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Course Title: Advanced Honors Algebra II, Grades 9 & 10, Level 5 Length of Course: One Year (5 credits) Prerequisites: Algebra, Geometry Description: Although traditional when considering topics, this course must be taught with any eye towards the future. It is in the study of Algebra II that key concepts and problem solving techniques need to be introduced and refined. Well placed questions, initiated now, will allow the students to begin to consider, on an intuitive basis, Calculus, Trigonometry and Discrete Math. It is also a goal of this course to provide students with the skills necessary to manipulate algebraic expressions expediently and accurately. This class must be taught following the Rule of Three. That is each topic must be considered numerically, graphically and algebraically. For example, we obtain solutions algebraically when that is the most appropriate technique to use, and we obtain solutions graphically or numerically when algebra is difficult to use. Students must be urged to solve problems by one method and then confirm their solutions by an alternative method. Students must begin preparing to see the value of each of these methods and must learn to choose the one that is most appropriate. This approach reinforces the idea that to understand the problem fully students need to understand it algebraically, as well as graphically and numerically. In order for students to pursue this investigation, we must provide them with the technology that will allow it. Therefore, in addition to teaching Algebra II, it is critical that students receive instruction in the graphing calculator. This class is the place to introduce the power of this instrument. We again must emphasize that even on a graphing calculator it is possible to consider algebraic as well as numerical thinking. This course strives to give students a proper balance between the mastery of skills and the comprehension of key concepts. With that in mind, this curriculum guide clearly defines the learning objectives for each unit in terms of the key skills and key concepts that must be mastered within each unit. Evaluation: Student performance will be measured using a variety of instructor-specific quizzes and chapter tests as well as a common departmental Quarter, Midterm and Final Exams. Assessments will equally emphasize measurement of the degree to which required skills have been mastered as well as how well key concepts have been understood. Text: Algebra 2, Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff, McDougal Littell, 2001 Topic What is the complex Number System Prerequisite Unit: The Complex Number System Learning Objectives: Key Definitions, Skills and Concepts What are natural (counting), whole, integers, rational, irrational, real, pure imaginary and complex numbers? Skills check, ability to: Identify the set(s) to which a given number belongs Identify subset relations among the sets of numbers Concept check: What is closure and how is this concept related to the expansion of the complex numbers? What is density? Which numbers are more dense: rationals or wholes? Explain. Topic 2.1 Functions & Their Graphs Unit 2: Linear Equations & Functions Learning Objectives: Key Definitions, Skills and Concepts What is the definition of a function? What is the domain & range of a function? What is a linear function? Skills check, ability to: Identify functions Graph relations Use the vertical line test Evaluate functions Graph a linear function Concept check: What does it mean to say f is a function of x? Is f ( x) the same as f x ? Are all functions Linear functions? Are all lines functions? 2.2 Slope & Rate of Change What is the slope of a line? What kinds of slopes do parallel and perpendicular lines have? Skills check, ability to: Correctly find the slope of a line given two points on the line, its equation, the equation of a parallel or perpendicular line. Concept Check: Do all lines have a slope? What causes a line to have no slope? Is slope a concept unique to a line? 2.3 Quick Graphs of Linear Equations How can a line be graphed using the slope-intercept method? How can a line be graphed using the intercept method? Skills check, ability to: Graph a line using slope-intercept and intercept methods Graph horizontal and vertical lines Concept check: Can all lines be graphed using slope-intercept or intercept methods? Explain. 2.4 Writing Equations of Lines What is the symbolic appearance of a linear equation? What is direct variation? Skills check, ability to: Write the equation of a line given its graph, two points, the equation of a parallel or perpendicular line, an anecdote Identify a linear function that is direct variation Concept check: What, in the equation of a line, distinguishes an oblique line from one that is horizontal or vertical? 