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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 Simplify: get the terms down to less complicated. Evaluate: solve, get the answer. a0 =1 20 = 1 Any number to the zero power is equal to 1. a =a Negative exponents X-3 = 1/X3 3-4 = 1/34 1 2=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. 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 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 6 5 4 30X +27X +27X -34X³-34X2 - 44X -32 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 -2.5 -3 -3.5 Series1 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. 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 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) Example 1: 3x -7y =8 5x -6y =10