Pre-Calculus At a Glance Approximate Beginning Date Test Date 1st 9 Weeks 2nd 9 Weeks 3rd 9 Weeks 4th 9 Weeks

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1 Pre-Calculus Curriculum Guide Page Pre-Calculus At a Glance Approximate Beginning Date Test Date 1 st 9 Weeks August 24 th September 11 th Ch 3: Systems of Equations and Inequalities September 14 th September 28 th Ch 4: Quadratic Functions and Relations September 29 th October 13 th Ch 5: Polynomials and Polynomial Functions 2 nd 9 Weeks Ch 6: Inverses and Radical Functions and Relations Ch 7: Exponential and Logarithmic Functions and Relations Ch 8: Rational Functions and Relations December 7 th, begin review for Fall Exam, Semester ends on December 18 th 3 rd 9 Weeks Ch 1: Functions from a Calculus Perspective Ch 2: Power, Polynomial, and Rational Functions Ch 3: Exponential and Logarithmic Functions Ch 4: Trigonometric Functions 4 th 9 Weeks The beginning of the 4 th 9 weeks will be used as time to complete concepts if we get behind in the other 9 weeks. Ch 7: Conic Sections and Parametric Equations Ch 10: Sequences and Series Ch 8: Vectors * Chapter 10 can be covered during Algebra 2 or Pre-calculus. If covered during Algebra 2, we will do a short review of these concepts in Pre-calculus. If not covered in Algebra 2, we will cover them in depth in Pre-calculus. During this course students will learn the properties, algebra, graphs, and language of functions. Students will also gain an understanding of vectors, sequences and series, and probability and statistics. This course is designed to build on Algebra 1, Geometry, and

2 Pre-Calculus Curriculum Guide Page 2 Algebra 2 concepts and apply these concepts to the real-world. Included in the following information are the key concepts students will learn throughout this course and the resources, materials, and activities that will enhance this learning experience. Also included is the standard, both Common Core Standards and TEKS. There is also an assessment stem for both low, medium, and high level questions. To show complete mastery a student should be able to answer all three levels of questioning. *This scope and sequence will only apply to the Pre-Calculus class Ongoing Unit and Standard Key Understanding Resources, Materials, and Activities N-Q 1: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. N-Q 2: Define appropriate quantities for the purpose of descriptive modeling. N-Q 3: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. -use units to aid in solving word problems and in creating and reading graphs/data displays -convert units when appropriate -solve word problems keeping in mind appropriate accuracy, units, etc. -determine the accuracy and unit necessary when answering word problems Assessment Stem R: Precalculus textbook L: using units in solutions when conversions are not necessary M/H: use conversions in order to determine a solution and understand when units are squared (area) or cubed (volume) R: Precalculus textbook L/M/H: understand the importance of units and appropriate rounding in solutions R: Precalculus textbook L/M/H: understand the importance of units and appropriate rounding in solutions 1 st 9 weeks Unit 1, Ch 3: Systems of Equations and Inequalities Wed, Aug 19 1 st day of school: class Key Understanding Resources, Materials, and Activities M: class structure page, Assessment Stem

3 Pre-Calculus Curriculum Guide Page 3 Wed, Aug 19- Fri, Aug 21 structure and calculator check out Review chapters 1 and 2 from Algebra 2 the summer math packet covered these chapters Mon, Aug 24 A-CED 3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context A-REI 11: Explain why the x- coordinates of the points where the graphs of the equations y- f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. A-REI 5: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the Solving Systems of Equations: -using algebra, graphs and tables -classify systems as consistent or inconsistent, and dependent or independent calculator check out sheet 3-1 L: Solve the system of equations by using a table, graphing substitution or elimination. M: Graph the system of equations and classify. H: From a word problem, write the system of equations and solve.

