Pre Calc—Chapter 1

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# Pre Calc—Chapter 1 - PowerPoint PPT Presentation

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

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

Compound Fractions
• Is a fraction in which the numerator, the denominator, or both, are themselves fractional expressions
Rationalizing the Denominator
• Multiply by the conjugate:
• Denominator:
• Conjugate:
Rationalizing the Numerator
• Used often in calculus type settings
• Same rules apply as rationalizing the denominator
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
• Zero Product Property
Completing the Square
• Rewrite equation so it is in the form:
• Example:
Using the Discriminant
• Determine how many solutions (roots) an equation has without actually solving
Consecutive Integers
• What does it mean to be consecutive?
• 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?
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 

Coordinate Geometry
• X-axis
• Y-axis
• Origin
• Ordered Pair
• Coordinates
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?
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

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