2.5 Correlation & Best Fitting Lines What is a line of best fit? What does it mean to say that two quantities have a correlation? Skill check, ability to: Describe a correlation given a scatterplot Approximate a line of best fit given a set of data Concept check: Is it always appropriate to model data with a line of best fit? Explain. What is extrapolation and why should this be considered in all modeling? Graphing Calculator Scatterplots and Linear Regression 2.6 Linear Inequalities How can a set of data be entered? How can a scatterplot be shown? How can the graphing calculator be used to obtain the linear model of the data? When is this reliable? How is a linear inequality graphed? Skill check, ability to: Graph a linear inequality Create a linear inequality from a real world problem Concept check: What is it about a linear inequality that determines the kind of boundary that is appropriate? Topic 2.7 Piecewise Functions Learning Objectives: Key Definitions, Skills and Concepts What is a piecewise function? How are these represented symbolically and graphically? What is a step function? Skill check: ability to: Recognize when a problem will result in a piecewise function Correctly model an anecdote with a piecewise function Correctly graph a piecewise function given its equation Correctly evaluate a piecewise function Concept check: Are all piecewise functions step functions? 2.8 Absolute Value Functions What is the absolute value function? What is its domain and range? Skills check, ability to: Identify the vertex of an absolute value function given its equation Identify whether the vertex is a “high” or “low” point on the graph given its equation Graph an absolute value function given its equation or an anecdote Concept check: Is an absolute value function a linear function? Explain. What characteristic of the equation controls the width of an absolute value function? Explain. Unit 3: Systems of Linear Equations & Inequalities Topic Learning Objectives: Key Definitions, Skills and Concepts 3.1 Solving Systems What is a system of equation? How is it solved graphically? By Graphing Skills check, ability to: Solve a system of linear equations by graphing Identify the graph of a system that has no solutions Identify the graph of a system that has an infinite number of solutions Correctly check the solution to a system of linear equations Concept check: Are all systems of equations linear? What is a 2x2 system, a 3x3 system, etc? What is the “dimension requirement” so that a unique solution exists? 5 x 6 y 24 Consider this system: . Explain how slopes can be used to 2 x 3 y 15 show that the system has only one solution. Graphing Calculator How is a linear system of equations solved on a graphing calculator? Skills check, ability to: Correctly enter equation Correctly identify a window that is appropriate Correctly use the intersect feature Correctly use the technology to determine when the lines are coinciding 3.2 Solving Linear Systems Algebraically What are the algebraic methods for solving s linear system? Skills check, ability to: Solve a linear system by substitution and elimination Concept check: Will algebraic methods always work when solving a linear system? Explain. What is the algebraic equivalent to a system of parallel lines, a system of coinciding lines? What about a given system should be considered when determining a method of solving? Write a system of linear equations that has (-4.3, 8) as its only solution 3.3 Graphing & Solving Systems of Linear Inequalities What does the solution to a system of linear inequalities look like? How is graphing used to solve a system of linear inequalities? Skills check, ability to: Correctly solve a system of linear inequalities by graphing Concept check: Is there an algebraic equivalent to a graphing solution for a system of linear inequalities? 3.4 Linear Programming What is an optimization problem? How is it solved using linear programming? What is an objective function? What are constraints? What is a feasible region? Skill check, ability to: Correctly interpret a real world situation into constraints and an objective function Correctly use linear programming techniques to optimize the objective function Concept check: Are all graphing solutions acceptable solution? What kinds of solutions are mandated by the context of the problem? How do you interpret an unbounded feasible region? Why is the optimal solution found at a vertex of the feasible region? 3.5 Graphing Three Equations in three Variables How do we graph in three dimensions? Skills check, ability to: Correctly plot points in three dimensions Correctly graph linear equations in three variables Correctly model an real world problem in three variables Concept check: What does the equation ax by cz d look like in three dimensions? 3.6 Solving Systems of Linear Equations in Three Variables What are the graphing and algebraic methods used to solve a 3x3 System? Skills check, ability to: Correctly solve a 3x3 system of equations Concept check: Topic Unit 4: Matrices & Determinants Learning Objectives: Key Definitions, Skills and Concepts What is a matrix? How is it described? When are two matrices considered to be equal? What are matrix operations? What is a scalar? 4.1 Matrix Operations Skills check, ability to: Correctly identify the dimensions of a matrix Correctly identify amn Correctly add and multiply a matrix by a scalar Concept check: Is it always informative to add rows or columns of a matrix? What conditions must be met so that two matrices can be added? Is matrix addition commutative and associative? Explain. True/False: If A and B have the same number of elements, then A+B exists True/False: If A+B exists then A-B exists. What is matrix multiplication? 