algebra 2 project
TRANSCRIPT
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.
Systems of Equations
A solution of a system of linear equations in two variables is an ordered pair (x,y) that satisfies each equation.
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
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.
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
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.
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
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.
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:
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.”
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