Advanced Algebra 2. I. Equations and Inequalities


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1 Advanced Algebra 2 I. Equations and Inequalities A. Real Numbers and Number Operations 6.A.5, 6.B.5, 7.C.5 1) Graph numbers on a number line 2) Order real numbers 3) Identify properties of real numbers 4) Perform operations with real numbers 5) Use unit analysis 6) Perform operations with real numbers in real life B. Algebraic Expressions and Models 6.A.5, 8.A.5, 8.C.5 1) Evaluate powers 2) Use order of operations 3) Evaluate algebraic expressions 4) Simplify expressions by combining like terms 5) Write and evaluate reallife models C. Solving Linear Equations 8.C.5, 9.A.5 1) Solve equations with a variable on one side 2) Solve equations with a variable on both sides 3) Use the distributive property 4) Solve equations with fractions 5) Write linear equations to model and solve reallife problems D. Rewriting Equations and Formulas 8.C.5, 9.A.5 1) Rewrite an equation with more than one variable 2) Calculate the value of a variable 3) Rewrite common formulas 4) Apply common formulas E. Problem Solving Using Algebraic Models 7.C.5, 8.C.5, 9.A.5 1) Write and use a formula 2) Write and use a model to solve reallife problems F. Solving Linear Inequalities 6.A.5, 8.C.5 1) Solve inequalities with a variable on one side 2) Solve inequalities with a variable on both sides 3) Solve compound inequalities
2 4) Write inequalities to model and solve reallife problems 2 G. Solving Absolute Value Equations and Inequalities 6.A.5, 6.C.5 1) Solve absolute value equations 2) Solve absolute value inequalities 3) Write and use absolute value models to solve reallife problems II. Linear Equations and Functions A. Functions and Their Graphs 8.B.5, 8.C.5 1) Identify functions 2) Graph functions 3) Evaluate functions 4) Use functions in real life B. Slope and Rate of Change 6.D.5, 8.C.5, 9.A.5 1) Find the slope of a line 2) Classify lines using slope 3) Compare steepness of lines using slope 4) Classify parallel and perpendicular lines 5) Find average rate of change 6) Use slope in real life C. Quick Graphs of Linear Equations 8.B.5, 8.C.5 1) Graph lines using slopeintercept form 2) Graph lines using x and yintercepts 3) Graph horizontal and vertical lines 4) Write and graph equations in two variables to model reallife situations D. Writing Equations of Lines 6.D.5, 8.B.5, 8.C.5 1) Write an equation given the slope and yintercept 2) Write an equation given the slope and a point 3) Write equations of perpendicular and parallel lines 4) Write an equation given two points 5) Write and use a linear model 6) Write and use a direct variation equation 7) Identify direct variation E. Correlation and BestFitting Lines 8.C.5, 10.A.5
3 1) Determine correlation 2) Fit a line to data 3 F. Linear Inequalities in Two Variables 8.B.5, 8.D.5 1) Check solutions of inequalities 2) Graph linear inequalities in one variable 3) Graph linear inequalities in two variables 4) Write and use linear inequalities G. Piecewise Functions 8.B.5, 8.D.5 1) Evaluate piecewise functions 2) Graph piecewise functions 3) Graph step functions 4) Write piecewise functions H. Absolute Value Functions 8.B.5, 8.D.5 1) Graph absolute value functions 2) Write absolute value functions III. Systems of Linear Equations and Inequalities A. Solving Linear Systems by Graphing 8.A.5, 8.B.5, 8.D.5 1) Check solutions of a linear system 2) Solve systems graphically B. Solving Linear Systems Algebraically 8.B.5, 8.C.5, 8.D.5 1) Solve systems using the substitution method 2) Solve systems using the linear combination method 3) Write and solve systems in reallife problems C. Graphing and Solving Systems of Linear Inequalities 8.B.5, 8.C.5, 8.D.5 1) Graph a system of two or more inequalities 2) Write and use a system of inequalities D. Linear Programming 8.B.5, 8.D.5 1) Find maximum and minimum values given the feasible region 2) Find maximum and minimum values after graphing the feasible region 3) Use linear programming to solve reallife problems