4.2 Multiplying Matrices Skills check, ability to: correctly identify when two matrices can be multiplies correctly multiply matrices Concept check: Is matrix multiplication commutative and associative? Explain. Graphing Calculator How are matrices entered on a graphing calculator? How are matrices added and multiplied on a graphing calculator? What error messages will appear if matrices cannot be combined or multiplied? 4.3 Determinants & Cramer’s Rule What is the determinant of a matrix? How can Cramer’s rule be used to solve systems of equations? Skills check, ability to: Correctly calculate the determinant of a 2x2 and 3x3 matrix Correctly use Cramer’s Rule to solve a system of equations Concept check: Do all matrices have determinants? Explain the circumstances in which a matrix fails to have a determinant. What is the “Cramer’s Rule” equivalent to a system that has either no solution or an infinite number of solutions? What about a system of equations would lead one to use Cramer’s rule as a method of solution? 4.4 Identity & Inverse Matrices What is the identity matrix? What is the inverse of a matrix? Skill check, ability to: Identify whether two matrices are inverses Find the inverse of a 2x2 matrix Solve matrix equations Concept check: Explain how you could tell if two matrices are inverses. How are inverse matrices used to solve systems of equations? 4.5 Solving Systems Using Inverse Matrices Skills check, ability to: Create a matrix equation from a system of equations Correctly use inverse matrices to solve the system of equations Concept check: Suppose you try to solve a system of equations using inverse matrices and your calculator. When you are done entering the coefficient matrix (A) and the constant matrix (B), you call for A 1 * B and your calculator reads: ERR: Singular Matrix. What is your calculator telling you? True/False: Every matrix has an inverse. Topic 5.1 Graphing Quadratic Functions Unit 5: Quadratic Functions Learning Objectives: Key Definitions, Skills and Concepts What is a quadratic function? How can a quadratic function be graphed? Skills check, ability to: Identify a quadratic function given its graph or equation Identify the vertex and line of symmetry of a parabola given its graph Identify whether a parabola opens up or down given its graph or equation Identify a quadratic equation in standard, intercept and vertex form Quickly sketch a quadratic function given in vertex form Quickly sketch a quadratic function given in intercept form Change from intercept form and vertex form to standard form Correctly solve a quadratic equation by graphing the associated quadratic function. Concept check: In what ways are quadratic functions the same/different than linear functions? Which coefficient(s) control opening up or down? By expanding f ( x) a( x h)2 k knowing that the vertex is at h, k , demonstrate that it is reasonable for the vertex of f ( x) ax 2 bx c to be b b found at , f ( ) . 2a 2a Why are the solutions to a quadratic equation found at the x-intercepts of the associated quadratic function? If a quadratic equation has no solutions, what does its graph look like? If a quadratic equation has one solution, what does its graph look like? Topic 5.2 Solving Quadratic Equations By Factoring Learning Objectives: Key Definitions, Skills and Concepts How is a quadratic equations solved by factoring? Skills check, ability to: Factor out the greatest common factor from a polynomial Factor the difference of two perfect squares Factor the sum/difference of two perfect cubes Group factor Factor quadratic trinomials Use the zero product property to solve a quadratic equation by factoring Concept check: Is it true that every quadratic equation has a solution? Explain. Is it true that every quadratic equation has two solutions? Explain. Is factoring a method that will work for all quadratic equations? Explain 5.3 Solving Quadratic Equations By Square Roots What are radicals, radicands and index numbers? How can a quadratic equation be solved using square roots? Skills check, ability to: Simplify radicals Identify which quadratic equations can be solved by square roots Solve a quadratic equation by square roots Concept check: What kinds of numerical solutions can be found using square roots that cannot be found using graphing or factoring? Graphing Calculator How can a quadratic equation be entered into the calculator? How can a graphing calculator be used to identify the vertex of a quadratic function? How can the zero feature be used to estimate the roots of a quadratic equation. After trying to use the zero feature to estimate the roots of a quadratic equation your calculator says: “ERR: NO SIGN CHNG”. What is the calculator telling you? Topic Learning Objectives: Key Definitions, Skills and Concepts What is i ? What is an imaginary number? What is a complex number? 5.4 Complex Numbers Skills check, ability to: Simplify pure imaginary numbers using i Add/subtract/multiply/divide complex numbers Solve a quadratic equation having complex roots Concept check: Describe the powers of i . Distinguish the powers of i from the powers of a real number. Describe how you can evaluate i n , n whole numbers . 1 What is the standard form of ? i Describe the nature of the graph of f ( x) ax 2 bx c when a,b, and c are real numbers and the equation ax 2 bx c 0 has nonreal complex solutions. How can the process of completing the square be used to solve quadratic equation? 