4 Pre-Calculus Curriculum Guide Page 4 other produces a system with the same solutions. A-REI 6: Solve systems of linear equations exactly and approximately (with graphs), focusing on pairs of linear equations in two variables. Tues, Aug 25 A-CED 3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context A-REI 12: Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Wed, Aug 26 A-CED 3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context Solving Systems of Inequalities by Graphing: -graph to find intersecting or separate regions -write a system and solve -find vertices of an enclosed region Optimization of Linear Programming: -find minimum and maximum values in bounded and unbounded regions -write a system of inequalities to solve an L: Solve the system of inequalities by graphing. M: Solve the absolute value system of inequalities by graphing. H: From the word problem, write the system of inequalities and solve. L: Graph the system of inequalities. Name the coordinates of the vertices and find the maximum and minimum values. M:From the word problem, write the system of inequalities. Name the coordinates of the vertices and find the maximum and minimum values. H: From the word problem, write the

5 Pre-Calculus Curriculum Guide Page 5 3A. analyze situations and formulate systems of equations in two or more unknowns or inequalities in two unknowns to solve problems Thurs, Aug 27 A-CED 3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context Review Systems on Fri, Aug 28 and Mon, Aug 31 Test on Tues, Sept 1 Wed, Sept 2 N-VM 6: Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. N-VM 7: Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. optimization problem Systems of Equations in Three Variables: -discuss the z plane -solve systems with one solution, no solution, and infinitely many solutions -write a system and solve Operations with Matrices: -analyze data -dimensions -add and subtract -multiply by a scalar -multi-step operations system of inequalities. Graph the system and make predictions from the data. L: Solve systems with the same format. M: Solve systems with various formats. H: Write systems from word problems and solve. L: Add, subtract, or multiply by a scalar the given matrices. M: Multiply matrices and perform multistep operations. H: Find the missing values in a matrix equation. N-VM 8: Add, subtract, and multiply matrices of appropriate dimensions. Thurs, Sept 3 N-VM 8: Add, subtract, and multiply matrices of Multiplying Matrices -dimensions 3-6 L: Multiply 2x2 matrices. M: Multiply matrices of varying

6 Pre-Calculus Curriculum Guide Page 6 Fri, Sept 4 appropriate dimensions. N-VM 9: Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. A-CED 3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context Tues, Sept 8 A-CED 3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context N-VM 10: Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. -multiply if possible -using the distributive property Solving Systems of Equations Using Cramer's Rule: -find determinants -use determinants to find the area -solve systems with Cramer's Rule Solving Systems of Equations Using Inverse Matrices: -identity matrix -verify and find inverses -use inverses to solve a system dimensions. H: Use matrices to determine if equations are true. L: Find the determinant. M: Use matrices to solve the systems of equations involving two equations. H: Use matrices to solve the systems of equations involving three equations. L: Determine if two matrices are inverses. M: Use matrices to solve the systems of equations involving three equations. H: Describe what a matrix equation with infinite solutions would look like.

7 Pre-Calculus Curriculum Guide Page 7 A-REI 9: Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3x3 or greater) Review Matrices on Wed, Sept 9 and Thurs, Sept 10 Test on Fri, Sept 11 1 st 9 weeks Unit 2, Ch 4: Quadratic Functions and Relations Mon, Sept 14 A-SSE 1: Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. F-IF 9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. F-IF 4:For a function that models a relationship between two quantities, interpret key Key Understanding Resources, Materials, and Activities Graphing Quadratic Functions: -graph using a table -determine the vertex, axis of symmetry, and maximum or minimum 4-1 Assessment Stem L: Graph the quadratic function. Find the coordinates of the vertex, the axis of symmetry, and the maximum or minimum. M: Determine if the function would have a max or min without graphing. H: Using the coordinates given in the table, and without graphing, find the vertex and maximum or minimum of the function.

8 Pre-Calculus Curriculum Guide Page 8 features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F-IF 7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. Tues, Sept 15 A-CED 2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. F-IF 4:For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and Solving Quadratic Equations by Graphing: -equations vs functions -graph and determine the roots (two real, one real, no real solutions) -solve with a calculator 4-2 L: Use the given graph and state the functions roots. M: Graph the function and find the roots. H: Interpret word problems involving quadratic functions.

