Equations:

Real Numbers and Number Operations

• Whole numbers

• 0, 1, 2, 3 ……..

• Integers

• ….., -3, -2, -1, 0, 1, 2, 3 …….

• Rational numbers

• Numbers that can be written as the ratio of two integers.

• Irrational Numbers

• Real numbers that are not rational.

Equations • Slope-intercept form of a linear equation y=mx + b

• Slope: m , y-intercept: b

• Standard form of a linear equation Ax + By = C

• A and B are not both zero.

Inequalities

Piecewise Functions

Absolute Value Equations

Systems of Equations

A solution of a system of linear equations in two variables is an ordered pair (x,y) that satisfies each equation.

Systems of Inequalities

Matrices

PARABOLA

Unit 4 Quadratic Functions (1)

• Terms:

• Quadratic function: form y=ax²+bx+c

• Parabola: u-shape

• Vertex: the lowest or the highest point

• Axis of symmetry: vertical line through the vertex

Unit 4 Quadratic Functions (2)

VERTEX

AXIS OF SYMMETRY

PARABOLA

y=x²

How to graph PARABOLAS

1. Find the vertex

a) Use (𝑥 = −𝑏

2𝑎) to find the x-value.

b) Find the y-value of vertex (fill in x-value.)

2. Choose value for x and find y.

3. Plot the mirror image point from number 2.

4. Sketch the curve.

5. May need to repeat number 2 and 3.

Factoring Patterns

1) Factoring Trinomials with Binomials

• x² + bx + c = (x + m)(x + n)

= x² + (m + n)x + mn

2) Factoring Difference of Squares

• a² - b² = (a + b)(a - b)

3) Factoring Perfect Square Trinomial

• a² + 2ab + b² = (a + b) ²

• a² - 2ab + b² = (a - b) ²

MATH & HISTORY

• The First Telescope was made in 1608 by a German guy named Hans Lippershey. • Refracting telescope: lenses magnify objects. • Reflecting telescope: magnify objects with parabolic mirrors. • Liquid telescope: made by spinning reflective liquids like mercury.

MATH & HISTORY (2)

Galileo first uses a refracting telescope

for astronomical purposes.

Isaac Newton builds first reflecting telescope.

Maria Mitchell is

first to use a telescope to

discover a comet.

Liquid mirrors are first used to do astronomical

research.

Fractals

• Definition

• A curve or geometric figure, each part of which has the same statistical character as the whole.

• A geometric pattern that is repeated at every scale.

• An object or pattern that is "self-similar" at all scales.

Examples of Fractals

WEBSITES

• http://mathforum.org/cgraph/history/glossary.html

• http://www.imemine98.com/edu7666/wq.html

• http://www.flixya.com/photo/2025935/Parabolic-Arch-of-Vijay-Vilas-Palace-of-Bhuj

• http://mentationaway.com/2010/08/03/idea-fractal/

• http://en.wikipedia.org/wiki/Palace_of_Ardashir

• http://womenworld.org/travel/san-francisco-s-top-10---architectural-highlights---top-10-public-art-sites.aspx

• http://www.pleacher.com/mp/mlessons/calculus/appparab.html

Complex Numbers

Polynomials

nth Roots and Rational Exponents

• Let n be an integer greater than 1 and let a be a real number.

• If n is odd, then a has one real nth root:

• If n is even and a > 0 , then a has two real nth roots:

• If n is even and a = 0, then a has one nth root:

• If n is even and a < 0, then a has no real nth roots.

Rational Exponenets

• Let a1/n be an nth root of a, and let m be a positive integer.

Properties of Exponents

Product of Powers Property

Power of A Power Property

Power of A Product Property

Negative Exponent Property

Zero Exponent Property

Quotient of Powers Property

Power of A Quotient Property

Scientific Notation

Polynomial function

End Behavior

Inverse Functions

• An inverse relation maps the output values back to their original input values.

• The domain of the inverse relation is the range of the original relation and that the range of the inverse relation is the domain of the original relation.

Radical Functions

Exponential Growth

• An exponential function involves the expression b^x where the base b is a positive number other than 1.

• ASYMPOTOTE: a line that a graph approaches as you move away from the origin.

• If a > 0 , and b > 1 , the function ab^x is an exponential growth function.

a = initial amount r = percent decrease 1+r = growth factor

Compound Interest

• Consider an initial principal P deposited in an account that pays interest at an annual rate r (expressed as a decimal), compounded n times per year. The amount A in the account after t years can be modeled by this equation:

Exponential Decay

• a = initial amount

• r = percent decrease

• 1-r = decay factor

The Number e

• Natural base e

• Euler number

• Discovered by Leonhard Euler (1707-1783)

• Natural base e is irrational.

• Defined as: As n approaches +, approaches

Logarithmic Functions

• Let b and y be positive numbers, b1. The logarithm of y with base b is denoted by and defined as follows:

The expression is read as “log base b of y.”

Special Logarithm Values

Common Logarithm

• Logarithm with base 10

• Denoted by…

Applications of Logarithm

• pH of solutions

• Decibels of sound

• Figuring interest

• Decaying radiation

• Short form for long numbers

• Signal decay

• Richter Scale – earthquakes

• F-stop in photography

• Oceanography

• Exponential growth

Right Triangles

http://blog.lib.umn.edu/stau0156/architecture/2006/11/

http://mathandreadinghelp.org/8th_grade_geometry.html

Right Triangle Trigonometry

Special Triangles