions 1 3

Algebraic Expressions—1.3


Scientific notation

Scientific Notation


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


Adding subtracting polynomials

Adding/Subtracting Polynomials


Product of polynomials

Product of Polynomials

  • FOIL


Product of polynomials1

Product of Polynomials

  • Examples:


Special product formulas

Special Product Formulas


Special factoring formulas

Special Factoring Formulas


Factoring

Factoring


Factoring1

Factoring


Factoring2

Factoring


Factoring3

Factoring


P 33 6 16 30 33

p.33 #6-16, 30-33


Factoring completely

Factoring Completely


Factoring4

Factoring


Factoring5

Factoring


Factoring6

Factoring


Factoring7

Factoring


Factoring by grouping

Factoring By Grouping


Factoring by grouping1

Factoring By Grouping


P 33 35 38 47 54 65 68 78 80

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


Fractional expressions 1 4

Fractional Expressions—1.4


Simplifying fractional expressions

Simplifying Fractional Expressions


Simplifying fractional expressions1

Simplifying Fractional Expressions


Simplifying fractional expressions2

Simplifying Fractional Expressions


Simplifying fractional expressions3

Simplifying Fractional Expressions


Simplifying fractional expressions4

Simplifying Fractional Expressions


Simplifying fractional expressions5

Simplifying Fractional Expressions


Compound fractions

Compound Fractions

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


Compound fractions1

Compound Fractions


Compound fractions2

Compound Fractions


Compound fractions3

Compound Fractions


Rationalizing the denominator2

Rationalizing the Denominator

  • Multiply by the conjugate:

    • Denominator:

    • Conjugate:


Rationalizing the denominator3

Rationalizing the Denominator


Rationalizing the denominator4

Rationalizing the Denominator


Rationalizing the numerator

Rationalizing the Numerator

  • Used often in calculus type settings

  • Same rules apply as rationalizing the denominator


Rationalizing the numerator1

Rationalizing the Numerator


Rationalizing the numerator2

Rationalizing the Numerator


P 43 25 27 29 41 42 51 59 68

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


Equations 1 5

Equations—1.5


Properties of equality

Properties of Equality


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 equations

Quadratic Equations


Quadratic equations1

Quadratic Equations

  • Zero Product Property


Quadratic equations2

Quadratic Equations


Completing the square

Completing the Square

  • Rewrite equation so it is in the form:

  • Example:


Completing the square1

Completing the Square


The quadratic formula

The Quadratic Formula


The discriminant

The Discriminant


Using the discriminant

Using the Discriminant

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


P 57 1 41 every other odd

p.57 #1-41 Every Other Odd


Equations 1 5 day 2

Equations—1.5 (Day 2)


Solving equations

Solving Equations


Solving equations1

Solving Equations


Solving equations2

Solving Equations


Solving equations3

Solving Equations


Solving equations4

Solving Equations


Solving equations5

Solving Equations


P 57 58 45 49 53 56 62

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


Modeling equations 1 6

Modeling Equations—1.6


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?


P 70 71 1 3 9 11 16 18 30 32

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


Inequalities 1 7

Inequalities—1.7


Linear inequalities

Linear Inequalities


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 


Non linear inequalities2

Non-Linear Inequalities

  • Example:


Non linear inequalities3

Non-Linear Inequalities

  • Example:


Absolute value1

Absolute Value


P 87 22 27 47 49

p.87 #22-27, 47-49


Coordinate geometry 1 8

Coordinate Geometry—1.8


Coordinate geometry

Coordinate Geometry

  • X-axis

  • Y-axis

  • Origin

  • Ordered Pair

  • Coordinates


Coordinate geometry1

Coordinate Geometry


Coordinate geometry2

Coordinate Geometry


Coordinate geometry3

Coordinate Geometry


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?


P 100 3 5 10 11 13 15 22 24 27 34 73 74

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


Lines 1 10

Lines—1.10


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


Standard form graphing

Standard Form--Graphing


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


P 124 1 31 odd 54 56 58 59

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


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