9 Pre-Calculus Curriculum Guide Page 9 sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. A-REI 7: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y=-3x and the circle x 2 +y 2 =3. A-REI 10: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line) A-REI 11: Explain why the x- coordinates of the points where the graphs of the equations y- f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately, e.g., using

10 Pre-Calculus Curriculum Guide Page 10 technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Wed, Sept 16 8D. solve quadratic equations and inequalities using graphs, tables, and algebraic methods A-SSE 2: Use the structure of an expression to identify ways to rewrite it. F-IF 8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Solving Quadratic Equations by Factoring: -write equations by foiling -factor using: GCF, perfect squares, difference of squares, and with trinomials -solve by factoring 4-3 L: Factor each polynomial. M: Solve each equation. H: Factor complex polynomials. Ex: 18a-24ay+48b-64by A-SSE 3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

11 Pre-Calculus Curriculum Guide Page 11 a. Factor a quadratic expression to reveal the zeros of the function it defines. A-REI 4: Solve quadratic equations in one variable. b. Solve quadratic equations by inspection (e.g., for x 2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Thurs, Sept 17 A-REI 7: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y=-3x and the circle x 2 +y 2 =3. N-CN 1: Know there is a complex number i such that i 2 = -1, and every complex number has the form a+bi with a and b real. N-CN 2: Use the relation i 2 = -1 and the commutative, associative, and distributive Complex Numbers: -define imaginary numbers -find the square root of negative numbers -multiply imaginary numbers -solve equations with imaginary solutions 4-4 L: Simplify the complex expression. M: Solve complex equations. H: Find the values of x and y that make each equation true. Ex: 2x+7+(3-y)i=-4+6i

12 Pre-Calculus Curriculum Guide Page 12 properties to add, subtract, and multiply complex numbers. N-CN 3: Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. -add, subtract, multiply, and divide complex numbers N-CN 8: Extend polynomial identities to the complex numbers. For example, rewrite x 2 +4 as (x+2i)(x-2i) Fri, Sept 18 N-CN 7: Solve quadratic equations with real coefficients that have complex solutions. F-IF 8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Completing the Square: -solve equations with rational and irrational roots -solve equations by completing the square, including equations with imaginary solutions 4-5 L: Solve the equation by completing the square. M: Solve more complex equations by completing the square. H: Find the value of c that makes each trinomial a perfect square. A-SSE 3: Choose and produce and equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

13 Pre-Calculus Curriculum Guide Page 13 b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. A-REI 4: Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x-p) 2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for x 2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Mon, Sept 21 N-CN 7: Solve quadratic equations with real coefficients that have complex solutions. A-SSE 1: Interpret expressions that represent a quantity in terms of its context. b. Interpret complicated expressions by viewing one or The Quadratic Formula and the Discriminant: -use the quadratic formula to solve quadratic equations with one, two, irrational, and complex roots 4-6 L: Solve the equation by using the quadratic formula. Find the value of the discriminant. M: Using the discriminant, describe the number and type of roots. Verify using the quadratic formula. H: Use the quadratic formula to answer word problems.

14 Pre-Calculus Curriculum Guide Page 14 more of their parts as a single entity. Ex: interpret P(1+r) n as the product of P and a factor not depending on P. -define discriminant and use it to find the number and type of roots A-REI 4: Solve quadratic equations in one variable. b. Solve quadratic equations by inspection (e.g., for x 2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Tues, Sept 22 F-IF 8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. F-BF 3: Identify the effect on the graph of replacing f(x) by f(x)+k, kf(x), f(kx), and f(x+k) for specific values of k (both Transformations of Quadratic Graphs: -write quadratic functions in vertex form -graph when in vertex form -write the equation from the graph 4-7 L: Graph an equation when in vertex form. M: Write a function in vertex form. H: Write the equation given the graph.

15 Pre-Calculus Curriculum Guide Page 15 positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Wed, Sept 23 A-CED 1: Create equations and inequalities in one variable and use them to solve problems. A-CED 3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context Review on Thurs, Sept 24 and Fri, Sept 25 Test on Mon, Sept 28 1 st 9 weeks Unit 3, Ch 5: Polynomials and Polynomial Functions Tues, Sept 29 A-APR 1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply Quadratic Inequalities: -graph an inequality -solve inequalities by graphing and algebraically 4-8 Key Understanding Resources, Materials, and Activities Operations with Polynomials: -properties of exponents -simplify expressions -degree of a polynomial 5-1 L: Solve the inequality algebraically and graphically. M: Write the quadratic inequality given the graph. H: Graph the intersection of two quadratic inequalities. Assessment Stem L: State if the expression is a polynomial and find the degree. M: Simplify the expression. H: Simplify more complex expressions and solve word problems.