5.5 Completing The Square Skills check, ability to: Correctly complete the square Solve a quadratic equation by completing the square Change the form of a quadratic function from standard or intercept into vertex Concept check: Of all the methods used to solve a quadratic equation outline the pros and cons of each. Are there some that can be used in all cases and some that have special requirements? Topic 5.6 The Quadratic Formula and the Discriminant Learning Objectives: Key Definitions, Skills and Concepts What is the discriminant of a quadratic equation? How is it used? How is a quadratic equation solved using the quadratic formula? Skills check, ability to: Solve a quadratic equation using the quadratic formula Calculate the discriminant of a quadratic equation Correctly interpret the nature of the roots based upon the value of the discriminant Concept check: If b 2 4ac 0 , what can you say about the zeros of the quadratic function f ( x) ax 2 bx c ? 5.7 Graphing & Solving Quadratic Inequalities What does the solution to a quadratic inequality with two variables look like? How can a quadratic inequality in one variable be solved by graphing and algebraically (using a number line)? Skills check, ability to: Graph a two variable quadratic inequality Graph of system of quadratic inequalities Solve a quadratic inequality by graphing and algebra Concept check: 5.8 Modeling With Quadratic Functions What is a quadratic model? How can a data set be modeled by a quadratic function? Skills check, ability to: Write a quadratic function in standard, vertex or intercept form given its graph, equation (in an alternate form), or three points on the graph Find a quadratic model for a given set of data Concept check: Is it always possible to create a quadratic model for a given data set? When is this appropriate and when is it not? Explain. Topic 6.1 Using Properties of Exponents Unit 6: Polynomials & Polynomial Functions Learning Objectives: Key Definitions, Skills and Concepts What are the properties of exponents and how can they be used to simplify expressions? Skills check, ability to: Evaluate numerical expressions Simplify algebraic expressions Use scientific notation 6.2 Evaluating & Graphing Polynomial Functions Concept check: Which of the properties of exponents can be called the ‘distributive property”? What is a polynomial function? What do the graphs of polynomial functions look like? Skills check, ability to: Identify polynomial functions, including degree, leading coefficient and constant Evaluate polynomial functions Identify end behavior for a polynomial function Graph polynomial functions Concept check: Are all linear functions polynomial functions? Are all polynomial functions linear functions? Is it possible for a third degree polynomial function to have no real zeros? Explain. Is it possible for a fourth degree polynomial function to have no real zeros? Explain. T/F: All polynomial functions have an x-intercept T/F: All polynomial functions have a y-intercept What are even and odd functions? Are all functions even or odd? 6.3 Adding, Subtracting and Multiplying Polynomials What are the operations on polynomials? Skills check, ability to: Add, subtract and multiply polynomials Recognize the sum/difference of perfect cubes Concept check: What are like terms? Must they have the same coefficients? 6.4 Factoring & Solving Polynomial Equations How are polynomial equations solved algebraically? How are polynomial equations solved graphically? Skill check, ability to: Factor polynomial expressions using special patterns Factor polynomial expressions using group factoring Factor polynomial expressions in quadratic form Solve a polynomial equation by factoring Concept check: Explain which factoring techniques should be used for polynomials that are binomials, trinomials and 4-term polynomials. 6.5 The Remainder & Factor Theorems How are polynomials divided? What is synthetic division and when is it appropriate? What is the remainder theorem and what does it reveal? Skills check, ability to: Divide polynomials using long division Divide polynomials using synthetic division Decide if one polynomial is a factor of another polynomial Find zeros of polynomial functions using division Concept check: Consider a polynomial function. What is the difference between the xintercepts and zeros? Consider a polynomial equation. What is the difference between the roots of the equation and the real zeros of the function? Fill-in: -4 is a __________of x 2 3 x 4 =0 -4 is an _________of f ( x) x 2 3 x 4 -4 is a __________of f ( x) x 2 3 x 4 x+4 is a _________of x 2 3 x 4 Explain how to carry out the following division using synthetic division. Work through the steps with complete explanations. Interpret and check your 4 x3 5 x 2 3x 1 result. 2x 1 Topic 6.6 Finding Rational Zeros Learning Objectives: Key Definitions, Skills and Concepts How are the rational zeros of a polynomial function found algebraically? What is the rational zero theorem? Skills check, ability to: List the possible rational zeros of a polynomial function Identify the rational zeros of a polynomial function using the Rational Zero Theorem Solve polynomial equations using the rational zero theorem Concept check: How can the graph of a polynomial function help to identify the rational zeros? 