4 E. Graphing Linear Equations in Three Variables 8.B.5, 8.C.5, 9.C.5 1) Plot points in three dimensions 2) Graph linear equations in three variables 3) Evaluate a function of two variables 4) Model reallife problems using systems of equations in three variables F. Solving Systems of Linear Equations in Three Variables 8.B.5, 8.C.5, 8.D.5, 9.C.5 1) Solve systems using the linear combination method 2) Solve systems using the substitution method 3) Write and solve linear systems to model reallife problems 4 IV. Matrices and Determinants A. Matrix Operations 6.A.5, 6.B.5, 8.C.5 1) Compare matrices 2) Add and subtract matrices 3) Multiply a matrix by a scalar 4) Solve a matrix equation 5) Use matrices to organize data B. Multiplying Matrices 6.A.5, 6.B.5, 8.C.5 1) Describe matrix products 2) Find the product of two matrices 3) Use matrix operations 4) Use matrix multiplication to solve reallife problems C. Determinants and Cramer s Rule 6.A.5, 6.B.5, 9.C.5 1) Evaluate determinants 2) Find the area of a triangle 3) Use Cramer s Rule to solve linear systems D. Identity and Inverse Matrices 6.B.5, 8.C.5 1) Find the inverse of a 2 x 2 matrix 2) Find the inverse of a 3 x 3 matrix 3) Solve matrix equations E. Solving Systems Using Inverse Matrices 6.B.5, 8.C.5, 8.D.5 1) Write a matrix equation 2) Use matrices to solve a linear system
5 5 V. Quadratic Functions A. Graphing Quadratic Functions 6.B.5, 8.C.5 1) Graph quadratic functions in standard form 2) Graph quadratic functions in vertex form 3) Graph quadratic functions in intercept form 4) Write quadratic functions in standard form B. Solving Quadratic Equations by Factoring 8.A.5, 8.C.5, 8.D.5 1) Factor trinomials 2) Solve quadratic equations by factoring C. Solving Quadratic Equations by Finding Square Roots 8.A.5, 8.C.5, 8.D.5 1) Simplify radicals 2) Solve quadratic equations by finding square roots D. Complex Numbers 6.A.5, 6.B.5, 8.A.5 1) Solve quadratic equations 2) Plot complex numbers 3) Perform operations with complex numbers 4) Find the absolute value of complex numbers E. Completing the Square 8.A.5, 8.B.5 1) Solve quadratic equations 2) Write quadratic functions in vertex form F. The Quadratic Formula and the Discriminant 8.A.5, 8.B.5, 8.D.5 1) Solve quadratic equations 2) Use the discriminant to find the number and type of solutions G. Graphing and Solving Quadratic Inequalities 8.B.5, 8.C.5, 8.D.5 1) Graph quadratic inequalities 2) Graph systems of quadratic inequalities 3) Solve quadratic inequalities H. Modeling with Quadratic Functions 8.B.5, 8.D.5 1) Write quadratic functions in vertex form 2) Write quadratic functions in intercept form 3) Write quadratic functions in standard form
6 6 VI. Polynomials and Polynomial Functions A. Using Properties of Exponents 6.A.5, 6.B.5, 7.C.5 1) Evaluating Numerical Expressions 2) Simplifying Algebraic Expressions 3) Using Scientific Notation B. Evaluating and Graphing Polynomial Functions 8.B.5, 8.C.5 1) Identifying Polynomial Functions 2) Using Synthetic Substitution 3) Evaluating a Polynomial Function in real life C. Adding, Subtracting, and Multiplying Polynomials 6.B.5, 6.D.5, 8.C.5 1) Add, Subtract, and multiply polynomials 2) Use polynomial operations in reallife problems D. Factoring and Solving Polynomial Equations 8.A.5, 8.D.5, 9.B.5 1) Factor polynomial expressions 2) Use factoring to solve polynomial equations E. The Remainder and Factor Theorems 6.B.5, 8.D.5 1) Divide polynomials and relate the result to the remainder theorem and the factor theorem 2) Use polynomial division in reallife problems F. Finding Rational Zeros 8.B.5, 8.C.5, 9.B.5 1) Find the rational zeros of a polynomial function 2) Use polynomial equations to solve reallife problems G. Using the Fundamental Theorem of Algebra 6.A.5, 6.C.5, 6.D.5, 8.A.5, 8.B.5 1) Use the fundamental theorem of algebra to determine the number of zeros of polynomial function 2) Use technology to approximate the real zeros of polynomial function H. Analyzing Graphs of Polynomial Functions 8.B.5, 8.D.5, 9.C.5 1) Analyze the graph of a polynomial function