16 Pre-Calculus Curriculum Guide Page 16 polynomials. Wed, Sept 30 A-APR 6: Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for for more complicated examples, a computer algebra system. Thurs, Oct 1 F-IF 4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. -multiplying polynomials Dividing Polynomials -divide polynomials by monomials -long division -synthetic division Polynomial Functions: -degrees and leading coefficients -evaluate polynomial functions -find function values -graph functions -describe the end behavior, odd or even degree and number of zeros L: Divide polynomials by monomials. M: Divide polynomials using long division and synthetic division. H: Analyze the division problems and determine the relationship between the degrees of the dividend, the divisor, and the quotient. L: Evaluate the function. Graph the function using a table. M: Use the degree and end behavior of the graph to match the polynomial equation to the correct graph. H: Use polynomial equations and graphs to solve word problems. L/M/H: Use the calculator to graph and solve polynomial equations. F-IF 7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and

17 Pre-Calculus Curriculum Guide Page 17 using technology for more complicated cases. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. A-APR 4: Prove polynomial identities and use them to describe numerical relationships. Ex: the polynomial identify (x 2 +y 2 ) 2 =(x 2 -y 2 ) 2 +(2xy) 2 can be used to generate Pythagorean triples. [Ex: Sum and Difference of Cubes: a 3 +/- b 3 = (a+/- b)(a 2 +/- ab + b 2 ] A-REI 10: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Fri, Oct 2 F-IF 4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in Analyzing Graphs of Polynomial Functions: -graph functions -identify zeros, 5-4 L: Graph the function using a table. M: Identify zeros, maximums and minimums of a graph. H: Use polynomial equations and graphs

18 Pre-Calculus Curriculum Guide Page 18 terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F-IF 7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. maximums and/or minimums to solve word problems. A-APR 4: Prove polynomial identities and use them to describe numerical relationships. Ex: the polynomial identify (x 2 +y 2 ) 2 =(x 2 -y 2 ) 2 +(2xy) 2 can be used to generate Pythagorean triples. [Ex: Sum and Difference of Cubes: a 3 +/- b 3 = (a+/- b)(a 2 +/- ab + b 2 ] Mon, Oct 5 A-CED 1: Create equations Solving Polynomial L: Solve by factoring.

19 Pre-Calculus Curriculum Guide Page 19 Tues, Oct 6 and inequalities in one variable and use them to solve problems. A-REI 11: Explain why the x- coordinates of the points where the graphs of the equations y- f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. A-APR 2: Know and apply the Remainder Theorem. F-IF 7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Equations: -factor: sum and difference of cubes, by grouping, difference of squares -solve functions by factoring -solve using quadratic form The Remainder and Factor Theorems: -Remainder Theorem; use synthetic substitution to find function values -Factor Theorem; use to find all factors of a polynomial 5-5 M: Solve using quadratic form. H: Factor complex polynomial equations with variables in the exponents. Ex: 36x 2n +12x n L: Use synthetic substitution for functions. M: Given a polynomial and one of its factors, find the remaining factors of the polynomial. H: Find the solutions of the polynomial function. Wed, Oct 7 A-APR 3: Identify zeros of Roots and Zeros: L: State the number and type of roots.

20 Pre-Calculus Curriculum Guide Page 20 polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. N-CN 9: Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Thurs, Oct 8 A-APR 3: Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. -use the Fundamental Theorem of Algebra to determine number and type of roots -use Descartes' Rule of Signs to find types of zeros -use synthetic substitution to find zeros -use the Complex Conjugates Theorem to write functions Rational Zero Theorem: -use the rational zero theorem to find zeros of a polynomial function 5-7 M: Using what you know about roots, solve the word problem. H: Write the equation of the polynomial function when given specific criteria. Ex:Write an equation of a polynomial function of degree 5 with 2 imaginary zeros, 1 nonintegral zero, and 2 irrational zeros. 5-8 L: Find all the zeros of the function. M/H: Use what you know about zeros to solve word problems. Review on Fri, Oct 9 and Mon Oct 12 Test on Tues, Oct 13 2 nd 9 weeks Unit 4, Ch 6: Inverses and Radical Functions and Relations Key Understanding Resources, Materials, and Activities Assessment Stem F-IF 9: Compare properties of Operations on L: Add, subtract, multiply, or divide

