# 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 real-life 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 real-life 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 real-life 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 real-life 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 real-life 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 slope-intercept form 2) Graph lines using x- and y-intercepts 3) Graph horizontal and vertical lines 4) Write and graph equations in two variables to model real-life situations D. Writing Equations of Lines 6.D.5, 8.B.5, 8.C.5 1) Write an equation given the slope and y-intercept 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 Best-Fitting 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 real-life 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 real-life 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 real-life 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 real-life 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 real-life 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

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 real-life 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 real-life 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 real-life 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 real-life 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 real-life 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 real-life 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 box-and-whisker 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 real-life 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 real-life 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 real-life 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 real-life 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 real-life 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 real-life 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 real-life situations B. Parabolas 8.A.5, 8.B.5, 8.D.5 1) Graph and write equations of parabolas 2) Use parabolas to solve real-life problems C. Circles 8.B.5, 8.D.5, 9.A.5 1) Graph and write equations of circles 2) Use circles to solve real-life 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 real-life situations E. Hyperbolas 8.B.5, 8.D.5, 9.A.5 1) Graph and write equations of hyperbolas 2) Use hyperbolas to solve real-life 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 real-life 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 real-life 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 real-life 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 real-life 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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