1 Write a paragraph describing how the zeros of f ( x) x 3 x 2 2 x 3 are 3 3 2 related to the zeros of g ( x) x 3x 6 x 9. In what ways does this example illustrate how the Rational Zeros Theorem can be applied to find the zeros of a polynomial with rational number coefficients? 6.7 Using the Fundamental Theorem of Algebra What is the Fundamental Theorem of Algebra and how does it relate to the number of zeros of a polynomial? Skills check, ability to: Find the number of solutions to a polynomial equation or the number of zeros of a polynomial function Find the zeros of a polynomial function Write a polynomial function/equation given the zeros/roots Concept check: If the coefficients of a polynomial function are real and rational, why must imaginary zeros come in pairs? Topic 6.8 Analyzing Graphs of Polynomial Functions Learning Objectives: Key Definitions, Skills and Concepts What is a turning point of a polynomial function? How many turning points will a polynomial function have? Skills check, ability to: Graph a polynomial function given its equation in factored form Graph a polynomial function given its equation in standard form Concept check: Are all turning points maximums? Do all polynomial functions have turning points? What is meant by multiplicity of factors? Suppose -3, 2 and 5 are the solution to a cubic polynomial equation. Sketch a graph of the polynomial function. Could there be more than one sketch? How many more? Explain. What is a local minimum? How is this different from an absolute minimum? 6.9 Modeling With Polynomial Function What does finite difference refer to and how does it help decide on an appropriate polynomial model? Skills check, ability to: Write the equation of a polynomial function given its graph Correctly identify the degree of a polynomial model given a set of data Create a polynomial model of a set of data Utilize the regression feature on a graphing calculator to find a polynomial model Concept check: Topic 7.1 nth Roots and Rational Exponents Unit 7: Powers, Roots & Radicals Learning Objectives: Key Definitions, Skills and Concepts What is the nth root of a number? How is it written? How are nth roots written using rational exponents? Skills check, ability to: Evaluate the real nth roots of numbers Evaluate a constant raised to a rational power Solve equations using nth roots Concept check: What is the principal root of a number? T/F: 9 3 1 T/F: 9 2 3 7.2 Properties of Rational Exponents What properties of exponents also apply to rational exponents? How are these properties used to simplify expressions? Skills check, ability to: Simplify expressions that have rational exponents Multiply radicals Rationalize an irrational denominator Add/subtract radicals Concept check: Under what circumstances can two radicals be added? Under what circumstances can two radicals be multiplied? 7.3 Power Functions & Function Operations What is a power function? What are function operations? Skill check, ability to: Identify a power function Distinguish a power function from a polynomial function Identify domain & range of a power function Compose functions Identify the domain of a composed function Concept check: Is f ( g ( x)) g ( f ( x)) ? Explain 7.4 Inverse Functions What makes two functions inverses? How do we create the inverse of a given function? Skills check, ability to: Identify when two functions are inverses given the equations and/or graphs Create the inverse of a given function Identify when the inverse will not be a function Use the horizontal line test to predict whether a function is invertible Concept check: In the expression f 1 ( x) is -1 an exponent? T/F: All functions have inverses Under what circumstances will a function be invertible? Explain T/F: All linear functions are invertible 7.5 Graphing Square Root & Cube Root Functions What does the graph of f ( x) x and f ( x) 3 x look like? What are their domain and range? Skills check, ability to: Graph the parent square root and cube root functions Graph the transformations of these parent curves Graph the transformations of other basic function (e.g. f ( x) x, f ( x) x2 , f ( x) x ) Identify domain and range of a transformed function Concept check: Given the graph of f ( x) sin x, sketch f ( x) 3sin( x 4) 7 Consider g ( x) af ( x b) c . Describe the effects a, b and c have on the graph of f ( x ). Topic 7.6 Solving Radical Equations Learning Objectives: Key Definitions, Skills and Concepts What is a radical equation and how are they solved? How are equations with rational exponents solved? What is an extraneous root? Skills check, ability to: Solve a radical equation algebraically Solve an equation containing rational exponents algebraically Identify extraneous root Concept check: How can a graph help identify when equations will have extraneous solutions? Is it always important to check solutions to equations? When is it necessary? (Turkey into a Pig) Topic 8.1 Exponential Growth Unit 8: Exponential & Logarithmic Functions Learning Objectives: Key Definitions, Skills and Concepts What is an exponential function? What is an asymptote? What does exponential growth look like algebraically and graphically? What is concavity? Skills check, ability to: Recognize an exponential function algebraically & graphically Identify the base and coefficient from an equation Identify an asymptote from a graph and equation State the domain and range of an exponential function given its