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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?

Rational Exponents

=????

Rational Exponents

- Examples:

Rationalizing the Denominator

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

Rationalizing the Denominator

- 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

Product of Polynomials

- 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

Quadratic Equations

- 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?
- 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?

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:

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

- Test:

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

- Test:

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

- Test:

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

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