Survey

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Survey

Document related concepts

Transcript

Chapter 1: Functions Vogler Algebra II Functions Functions give a one to one relationship between two variables: Y=2x, z=5+u, Pnuts+Bter=PB You get the idea. We can tell that a graph is a function by using the vertical line test: Functions Functions Since functions give a one to one relationship between two variables, then we can also identify functions on tables: There it is!! x 2 4 6 8 y 1 7 9 10 Functions Domain: all possible values of x. Range: all possible values of y. Stated as: all reals or with notation {∞<x<∞} Stated as: all reals or with notation {∞<x<∞} The domain and range depend on the graph. Types of Functions 16 18 16 14 14 12 12 10 10 8 8 6 6 4 4 2 2 0 0 1 2 3 4 -5 5 0 Quadratic 3.5 Linear 3 2.5 2 1.5 1 0.5 Absolute 0 -4 -2 5 0 2 4 Functions Function notation is just a fancy way to write y=: y=3x-7 f(x)=3x-7 t: x3x-7 It gives us a short hand way of telling us to evaluate functions for certain numbers: Evaluate y=x2+4 for x=3 f(3)=x2+4 t: 3x2+4 Distributive Property The distributive property helps us to simplify a(b+c)=ab+bc 4(2+x) 4•2+4x 8+4x Simplifying Expressions Simplified expressions are easier: 4x+3x+2+1 7x+3 Combine like terms Apply the distributive property: 7(3x+4)+5 21x+28+5 21x+33 Solving Multi-Step Equations 2 cars are traveling towards each other. Car A is going 50 MPH. Car B is going 60 MPH. They started 120 miles from each other, how long before they pass? The closing speed is an increase: Add 50+60 110 MPH d=rt formula: 110t=120 T=120/110 1.09 hours (about 1 hour and 6 minutes) Car B (60 MPH) Car A (50 MPH) Solving Multi-Step Equations 2 cars are traveling towards each other. Car A is going 50 MPH. Car B is going 60 MPH. They started 120 miles from each other, how long before they pass? Algebraically: 50t+60t=120 Combine like terms: 110t=120 Divide: t=120/110 t=1.09 hours Solving Multi-Step Equations Method 1: 2(3x+4)=20 Since 2 goes into 20 evenly, divide it first: 3x+4=10 Method 2: 2(3x+4)=20 Distributive property 6x+8=20 3x=6 6x=12 x=2 x=2 Clearing Fractions Sometimes, fractions suck… Get rid of them, multiply by the reciprocal: X/4=20 Multiply both sides by 4 4(x/4)=4(20) X=80 Solving Multi-Step Equations Same direction travel is a decrease: A B Two runners are running in the same direction. Runner A is going 10 MPH. Runner B is going 8 MPH and is 0.5 miles ahead. How long does it take runner A to overtake runner B? Runner A is running 2 MPH faster than B: Think d=rt: 10-8=2 2t=0.5 t=0.5/2 Decimals? No problem: t=5/20 or 1/4 of an hour 2 MPH Solving Multi-Step Equations Two runners are running in the same direction. Runner A is going 10 MPH. Runner B is going 8 MPH and is 0.5 miles ahead. How long does it take runner A to overtake runner B? Algebraically 10t=8t+0.5 Combine like terms: 10t-8t=8t-8t+0.5 2t=0.5 2t/2=0.5/2 t=5/20 t=1/4 Solving equations Work backwards through the order of operations: 1. 2. 3. 4. Parentheses Exponents Multiply/Divide Add/Subtract Combine like terms Clear fractions Apply the distributive property You made $250. Each hour, you make $2.90 of base pay plus about $32.50 in tips. You also get a bonus of $50 if you make more than $200. Solving equations cont. Create an equation and solve 250=h(2.90+32.50)+50 200=h(35.4) 5.6=h You worked 5.6 hours. In word problems, use context to check. Sequences: explicit and recursive formulas Recursive formulas refer to previous numbers in a pattern to find the next one: 2, 4, 6, 8, … First term: a1 All other terms: an Therefore: a1=2 an=an-1+2 Sequences cont. Explicit formulas give the resulting number from a pattern without regard for previous numbers: 2, 4, 6, 8, … Term Value 1 2 2 4 3 6 a(n)=2n The difference between terms is the slope in the equation y=mx+b b=the term before the 1st term Sequences Write the recursive and explicit formulas for the following sequence: 8, 12, 16… Find the first term: a1 Find the difference between each successive term