equation and/or graph Recognize an exponential growth function from a graph, equation and anecdote Create an exponential growth function from an anecdote Graph a transformed exponential faction Use the compound interest formula Concept check: Are all exponential functions ? Are all exponential functions concave up? Why can’t the base of an exponential function be a negative number? Write an problem that would have the following as its answer: P 100(5) x Topic 8.2 Exponential Decay Learning Objectives: Key Definitions, Skills and Concepts What does exponential decay look like algebraically and graphically? Skills check, ability to: Identify the base and coefficient from an equation Identify an asymptote from a graph and equation State the domain and range of an exponential decay function given its equation and/or graph Recognize an exponential decay function from a graph, equation and anecdote Create an exponential decay function from an anecdote Graph a transformed exponential function Concept check: 2 1 You are asked to sketch f ( x) ( ) x 4 . Explain how you would do this. 3 8 Describe the effects of different values of a, b and k in the function f ( x) abkx . Write two exponential functions, having the same base, with one being growth and one being decay. Without using formulas or graphs, compare and contrast exponential and linear functions. 8.3 The Number e What is e? What is it used for? Skill check, ability to: Sketch f ( x) e x Sketch transformed functions using e Simplify expressions using powers of e Use the continuous interest formula Concept check: What kind of number is e? Consider the function f ( x) e x . Identify its domain, range, asymptotes, increasing/decreasing, concavity. 8.4 Logarithmic Functions What is a logarithmic function and why is it necessary? How are logarithmic and exponential form related? What are common and natural logs? Skills check, ability to: Change from exponential form to logarithmic form Change from logarithmic form to exponential form Evaluate logarithmic expressions (all bases) Graph logarithmic functions (all bases) Graph transformed logarithmic functions Find the inverse of a logarithmic function Concept check: Evaluate: log 3 9 . Explain your answer. Consider the function f ( x) ln x . Identify its domain, range, asymptotes, increasing/decreasing and concavity. T/F: the logarithm of a positive number is positive. Topic 8.5 Properties of Logs Learning Objectives: Key Definitions, Skills and Concepts What are the properties of logs and how are they related to the rules of exponents? Skills check, ability to: Expand logarithmic expressions using the properties of logs Condense logarithmic expressions using the properties of logs Use the change of base formula Concept check: Write log 64 in four different ways Can you write log3 6 log 2 5 as a single log? Explain. Can you expand log3 (2 x 1) ? Explain. If you knew the value of log 2 and log 5, how could you evaluate log 20 without using a calculator? log b x log b x log b y T/F: log b y You wish to graph f ( x) log5 x on your graphing calculator. You know the calculator will not work using a base of 5. Write two other functions that are equivalent and can be used on your calculator. Describe how to transform the graph of f ( x) log x into the graph of g ( x) log.1 x. Topic 8.6 Solving Exponential & Logarithmic Equations Learning Objectives: Key Definitions, Skills and Concepts How do you solve exponential & logarithmic equations algebraically? Skills check, ability to: Solve exponential equations when the bases are or can be the same Solve exponential equations when the bases cannot be the same Solve a logarithmic equation with only one log Solve a logarithmic equation with two or more logs that have the same base Concept check: Suppose you are solving an equation containing exponents. You find that 5 x 3 k . What is the next step? Suppose you are solving a radical equation. You find that 6 10 y 3 . What is the next step? Why is it important to check the solutions to exponential and logarithmic equations? Your friend solves a logarithmic equation as follows: 2 log x log 3 2 log( x2 )2 3 x2 102 3 x2 100 3 x 2 300 x 300 x 10 3 Is he correct? If not explain the error. Topic 8.7 Modeling With Exponential & Power Functions Learning Objectives: Key Definitions, Skills and Concepts How can you write an exponential function given two points? How can you write a power function given two points? How can you decide if a data set should be modeled by an exponential or power function? How can you use a graphing calculator to find a model? Skills check, ability to: Create an exponential and power model given two points Decide which is an appropriate model Use a calculator to find a model Use a model to predict (interpolation v. extrapolation) Concept check: 8.8 Logistic Growth Functions What is a logistic curve and what type of data does it accurately model? What is a point of inflection? Skills check, ability to: Evaluate a logistic function Graph a logistic function given its equation Create a logistic model given a carrying capacity and two points Solve a logistic equation Concept check: What does the inflection point indicate about the growth? The formula for the inflection point of the function c ln a c f ( x) is , . Explain how this formula is derived. bx 1 ae b 2 Is it important to check the solutions