7 2) Use the graph of a polynomial function to answer questions about reallife situations 7 I. Modeling with Polynomial Functions 8.B.5, 8.D.5, 10.A.5 1) Use finite differences to determine the degree of a polynomial function that will fit a set of data 2) Use technology to find polynomial models for reallife data VII. Powers, Roots, and Radicals A. nth Roots and Rational Exponents 6.A.5, 6.B.5, 6.C.5, 7.A.5 1) Evaluate nth roots of real numbers using both radical notation and rational exponent notation 2) Use nth roots to solve reallife problems B. Properties of Rational Exponents 6.B.5, 7.A.5, 7.C.5 1) Use properties of rational exponents 2) Properties of radicals 3) Adding and subtracting roots and radicals 4) Simplifying Expressions Involving Variables 5) Modeling and using properties of rational exponents with variables C. Power Functions and Function Operations 6.D.5, 8.B.5, 8.C.5, 8.D.5 1) Perform function operations 2) Adding and subtracting functions 3) Multiplying and dividing functions 4) Finding the composition of functions 5) Using composition of functions D. Inverse Functions 6.C.5, 8.B.5, 8.C.5, 8.D.5 1) Finding an inverse relation 2) Verifying inverse Functions 3) Writing an inverse model 4) Finding an inverse power function 5) Writing an inverse model
8 8 E. Graphing Square Root and Cube Root Functions 6.A.5, 8.B.5, 8.C.5, 8.D.5 1) Graphing radical functions 2) Graphing a square root function 3) Graphing a cube root function 4) Finding domain and range 5) Modeling square root and cube root functions F. Solving Radical Equations 8.B.5, 8.D.5 1) Solving a simple radical equation 2) Solving and equation with rational exponents 3) Solving and equation with one radical 4) 2 radials 5) Equations with extraneous solutions G. Statistics and Statistical Graphs 6.A.5, 8.B.5, 10.A.5 1) Finding measures of central tendency 2) Finding ranges of data sets 3) Finding standard deviation of data sets 4) Drawing boxandwhisker plots VIII. Exponential and Logarithmic Functions A. Exponential Growth 8.A.5, 8.B.5, 8.C.5, 8.D.5 1) Graphing exponential growth functions 2) Graphing exponential function of the form y = ab x 3) Graphing general exponential functions 4) Modeling exponential growth B. Exponential Decay 8.B.5, 8.C.5, 8.D.5 1) Recognizing exponential growth and decay 2) Graphing exponential functions of the form y = ab x 3) Modeling exponential decay C. The Number e 6.B.5, 6.D.5, 8.B.5, 8.C.5, 8.D.5 1) Simplifying natural base expressions 2) Evaluating 3) Graphing natural base functions
9 D. Logarithmic Functions 8.B.5, 8.D.5 1) Evaluate logarithmic functions 2) Graph logarithmic functions 9 E. Properties of Logarithms 6.A.5, 6.B.5, 6.D.5, 8.D.5 1) Use properties of logarithms 2) Use properties of logarithms to solve reallife problems F. Solving Exponential and Logarithmic Equations 8.A.5, 8.B.5, 8.D.5 1) Solve exponential equations 2) Solve logarithmic equations G. Modeling with Exponential and Power Functions 8.A.5, 8.B.5, 10.A.5 1) Model data with exponential functions 2) Model data with power functions H. Logistic Growth Functions 8.B.5, 8.C.5, 8.D.5 1) Evaluate and graph logistic growth functions 2) Use logistic growth functions to model reallife quantities IX. Rational Equations and Functions A. Inverse and Joint Variation 8.B.5, 8.D.5 1) Write and use inverse variation and joint models B. Graphing Simple Rational Functions 8.B.5, 8.C.5 1) Graph simple rational functions 2) Use the graph of a rational function to solve reallife problems C. Graphing General Rational Functions 7.C.5, 8.B.5, 8.C.5, 8.D.5 1) Graph general rational functions 2) Use a graph of a rational function to solve reallife problems D. Multiplying and Dividing Rational Expressions 6.A.5, 6.B.5, 7.C.5 1) Multiply and divide rational expressions 2) Use rational expressions to model reallife quantities
