Aldine I.S.D. Benchmark Targets/ Algebra 2 SUMMER 2004

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1 ASSURANCES: By the end of Algebra 2, the student will be able to: 1. Solve systems of equations or inequalities in two or more variables. 2. Graph rational functions and solve rational equations and inequalities. 3. Solve quadratic equations and inequalities with real and complex solutions. 4. Identify and graph conic sections. 5. Solve exponential and logarithmic equations. Problem solving applications must be included in daily instruction. The teacher needs to pace instruction so students are able to develop a conceptual understanding. Therefore, the timeline is flexible. It is critical that the objectives for each quarter are completed by the end of that quarter. The teacher should incorporate some of the following TEKS in every lesson: A Identify and apply mathematics to everyday experiences, to activities in and out of school, with other disciplines, and with other mathematical topics. (TEKS b8.14a) B Use a problem solving model that incorporates understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness. (TEKS 8.14B) C Select or develop an appropriate problem-solving strategy from a variety of different types, including: work backwards guess and check make a table draw a picture look for a pattern work a simpler problem acting it out D Communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or algebraic models. (TEKS 8.15A) E Use logical reasoning to make conjectures from patterns or sets of examples and nonexamples (TEKS 8.16 A) F Validate conclusions using mathematical properties and relationships. (TEKS 8.16 B) Sara Ptomey Algebra 2, Page - 1

2 associative property commutative property distributive property equality property equation expression formula identity property inequality Aldine I.S.D. Benchmark Targets/ Algebra 2 integer numbers inverse operations irrational numbers natural numbers order of operations rational numbers real numbers reflexive property solution u substitution symmetric property transitive property variable whole numbers Use order of operations to evaluate and simplify expressions. Use properties of real numbers to simplify expressions. Determine the classification of Real numbers. Solve and graph equations and inequalities in one and two variables. Algebra 2a PARENT FUNCTIONS (Special Topic: This may be presented as a separate unit or integrated throughout the curriculum as corresponding topics are introduced.) exponential function function inverse function linear function logarithmic function parent function quadratic function square root function 1. Identify and sketch graphs of parent functions, including linear (y = x), quadratic (y = x 2 ), square root (y = x ), inverse (y = 1/x), exponential (y = a x ), and logarithmic (y = log a x) functions. (TEKS c1a) 2. Extend parent functions with parameters such as m in y = mx and describes parameter changes on the graphs of parent functions. (TEKS c1b) Sara Ptomey Algebra 2, Page - 2

3 LINEAR FUNCTIONS best-fit/regression line consistent system constant function continuous function coordinate system dependent variable discrete function domain inconsistent system independent system dependent system Aldine I.S.D. Benchmark Targets/ Algebra 2 independent variable inverse operations linear equation linear function ordered pairs origin parallel parent graph/function perpendicular quadrant range regression equation relation scatter plot slope vertical line test x-axis x-intercept y-axis y-intercept 3. Identify the mathematical domains and ranges and determine reasonable domain and range values for given situations. (TEKS b1a) 4. Connect the function notation of y = and f(x) =. (TEKS b2c, c1a, c1b) 5. Simplify expressions and transform and solve equations and inequalities.(teks b2a, b2) 6. Make scatter plots, fit the curves to the appropriate parent function, model, predict, and make decisions. (TEKS b1b) 7. Recognize the inverse relationship from a set of equations or graphs. Write the inverse of a given equation and graph inverse relationships. (TEKS c1c) 8. Formulate systems of equations or inequalities in two or more unknowns to solve problems. (TEKS b3a) 9. Solve systems of equations or inequalities using graphs, algebraic methods, tables, and matrices. (TEKS b3b) 10. Determine the reasonableness of solutions to systems of equations or inequalities. (TEKS b3c) Sara Ptomey Algebra 2, Page - 3

4 POLYNOMIAL AND RATIONAL EXPRESSIONS Prerequisites: Multiply and divide monomials. Add, subtract, and multiply polynomials. Multiply and divide rational expressions. Add and subtract rational expressions. Factor polynomials. binomial coefficient complex fraction degree of polyniomial extraneous solution foil method monomial polynomial power rational equation rational expression rational function term factors trinomial 11. Use factoring to simplify polynomials. (TEKS b2a) 12. Solve rational equations and inequalities using graphs, tables, and algebraic methods such as factoring and quadratic formula. (TEKS e4) 13. Analyze a situation modeled by a rational function, formulate an equation or inequality composed of a linear or quadratic function, and solve the problem. (TEKS e5) RADICAL FUNCTIONS direct variation inverse relationship inverse variation parameter changes quadratic function square root function 14. Recognize inverse relationships. (TEKS c1c) 15. Express inverses of quadratic functions using square root functions. (TEKS d4f) 16. Investigate, describe, and predict the effects of changes in a, h, and k on the graphs of y = a x + h + k. (TEKS c1a, c1b, d4a) 17. Determine the reasonable domain and range values of square root functions; interpret and determine the reasonableness of solutions to square root equations and inequalities. (TEKS d4c) 18. Analyze situations modeled by square root functions, formulate equations or inequalities, select a method, and solve problems. (TEKS d4e) Sara Ptomey Algebra 2, Page - 4

