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Algebra II Math Methods II Visual Mathematics QuickTime™ and a decompressor are needed to see this picture. Functions: CCSS ❖ ◦ ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦ ◦ Interpreting Functions Understand the concept of a function and use function notation Interpret functions that arise in applications in terms of the context Analyze functions using different representations Building Functions Build a function that models a relationship between two quantities Build new functions from existing functions Linear, Quadratic, and Exponential Models Construct and compare linear and exponential models and solve problems Interpret expressions for functions in terms of the situation they model Trigonometric Functions Extend the domain of trigonometric functions using the unit circle Model periodic phenomena with trigonometric functions Prove and apply trigonometric identities Esrevni Functions ❖ F-BF.4 Build new functions from existing functions. Find inverse functions. ❖ Visually show the concept with multiple representations ❖ Inverse ❖ The of the point (2,-3) is (-3,2) DOMAIN and RANGE of the function become the RANGE and DOMAIN of the Inverse Inverse Functions ❖ 8.G.1 Verify experimentally the properties of rotations, reflections, and translations ❖ The Inverse of y = x2 becomes x = y2 Inverse Functions Painted Cube Problem ❖ F-LE Construct and compare linear, quadratic, and exponential models and solve problems. ❖ Different size cubes are constructed from unit cubes, the surface areas of the resulting larger cubes are painted, and then each of the larger cubes is disassembled into its original unit cube? ❖ How many of the unit cubes are painted on 3 faces? 2 faces? 1 face? 0 faces? ❖ Use tables, graphs, and equations to explore. Cubes & Spatial Thinking Functions ❖ Problem 1 ❖ The 10th grade class of Taft High is planning a day to a local amusement park. It will cost $350 to rent a bus for one day. How will the size of each person’s contribution depend upon the size of the group going on the trip? Functions ❖ Problem 2 ❖ The following chart appeared in a mail order catalog. Use this chart to determine your shipping charges. ❖ Merchandise Total ❖ Up to $15.00 ❖ From $15.01 to $25.00 add $5.00 ❖ From $25.01 to $50.00 add $7.00 ❖ Over $50.00 add $3.00 add $9.00 Functions ❖ Problem 3 ❖ The purchasing agent for the school store wants to buy 1200 pencils imprinted with the school logo. The pencils can be purchased in packages of 12, 24, 48, 100, and 200 pencils but package sizes cannot be combined. Show the function, in any form, that assigns to each possible package size the number of packages that must be purchased. Functions ❖ Problem 4 ❖ A, B, C are points in the coordinate plane. How many functions can be drawn that include all three points A, B, C? Explain. ❖ ❖ Problem 5 Functions A function is defined by the following recursion formula: ❖ f1=1; fn=√(2*fn-1) ❖ The first six values of the function are ❖ 1 ❖ ≈1.4142 ❖ ≈1.6818 ❖ ≈1.8340 ❖ ≈1.9152 ❖ ≈1.9571 Functions ❖ Problem 6 ❖ Show a different representation of the following function: Space Math: NASA Conversions (1.1.1) ❖ http://spacemath.gsfc.nasa.gov/algebra2/Algebra2V3.pdf ❖ 6.RP.3d Use ratio reasoning to convert measurement units. ❖ The Space Shuttle used 800,000 gallons of rocket fuel to travel 400 km into space. If one gallon of rocket fuel has the same energy as 5 gallons of gasoline, what is the equivalent gas mileage of the Space Shuttle in gallons of gasoline per mile? ❖ 16,000 gallons/mile Space Math: NASA Pythagorean Theorem 3D (1.2.1) 8.G.6-8 Explain and apply Pythagorean Theorem *Distance in light years Space Math: NASA The Hunt for Higgs Boson (1.6.3) ❖ A-CED Create equationsconcluded & inequalities Fermilab's Tevatron accelerator experiments that it must either be more massive than 170 GeV or less massive than 160 GeV. ❖ CERN's LEP accelerator concluded after years of searching that the Higgs Boson must be more massive than 115 GeV ❖ The Standard Model, which describes all that is currently known about the interactions between nuclear elementary particles, provided two constraints depending on the particular assumptions used: The Higgs Boson cannot be more massive than 190 GeV, and it has to be more massive than 80 GeV but not more than 200 GeV. ❖ From all these constraints, what is the intersection of possible masses for the Higgs Boson that is consistent with all of the constraints? Space Math : NASA Solving Systems of Linear Equations (3.1.1) A-CED.3 Represent & solve systems of linear equations ❖ Studies of the number of craters on Venus and Mars have determined that for Venus, the number of craters with a diameter of D kilometers is approximated by N = 108 – 0.78D while for Mars the crater counts can be represented by N = 50 – 0.05 D. ❖ Graphically solve these two equations to determine for what crater diameter the number of craters counted on the two planets is the same over the domain D:[0,100 km]. Space Math: NASA Logs (8.6.1) F-BF.5 Understand logarithms ❖ ❖ ❖ For stars, the apparent brightness or ‘magnitude’ of a star depends on its distance and its luminosity, also called its absolute magnitude. What you see in the sky is the apparent brightness of a star. The actual amount of light produced by the surface of the star is its absolute magnitude. A simple equation, basic to all astronomy, relates the star’s distance in parsecs, D, apparent magnitude, m, and absolute magnitude, M as follows: M = m + 5 - 5log(D) Problem 1 – The star Sirius has an apparent magnitude of m = 1.5, while Polaris has an apparent magnitude of m = +2.3, if the absolute magnitude of Sirius is M = +1.4 and Polaris is M = -4.6, what are the distances to these two stars? Space Math: NASA Logs (8.4.1) F-BF.5 Understand logarithms ❖ Log N = -0.0003 m^3 + 0.0019 m^2 + 0.484 m - 3.82 ❖ A small telescope can detect stars as faint as magnitude +10. If the human eye-limit is +6 magnitudes, how many more stars can the telescope see than the human eye? Trig Apps Trig Apps Trig Apps Trig Apps