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Algebra 2 L to J Essential Learning 2008-2009 1 Absolute value Alg II a a, a 0 2 Arithmetic sequence Alg II A pattern of numbers where the common difference of consecutive terms is the same 3 Arithmetic series Alg II The sum of the elements of an arithmetic sequence 4 Asymptote Alg II The line that a curve approaches 5 Axis of Symmetry Alg II The line that splits a graph into symmetrical parts 6 Complex conjugate Alg II 7 Complex Fraction Alg II A fraction that contains a fraction in the numerator, denominator or both 8 Complex Number Alg II Any number of the form 9 Composition Alg II a, a 0 (a bi ) and (a bi ) a bi where a,b are real numbers f g ( x) f [ g ( x)], and , g f ( x) g[ f ( x)] 10 Consistent Dependent System Alg II A system of equations with an infinite number of solutions 11 Consistent Independent System Alg II A system of equations with exactly one solution 12 Constraints Alg II Boundaries used in Linear Programming 13 Direct Variation Alg II 14 Discriminant Alg II 15 Feasible region Alg II The area of intersection of a system of inequalities 16 Geometric Sequence Alg II A sequence of numbers where each term is found by multiplying by a constant 17 Geometric Series Alg II The sum of n terms of a geometric sequence 18 Imaginary Number Alg II 19 Inconsistent System Alg II A system of equations with no solution y kx , x and y increase or decrease together b2 4ac , used to determine the number and type of solutions to a quadratic equation i 1 20 Inverse functions Alg II A reflection over the line y = x 21 Inverse Variation Alg II 22 Iteration Alg II A function in which the output becomes the next input 23 Linear Regression Alg II A line of best fit 24 Matrix Alg II An organization of numbers or variables in columns and rows 25 Objective Function Alg II The function being maximized or minimized in Linear Programming 26 Parabola Alg II 27 Radical Expression Alg II An expression with variables inside/under the radical 28 Rational Expression Alg II A polynomial divided by a polynomial 29 Recursive Formula Alg II In a sequence, shows how to find the nth term from the terms before or after it 30 Vertex Alg II Maximum or minimum of a parabola 31 Zero product property Alg II If ab=0, then a=0, b=0, or both a and b = 0 32 Zeros/Roots Alg II The x - intercepts of a function y k x as x increases y decreases and vice versa y a( x h)2 k or x a( y k )2 h L to J Concepts Algebra II Essential Learning Example Problem with Solution 1. Point slope form of a line y y1 m( x x1 ) 2. Write the equation of a line given two points. ( 2, 4 ) and ( 6, - 8) y – 4 = -3( x – 2 ) 3. Write the equation of a line given one point and slope. m = 8 point ( 3 , -5 ) (y + 5) = 8(x– 3 ) 4. Direct Variation y varies directly as x, when x = 5, y = 25 find y when x = 7 y 5 x m = -3 y 35 5. Inverse Variation x varies inversely as y, when y = 12, x = 10 find x when y = 4 6. Linear Regression Write equation of line of best fit through (1,1), (2,1),(3,3), and (4,4) yx y 5 7. Graphing Linear Equations 3 y x5 4 4 3 2 1 x -5 -4 -3 -2 -1 -1 1 2 3 4 5 -2 -3 -4 -5 8. Graphing Linear Inequalities 9. Graphing Greatest Integer Functions y 2x 3 2 y x 1 2 y 10. Graphing Absolute Value Functions 5 y x 3 4 3 2 1 -6 -5 -4 -3 -2 -1 -1 x 1 2 3 4 5 6 7 -2 -3 -4 -5 11. Solve Compound Inequalities 8 2x 4 10 12. Solve Absolute Value Equations 2 x 3 18 13. Solve for Specified Variable s1v 14. Solve systems of equations by substitution. 