5 Algebra 2b QUADRATICS completing the square complex number conjugate discriminant rationalize denominators imaginary number principal root quadratic function radical simplify radicals radical equation rational exponent roots square root zeros 19. Relate representations of square root functions, such as algebraic, tabular, graphical, and verbal descriptions. (TEKS d4b) 20. Write and evaluate expressions with rational exponents in simplest radical form and vice versa. (TEKS b2a) 21. Solve square root equations and inequalities using graphs, tables, and algebraic methods. (TEKS d4d) 22. Use complex numbers to describe the solutions of quadratic equations. (TEKS b2b) 23. Investigate, describe, and predict the effects of changes in a, h, and k on the graphs of y = a(x h) 2 + k. (TEKS d2b, c1a, c1b) 24. Determine the reasonable domain and range values of quadratic functions. (TEKS d1a) 25. Relate representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions. (TEKS d1b) 26. Solve quadratic equations and inequalities by graphing, factoring, completing the square, and using the quadratic formula. (TEKS d3a, d3b, d3d) 27. Determine a quadratic function from its roots or a graph. (TEKS d1c) 28. Connect between the y= ax 2 + bx + c and the y= a(x - h) 2 + k. (TEKS d2a, c1a, c1b) 29. Compare and translate between algebraic and graphical solutions of quadratic equations. (TEKS d3c) 30. Analyze and interpret the solutions of quadratic equations using discriminants. (TEKS d3b) 31. Formulate quadratic equations or inequalities to solve problems. (TEKS d3a) 32. Determine the reasonableness of solutions to quadratic equations and inequalities. (TEKS d1a) Sara Ptomey Algebra 2, Page - 5

6 CONICS Prerequisites: Find the distance between two points in the coordinate plane. Find the midpoint of a line segment in the coordinate plane. asymptotes axis of symmetry center circle conic section ellipse hyperbola major axis conjugate axis vertex minor axis parabola radius transverse axis 33. Describe a conic section as the intersection of a plane and a cone. (TEKS c2a) 34. Relate simple parameter changes in the equation to corresponding changes in the graph. (TEKS c1a, c1b, c2b) 35. Identify the conic sections from a given equation. (TEKS c2d) 36. Use the method of completing the square to write the equation of a conic in standard form. (TEKS c2e) 37. Graph conic sections. (TEKS c2b) 38. Identify symmetries from graphs of conic sections. (TEKS c2c) RATIONAL FUNCTIONS asymptote direct variation domain inverse variation inversely proportional limits range rational function 39. Use quotients to describe the graphs of rational functions, describe limitations on the domains and ranges, and examine asymptotic behavior. (TEKS c1a, c1b,e1) 40. For given contexts, determine the reasonable domain and range values of rational functions; interpret and determine the reasonableness of solutions to rational equations and inequalities. (TEKS e3) 41. Analyze various representations of rational functions with respect to problem situations. (TEKS e2) 42. Use direct and inverse variation functions as models to make predictions in problem situations. (TEKS e6) Sara Ptomey Algebra 2, Page - 6

7 LOGARITHMS antilogarithm change of base formula common logarithms exponential equations exponential growth exponential growth rate Aldine I.S.D. Benchmark Targets/ Algebra 2 growth and decay formulas logarithm logarithmic function natural logarithms power property of logarithms product property of logarithms property of equality for exponential functions property of equality for logarithmic functions quotient property of logarithms 43. Develop the definition of logarithms by exploring and describing the relationship between exponential functions and their inverses. (TEKS f1) 44. Use the parent function to investigate, describe, and predict the effects of parameter changes on the graphs of exponential and logarithmic functions, describe limitations on the domains and ranges, and examine asymptotic behavior. (TEKS c1a, c1b, f2) 45. For given contexts, determine the reasonable domain and range values of exponential and logarithmic functions; interpret and determine the reasonableness of solutions to exponential and logarithmic equations and inequalities. (TEKS f3) 46. Solve exponential and logarithmic equations and inequalities using graphs, tables, and algebraic methods. (TEKS f4) 47. Analyze a situation modeled by an exponential function, formulate an equation or inequality, and solve the problem. (TEKS f5) POLYNOMIAL FUNCTIONS long division relative extrema polynomial roots polynomial function synthetic division zeroes 48. Divide polynomials by long and synthetic division. 49. Rewrite polynomial functions as a product of binomials. 50. Find the zeros of polynomial functions. 51. Given the zeros be able to graph the polynomial. 52. Use the calculator to find the intervals where the functions is increasing / decreasing. 53. Use the graphing calculator to find relative extrema on an interval. Sara Ptomey Algebra 2, Page - 7

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