to a logistic equation? Explain. Without using formulas or graphs, compare and contrast exponential and logistic functions. Topic 9.1 Inverse & Joint Variation Unit 9: Rational Equations & Functions Learning Objectives: Key Definitions, Skills and Concepts What is inverse and joint variation and what situations do they model? What is the constant of variation? Skills check, ability to: Classify variation as direct or inverse Write an inverse variation model given two points Write an inverse or joint variation model given an anecdote Write a direct or inverse variation model using powers State the domain of an inverse variation function Concept check: Explain the differences among data that vary directly, inversely, and inversely with the square. Include an example of each. 9.2 Graphing Simple Rational Functions What is a rational function? What is the graph of a rational function called? What is a discontinuous function? What is a limit? Skills check, ability to: 1 x State the domain, range, equations of asymptotes of a rational function Graph a rational function (numerator and denominator of degree 1) by finding its vertical and horizontal asymptotes and then a few key points Identify when a rational function will have no asymptotes Identify when a rational function will have holes Create a rational function that models an anecdote Graph a simple rational function using transformations on f ( x ) Concept check: Write any rational function that has a vertical asymptote at x=-7 and a horizontal asymptote at y=4. Is this unique? Polynomial functions have a characteristic known as end behavior. Rational functions have limits as x . How are these the same and how are they different? Topic 9.3 Graphing General Rational Functions Learning Objectives: Key Definitions, Skills and Concepts How will the degrees of numerator and denominator affect the graph of a rational function? What are local extrema? Skills check, ability to: Identify x and y intercepts, vertical and horizontal asymptotes, holes, domain and range of rational functions Graph rational functions Create rational functions given an anecdote Identify local and absolute extrema given a graph Concept check: Write a rational function that has a vertical asymptote at x=8. Write another rational function that has a hole at x=8. How are rational expressions multiplied and divided? 9.4 Multiplying & Dividing Rational Expressions Skills check, ability to: Multiply and divide rational expressions Concept check: x 2 10 x 25 x 5 T/F: 2 x 9 x 20 x 4 How can you tell when a rational expression is in simplest form? x Write three rational expressions that simplify to x 1 9.5 Addition, Subtraction and Complex Fractions How are rational expressions combined? How are complex fractions simplified? Skills check, ability to: Combine rational expressions Simplify complex fractions Concept check: T/F: The least common denominator of two rational expressions is always the product of their denominators T/F: The LCD of two rational expressions will have a degree greater than or equal to that of the denominator with the highest degree Explain how factoring is used in finding the LCD when combining two rational expressions. Topic 9.6 Solving Rational Equations Learning Objectives: Key Definitions, Skills and Concepts What is a rational equation and how is it solved algebraically? Skills check, ability to: Solve rational equations and identify extraneous solutions Concept check: Explain why the equation x 1 2 has no solutions. x2 2 x2 Topic 10.1 The Distance & Midpoint Formulas Unit 10: Quadratic Relations & Conic Sections Learning Objectives: Key Definitions, Skills and Concepts How can you find the distance between two points, the midpoint of a line segment and the equation of a perpendicular bisector of a line segment? Skills check, the ability to: Find the distance between two points The midpoint of a line segment The equation of the perpendicular bisector of a lone segment Concept check: How is the distance from a point to a line measured? 10.2 Parabolas (In this section, the vertex of the parabola is at the origin) What is the ‘conic section’ definition of a parabola? What is a focal point and a directrix? Skills check, ability to: Graph a parabola given its equation Identify the focus and directrix of a parabola given its equation/graph Write the equation of a parabola given its graph or focus/directrix Concept check: Are all parabolas functions? Explain. Consider the parabola f ( x) ax 2 . What is the effect of increasing a ? Explain how to find the distance from the focus to the directrix for the parabola f ( y) 2 y 2 . What is the reflective property of a parabola and what is its primary use? Topic 10.3 Circles (In this section the center of the circle is at the origin) Learning Objectives: Key Definitions, Skills and Concepts What is the equation of a circle? What is a tangent line drawn to a circle? Skills check, ability to: Graph a circle given its equation Write the equation of a circle given its graph or radius Write the equation of a tangent line drawn to a circle given the circle and point of tangency Concept check: Can a circle have a radius of -5? How can a circle be graphed on a graphing calculator? Explain why x 2 y 2 0 does not represent a circle. What does the equation represent? Write the equation of a circle that does not intersect the