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Algebra 2 EXPONENT RULES 11/10/08 Product of Powers X3 ● X5 = X3+5 = X8 34 ● 35 = 34+5 = 39 Quotients of Powers X8 = X8-3 = X3 X3 49 = 49-3 = 46 43 Power of a Power (X4) 6 = X4●6 = X24 (53) 4 = 53●4 = 512 = 244140625 (XY) 6 = X6 Y6 (3X) 7 = 37 X7 Power of a Quotient b b8 7 ( )8 = ( )4 a a8 3 Rational Exponents 1 n 4 C C n 1 X7 7 X4 Simplify: get the terms down to less complicated. Simplify: 3X²Y²(-2X³Y4) (3●-2)(X²●X³)(Y²●Y4) = -6X5Y6 y 7 4 2 z 13 y 3 = y 7*4 y 28 2 4 z 13*4 y 3*4 16 z 52 y 12 Evaluate: solve, get the answer. a0 =1 20 = 1 Any number to the zero power is equal to 1. a1 = a Negative exponents X-3 = 1/X3 3-4 = 1/34 21 = 2 1 11/14/08 Operations with Functions Function: Relationship between two variables. Relation: For every X you will have a Y. Domain (X) Range (Y) Are these functions? Domain Range Domain Range X Y X Y 2 3 3 2 3 4 5 3 4 8 6 3 5 10 6 5 6 12 7 6 Function Not a function b/c X repeats. Vertical Line Test : Use to determine if you have a function or not. If you draw a vertical line that crosses the graph at two points then you do not have a function. This isn’t a function. This IS a function. 2 FUNCTION NOTATION: Helps you know the value of your function. Y = 3X + 5 Y = 7X + 10 f(x) = 3X +5 g(X) = 7X + 10 f(6) = 3(6) +5 g(6) = 7(6) + 10 f(6) = 18 + 5 = 23 11/14/08 g)6) = 42 + 10 = 52 Addition & Subtraction f(X) + g(X) f(X) = 5X² -2X +3 Add vertically g(X) = 4X²+ 7X -5 5X² -2X +3 + 4X²+ 7X -5 9X² +2X -2 Subtract vertically Note: By subtracting the – changes all signs. 5X² -2X +3 5X² -2X +3 -(4X²+ 7X -5) becomes -4X² -7X + 5 X² -9X + 8 Example 1: Add: f(X) = 5X³ + 6X² -7X +12 g(X) = 7X³ + 5 12X³ + 6X² -7X + 17 Subtract: 5X³ + 6X² -7X +12 - 7X³ - 5 - 2X³ +6X² -7X +7 Multiplication/Division Example 1: f(X) = 5X² g(X) = 3X -1 (5X²)(3X-1) = (5●3)(X²●X1)(5X²● -1) = 15X³- 5X² Example 2: f(X)= 6X4 -3X³-2X -4 3 g(X) = 5X²+7X +8 (5X²+7X +8)( 6X4 -3X³-2X -4) Take each term in the first function and multiply everything in the second function. 5X²(6X4 )+5X²(-3X³)+5X²(-2X)+ 5X²(-4)= 30X6-15X5 7X(6X4 )+7X(-3X³)+7X(-2X)+7X(-4) = -10X³-20X² 42X5-21X4 8(6X4)+8(-3X³)+8(-2X)+8(-4) = Answer: -14X²-28X 48X4-24X³ -16X -32 30X +27X +27X4 -34X³-34X2 - 44X -32 6 5 Division f ( x) 3 x 2 3x 3 2 x 4 .5 .25 .4 .90 .25 .65 g ( x) 6 x 2 16 x 10 f ( x) 6 x 4 3x 3 2 x 4 1.2 x 2 .43x 2 2 x .5 2 g ( x) 5x 7 x 8 11/18/08 Solving Systems of Equations by Graphing or Substitution y x 3 y 3 x 5 Y = mx + b Y = dependent X = independent Remember m = slope b = y-intercept Y-intercept is where the line crosses the Y axis. For the first equation, Y=x-3, the line should look like this: 0 -0.5 0 1 2 3 The line crosses the y-axis 4 at -3. -1 -1.5 -2 Series1 -2.5 -3 -3.5 4 11/21/08 The solution of a system is where the two graphs of the equations intersect. This graph has only one solution. It is CONSISTENT and INDEPENDENT. These are parallel lines. THEY DO NOT INTERSECT. No solution. It is INCONSISTENT. Two equations with the same line. Multiple solutions. It is CONSISTENT and DEPENDENT. 5 Example 1: Graph the system x y 5 x 5 y 7 Make a table for both equations. X+y = 5 x-5y = 7 X Y X Y 5 2 3 2 1.4 4 4 1 4 2.2 3 X+Y=5 2 X-5Y=-7 1 0 The graph has one solution. 