2hf 5 s2 1 x3 2 y 2x 1 y 6 x 3 x 15 or 2 21 2 solve for f f s1s2 v 5s2 2h 8 13 ( , ) 3 3 y 15. Solve systems of equations by graphing. And identify the type of solution 5 y x3 4 3 3 x 3 y 9 y x3 2 1 x -5 -4 -3 -2 -1 -1 1 2 3 4 5 Consistent Dependent -2 -3 -4 -5 y x 2y 1 5 2x y 5 3 4 2 1 x -5 -4 -3 -2 -1 -1 1 2 3 4 (3,-1) Consistent Independent 5 -2 -3 -4 2 y 6x 6 -5 y y 3x 1 5 4 3 2 1 -5 -4 -3 -2 -1 -1 x 1 2 3 4 5 No solution Inconsistent -2 -3 -4 -5 16. Solve systems of equations by elimination. 17. Graphing Systems of Inequalities 6 x 5 y 27 (2, 3) 3 x 10 y 24 x 1 y3 x y 5 18. Simplifying with Rule of Exponents [(6 x 2 y 3 )]2 [(3x 1 y 2 )2 ]1 19. Multiply Polynomials ( x 4)( x 2)( x 1) 20. Divide Polynomials ( x 2 5x 6) ( x 3) 21. Factor Polynomials 6m 2 7 m 3 y10 4 x6 x3 7 x 2 14 x 8 x2 (3m 1)(2m 3) 5(3v 5)(v 2) 15v 2 55v 50 x3 8 ( x 2)( x 2 2 x 4) 49 x6 625 y 4 22. Add/Subtract Polynomials (7 x3 25 y 2 )(7 x3 25 y 2 ) 6 x3 x 2 3x 2 (3x 2 4 x 5) (6 x3 2 x 2 x 3) 23. A third degree equation x3 81 x 33 3 24. A fourth degree equation x 4 625 x 5 25. Simplification of complex number 25 5i 20 2i 5 3 72 18i 2 26. Addition of complex numbers (2 + 3i ) + ( 5 + 2i ) = 7 + 5i (3 – 5i ) + ( 6 + i ) = 9 – 4i 27. Subtraction of complex numbers (2 + 3i ) - ( 5 + 2i ) = -3 + i (3 – 5i ) - ( 6 + i ) = -3 – 6i 28. Multiplication of complex numbers ( 6 + 2i )( 3 – 5i ) = 28 - 4i ( 3 + 2i ) 2 = 5 + 12i 29. Division of complex numbers 6 2i (6 2i)(3 i) 8 6i 3i (3 i)(3 i) 5 30. Cyclic Powers of i 31. Solve by completing square i 7 2 i 3x 6 x 14 32. Solve by using square roots x2 4 0 33. Solving by quadratic formula x2 8x 5 0 x 34. Solve using Quadratic Techniques x4 5x2 6 0 x i 2 or x i 3 35. Use the Discriminant to Find the Type of Roots and Number x x 2i or 2i 5 x 2 7 x 12 36. Graph y = 2(x – l ) -3 5 4 3 2 1 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 x 1 2 3 8 44 4 11 2 discriminant = -191 2 imaginary roots y 2 1 51 3 4 5 37. Determine if a relation is a function 38. Find f g (3) if f ( x) x 2 and g ( x) x 1 39. Identify the domain and range of a function {(0,2), (3,4), (0,5), (-12,-1)} --no y = 3x – 1 --yes vertical line test for functions f g (3) 4 y x 2 1 Domain {x : x all reals} Range { y : y 1} y 5 4 3 40. Determine if a function has an inverse, if so find it and graph it y = 3x+2 1 2 f 1 ( x) x 3 3 y=3x+2 2 y = 1/3x - 2/3 1 -5 -4 -3 -2 -1 -1 x 1 2 3 4 5 -2 -3 -4 -5 41. Recursion and Iteration 42. Addition of radicals Find the first three iterates x1 16, x2 142, x3 1276 f ( x) 9 x 2, x0 2 34 3 5 3 6 5 2 5 4 5 8 18 2 2 3 2 5 2 43. Subtraction of radicals 3 4 3 3 3 6 5 2 5 4 5 44. Multiplication of radicals 8 18 2 2 3 2 1 2 3 * 5 15 3 2 *5 7 15 14 45. Division of radicals 6 3 2 7 21 3 3 46. Radical equations 2x 3 6 2x 3 6 47. Simplify Rational Expressions 2x x 3 1 x 3 x 3 x 2 3x 6 14 x 14 7 x 7 5 x 10 x2 2x 8 x 2 x 2 4 x 3 3x 3 x = 33/2 x = 9/2 x = 4 or 1 x2 5x 9 ( x 3)( x 2) 3( x 4) x3 6 5 48. Rational Equations 2x 2 5 x 5 5 49. Arithmetic Sequences an a1 (n 1)d 50. Arithmetic Series 51. Geometric Sequence 52. Geometric Series sn x 1 a1 5, d , n 12, an ? 3 26 a12 3 7 14 21 28 ... 98 n (a1 an ) 2 S n 735 an a1r n 1 or an an 1r a1 (1 r n ) Sn 1 r 15 13 1 , n 9, r 4, an ? 64 a9 1024 a1 a1 5, r 2, n 14, Sn ? Sn 81,915