10 E. Addition, Subtraction, and Complex Fractions 6.A.5, 6.B.5, 6.C.5 1) Add and subtract rational expressions 2) Simplify complex fractions 10 F. Solving Rational Equations 8.A.5, 8.C.5, 8.D.5 1) Solve rational equations 2) Use rational equations to solve reallife problems X. Quadratic Relations and Conic Sections A. The Distance and Midpoint Formulas 6.B.5, 9.A.5, 9.B.5 1) Find the distance between two points and find the midpoint of the line segment joining two points 2) Use the distance and midpoint formulas in reallife situations B. Parabolas 8.A.5, 8.B.5, 8.D.5 1) Graph and write equations of parabolas 2) Use parabolas to solve reallife problems C. Circles 8.B.5, 8.D.5, 9.A.5 1) Graph and write equations of circles 2) Use circles to solve reallife problems D. Ellipses 7.C.5, 8.B.5, 8.D.5, 9.A.5 1) Graph and write equations of ellipses 2) Use ellipses in reallife situations E. Hyperbolas 8.B.5, 8.D.5, 9.A.5 1) Graph and write equations of hyperbolas 2) Use hyperbolas to solve reallife problems F. Graphing and Classifying Conics 8.B.5, 8.C.5, 8.D.5, 9.B.5 1) Write and graph an equation of a parabola with its vertex at (h, k) and an equation of a circle, ellipse, or hyperbola with its center at (h, k) 2) Classify a conic using its equation
11 G. Solving Quadratic Systems 6.B.5, 8.D.5 1) Solve systems of quadratic equations 2) Use quadratic systems to solve reallife problems 11 H. Eccentricity of Conic Sections 8.C.5 1) Find the eccentricity of a conic section XI. Sequences and Series (As Much as Time Allows) A. An Introduction to Sequences and Series 6.B.5, 8.A.5, 8.C.5, 8.D.5 1) Use and write sequences 2) Use summation notation to write series and find sums of series B. Arithmetic Sequences and Series 6.A.5, 8.A.5, 8.C.5 1) Write rules for arithmetic sequences and find sums of arithmetic series 2) Use arithmetic sequences and series in reallife problems C. Geometric sequences and Series 6.A.5, 6.D.5, 8.A.5, 8.D.5 1) Write rules for geometric sequences and find sums of geometric series 2) Use geometric sequences and series to model reallife quantities D. Infinite Geometric Series 8.A.5, 8.C.5, 8.D.5 1) Find sums of infinite geometric series 2) Finding common ratio E. Recursive Rules for Sequences 8.A.5, 8.C.5, 8.D.5 1) Evaluate and write recursive rules for sequences 2) Use recursive rules to solve reallife problems XII. Probability and Statistics (If Time Allows) A. The Fundamental Counting Principle and Permutations 6.B.5, 10.A.5 1) Use the fundamental counting principle and permutations to count the number of ways an event can happen
12 12 B. Combinations and the Binomial Theorem 6.B.5, 10.A.5 1) Use combinations to count the number of ways an event can happen 2) Use the binomial theorem to expand a binomial that is raised to a power C. An Introduction to Probability 6.A.5, 6.B.5, 8.A.5 1) Find theoretical and experimental probabilities 2) Find geometric probabilities D. Probability of Compound Events 6.C.5, 10.C.5 1) Find probabilities of unions and intersections of two events 2) Use complements to find the probability of an event E. Probability of Independent and Dependent Events 10.C.5 1) Find the probability of independent and dependent events F. Binomial Distributions 10.C.5 1) Find binomial probabilities and analyze binomial distributions 2) Test a hypothesis G. Normal Distributions 8.B.5, 10.C.5 1) Calculate probabilities using normal distributions 2) Use normal distributions to approximate binomial distributions
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