x-Axis. 10.4 Ellipses (In this section, the center of the ellipse is at the origin) What is an ellipse? What is its equation? What are its vertices, covertices, focal points, center, major and minor axes? Concept check, ability to: Find the vertices of an ellipse given its equation Find the equation of an ellipse given its focal points and another piece of information that fixes the ellipse (e.g. minor axis length, a point on the ellipse, etc.) Graph ellipses given either equation and/or enough information to fix the ellipse Concept check: What is a whispering gallery and how is this related to an ellipse? What does the eccentricity of an ellipse measure? What is the eccentricity of a circle? What is the maximum value of the eccentricity of an ellipse? What is the least it can be? Explain why a circle is also an ellipse. The area of a circle is r 2 . The area of an ellipse is ab . Explain the connection. Topic 10.5 Hyperbolas (In this section, the center of the hyperbola is at the origin) Learning Objectives: Key Definitions, Skills and Concepts What is a hyperbola? What is its equation? What are its vertices, focal points, center, transverse axis and asymptotes? Skills check, ability to: Graph a hyperbola given its equation Write the equation of a hyperbola given enough information to fix the hyperbola Concept check: Are any hyperbolas functions? If yes, sketch one. 10.6 Graphing & Classifying Conics What is the equation of a parabola with vertex (h,k)? What is the equation of a circle, ellipse and hyperbola with center (h,k)? Skills check, ability to: Write the equation of and graph a parabola whose vertex is not at the origin Write the equation of and graph circles, ellipses and hyperbolas whose center is not at the origin Classify a conic given its equation in standard form Graph a conic given its equation in standard form Concept check: How does the translation of an ellipse or hyperbola from center (0,0), to center (h,k) affect the coordinates of the vertices and foci? How does the translation of an ellipse affect the length of it major and minor axes? Describe how the translation of a hyperbola will affect its asymptotes. 10.7 Solving Quadratic Systems How can a quadratic system of equations be solved algebraically? Skills check, ability to: Solve a quadratic system of equations using substitution and elimination Concept check: Graph a nonlinear system that has three solutions Describe the circumstance under which substitution would be better than elimination to solve a nonlinear system. Describe the circumstances under which elimination would be better than substitution to solve a nonlinear system. Recommended Unit Sequencing and Pacing Guide Timeframe Q1 Unit Prerequisite: The Complex Number System Unit 1: Linear Equations & Functions 1.1 Functions and their Graphs 1.2 Slope & Rate of Change 1.3 Quick Graphs of Linear Equations 1.4 Writing Equations of Lines 1.5 Correlation & Best Fitting Lines 1.6 Linear Inequalities in Two Variables 1.7 Piecewise Functions 1.8 Absolute Value Functions Unit 2: Systems of Linear Equations & Inequalities 2.1 Solving Linear Systems by Graphing 2.2 Solving Linear Systems Algebraically 2.3 Graphing & Solving systems of Linear Inequalities 2.4 Linear Programming 2.5 Solving Systems of Linear Equations in Three Variables Unit 3: Matrices & Determinants 3.1 Matrix Operations 3.2 Multiplying Matrices Timeframe Q2 Unit 3: Matrices & Determinants 3.3 Determinants and Cramer’s Rule 3.4 Identity & Inverse Matrices 3.5 Solving Systems Using Matrices Unit 4: Quadratic Functions 4.1 Graphing Quadratic Functions 4.2 Solving Quadratic Equations By Factoring 4.3 Solving Quadratic Equations by Finding Square Roots 4.4 Complex Numbers 4.5 Completing The Square 4.6 The Quadratic Formula & The Discriminant 4.7 Graphing & Solving Quadratic Inequalities 4.8 Modeling With Quadratic Functions Unit 5: Polynomials & Polynomial Functions 5.1 Using Properties of Exponents Midterm Timeframe Q3 Unit 5: Polynomials & Polynomial Functions 5.2 Evaluating & Graphing Polynomial Functions 5.3 Adding, Subtracting and Multiplying Polynomials 5.4 Factoring & Solving Polynomial Equations 5.5 The Remainder & Factor Theorems 5.6 Finding Rational Zeros 5.7 Using The Fundamental Theorem of Algebra 5.8 Analyzing Graphs of Polynomial Functions 5.9 Modeling With Polynomial Functions Unit 6: Powers, Roots & Radicals 6.1 nth Roots and Rational Exponents 6.2 Properties of Rational Exponents 6.3 Power Functions and Function Operations 6.4 Inverse Functions 6.5 Graphing Square Root & Cube Root Functions 6.6 Solving Radical Equations Unit 7: Exponential and Logarithmic Functions 7.1 Exponential Growth 7.2 Exponential Decay 7.3 The Number e 7.4 Logarithmic Functions 7.5 Properties of Logarithms 7.6 Solving Exponential & Logarithmic Equations 7.7 Modeling With Exponential & Power Functions 7.8 Logistic Growth Functions Timeframe Q4 Unit 8: Rational Equations & Functions 8.1 Inverse & Joint Variation 8.2 Graphing Simple Rational Functions 8.3 Graphing General Rational Functions 8.4 Multiplying & Dividing Rational Expressions 8.5 Addition, Subtraction and Complex Fractions 8.6 Solving Rational Equations Unit 9: Quadratic Relations & Conic Sections 9.1 The Distance & Midpoint Formula 9.2 Parabola 9.3 Circles 9.4 Ellipses 9.5 Hyperbolas 9.6 Graphing & Classifying Conics 9.7 Solving Quadratic Systems Final