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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.

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?

=????

• Examples:

### Rationalizing the Denominator

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

• Examples:

### 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

• FOIL

• Examples:

### 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 

• Example:

• Example:

• 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?

• Graph:

• 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

• 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