Algebra 2 project
- 2. Equations:
- 3. 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.
- 4. 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.
- 5. Inequalities
- 8. Systems of Equations A solution of a system of linear equations in two variables is an ordered pair (x,y) that satisfies each equation.
- 10. Matrices
- 11. PARABOLA
- 12. 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
- 13. Unit 4 Quadratic Functions (2) VERTEX AXIS OF SYMMETRY PARABOLA y=x²
- 14. 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.
- 18. 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
- 19. 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.
- 20. 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.
- 21. 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.
- 23. 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.ht ml
- 24. Complex Numbers
- 25. Polynomials
- 26. 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.
- 27. Rational Exponenets • Let a1/n be an nth root of a, and let m be a positive integer.
- 28. 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
- 31. End Behavior
- 32. 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.
- 34. 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
- 35. 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:
- 36. Exponential Decay • a = initial amount • r = percent decrease • 1-r = decay factor
- 37. 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
- 38. 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.”
- 39. Special Logarithm Values Common Logarithm • Logarithm with base 10 • Denoted by…
- 40. 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