1 2 3 4 (3,2) It is consistent and independent. Solving with substitution x y 5 x 5 y 7 1) First take the first equation and solve for X X + Y =5 -Y -Y X = 5-Y Replace this value for X in second equation and solve for Y. 2) X-5Y=-7 5-Y -5Y =-7 5 – 6Y = -7 -5 -5 -6Y = -12 -6 -6 Y = 2 Take your value for Y, 2 and replace it in the first equation and solve for x. 3) X + Y = 5 X+2=5 -2 -2 6 X=3 Solution of the system is (3,2) Example 2: 2X + Y = 3 3X – 2Y = 8 1) 2X + Y =3 -2X -2x Y = -2X +3 2) 3X – 2(-2X + 3) = 8 3X + 4X – 6 = 8 7X -6 = 8 + 6 +6 7X = 14 7 7 X=2 3) 2(2) + Y = 3 4+Y=3 -4 -4 Y = -1 Solution (2,-1) 11/25/08 2x+5y=15 To solve this you want to be able to cancel out one of the -4x+7y=-13 variables. Find which one you could cancel by multiplying one of the equations. 2(2x+5y)=15(2) Multiply both sides by 2, now you can cancel out 4x+10y = 30 the x. -4x+7y = -13 17y = 17 17 17 Y=1 Now put your value for y into the first equation and solve for X. 2x+5(1) = 15 2x +5 = 15 -5 -5 2x = 10 2 2 X=5 Solution for this system is (5,1) 7 Example 1: 3x -7y =8 5x -6y =10 To cancel the Xs out you need to multiply by -3and 5. Then solve for Y. (5)3x -7y = (8)(5)= 15x-35y=40 (-3)5x -6y =(10)(-3) = -15x+18y=-30 0 -17y = 10 -17 -17 Y = -.58 Now replace the value of Y in the first equation and solve for X 3x – 7(-.58) = 8 3x + 4.06 = 8.00 - 4.06 -4.06 3x = 3.94 3 3 X = 1.31 Solution is (1.31, -.58) Example 2: 2x + 5y = 15 2x - 5y = -13 4x =2 4 4 X = ½ = .5 The Ys will cancel out. 2(.5) + 5y = 15 1 + 5y = 15 -1 -1 5y = 14 5 5 Y = 2.8 Solution (.5, 2.8) 8 12/1/08 EXPONENTIAL FUNCTION Bacteria cells double (2) every hour. You start with 25 cells. Hrs. Cells 1800 0 25 1 50 1400 2 100 1000 3 200 600 4 400 5 800 6 1600 1600 1200 Series1 800 400 200 0 0 The function is f(x) = 25(2)x 5 10 f(x) = ab x a - is what you start with. b – is the multiplier x = time ( secs., hrs. ,days, years etc.) Example 1 Population in the US 1990 248,718,301 Population grows 8% per decade (every 10 years). In 2025 what will the population be. 2025- 1990 = 35 years or 3.5 decades Rules for % Multiplier Growth: If you have a % 1st you change the percent to a decimal. 8% = .08 (Count 2 spaces to the left from end, add zeros if needed) Next add 1: 1 + .08 = 1.08 f(x) = 248718301 (1.08)3.5 f(x) = 325,604,866 Decay: Change the percent then SUBTRACT from 1. Example; 15% decay rate, 15% = .15 1-.15 = .85 9 12/5/08 Compounded Interest Formula A = P (1 + r nt ) n r= interest rate n = how it is compounded t = time Compounding terms: Annually = 1 Semi-annually = 2 Quarterly = 4 Daily = 365 Weekly = 526 Example 1. P = $250 A = 250 (1 + 4% compounded annually for years. .04 1*6 ) 1 A = 250(1.04)6 A = $316.33 A = 250(1.01)24 A = $317.43 Compounded quarterly A = 250 (1 + f(x) = bx .04 4*6 ) 4 b is the multiplier when 0<b<1 then you have a decay rate. When b is greater than 0 and less than 1 – decay 12/8/08 Logarithms Exponential Logarithm bx = y x = logb y 7³ = 343 4 = log7 343 35 = 243 5 = log3 243 105 = 100000 5 = log10 100000 Exponent b x =y base Exponent x = logb y base 10 One-to-One Property Exponents are the same bx= by b3= b3 x=y 5 = log2 32 25 = 32 4 log 3V = 34=V 25 = 25 V = 81 12/9/08 Product Property 34 * 36 346 310 log( m * n) log m log n .47 + .60 = 1.07 Log 36 = log(6●6) = log 6 + log 6 1.55 + 1.55 = 3.1 Quotient Property m log m log n n 3 log log 3 log 2 2 3 log .47 .30 .17 2 log 5 log 5 log 2 2 log 2.5 .69 .30 .39 log 2.5 log Power Property log P m m log P log 34 4 log 3 log 34 4(.47) 1.08 Exponential – Logarithmic Inverse Property log b b x X b log X x 3log4 4 log 34 4 11 Change of Base log x log b X log b log 4 .60 log 3 4 .27 log 3 .47 Solve log 3 ( x 2 7 x 5) log 3 (6 x 1) x²+7x-5= 6x+1 -6x -6x x²+1x-5= 1 -1 -1 x²+1x-6=0 Factor (x+3)(x-2) x+3=0 x-2=0 -3 -3 +2 +2 X = -3 x=2 Use positive solution first and replace in equation log 3 (2 2 7(2) 5 log 3 6(2) 1 log 3 (4 14 5) log 3 (12 1) log 3 13 log 3 13 True answer If it wasn’t true you would then use the negative solution. 12