21 Pre-Calculus Curriculum Guide Page 21 two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Functions: -add, subtract, multiply and divide functions -compose functions -identify domain and range 6-1 functions. M: Perform a composition of functions. State the domain and range of the composed function. H: Perform a composition of functions when the equations contain square roots. F-BF 1: Write a function that describes a relationship between two quantities. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. c. Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the geight of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. F-IF 4: For a function that models a relationship between two quantities, interpret key Inverse Functions and Relations: -find and graph an 6-2 L: Determine if the pair of functions are inverse functions. M: Determine if the inverse of the

22 Pre-Calculus Curriculum Guide Page 22 features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. inverse -verify inverses function is also a function. H: Find an example of a function that is its own inverse. F-BF 4: Find inverse functions. a. Solve an equation of the form f(x)=c for a simple function f that has an inverse and write an expression for the inverse. Ex: f(x)=2x 3 or f(x)=(x+1)/(x-1) for x 1. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. Produce an invertible function from a non-invertible function by restricting the domain. F-IF 7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and Square Root Functions and Inequalities: -parent function -graph functions 6-3 L: Graph square root functions. M: Identify domain and range from a graph. H: Identify domain and range from a

23 Pre-Calculus Curriculum Guide Page 23 using technology for more complicated cases. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. -transformations -identify domain and range -graph inequalities function without graphing. F-BF 3: Identify the effect on the graph of replacing f(x) by f(x)+k, kf(x), f(kx), and f(x+k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. A-REI 10: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). A-SSE 2: Use the structure of an expression to identify ways to rewrite it. Nth Roots: -parts of a radical -find and simplify roots, include using absolute value 6-4 L: Use a calculator to approximate the expression. M: Simplify expressions H: Explain when and why absolute value symbols are needed when taking an nth root.

24 Pre-Calculus Curriculum Guide Page 24 A-SSE 2: Use the structure of an expression to identify ways to rewrite it. Operations with Radical Expressions: -simplify expressions -add, subtract, and multiply radicals -rationalize denominators 6-5 L: Simplify the expression. M:Simplify expressions with addition, subtraction, and multiplication H: Simplify expressions with division. N-RN 1: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. Ex: 5 1/3 = the cube root of 5 Rational Exponents: -radical and exponential form -evaluate expressions -solve equations 6-6 L: Convert expressions between radical and exponential form. M: Evaluate expressions. H: Simplify expressions. N-RN 2: Rewrite expressions involving radicals and rational exponents using the properties of exponents. A-REI 2: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 2 nd 9 weeks Unit 5, Ch 7: Exponential and Logarithmic Functions and Relations F-IF 7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and Solving Radical Equations and Inequalities: -solve various types of radical equations and inequalities 6-7 Key Understanding Resources, Materials, and Activities Graphing Exponential Functions: -parent functions -graph exponential 7-1 L: Solve equations with square roots M: Solve equations with rational exponents. H: Solve radical inequalities. Assessment Stem L: Graph exponential functions. M: State the domain and range. Identify transformations in the function. H: Write exponential growth and decay

25 Pre-Calculus Curriculum Guide Page 25 using technology for more complicated cases. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude growth and decay functions functions from word problems. F-IF 8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions and classify them as representing exponential growth or decay. A-REI 10: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). A-REI 11: Explain why the x- coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are

26 Pre-Calculus Curriculum Guide Page 26 the solutions of the equation f(x)=g(x); find the solutions approximately, ex: using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. F-BF 3: Identify the effect on the graph of replacing f(x) by f(x)+k, kf(x), f(kx), and f(x+k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. F-LE 1: Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal

27 Pre-Calculus Curriculum Guide Page 27 factors over equal intervals. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. F-LE 2: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two inputoutput pairs (including reading these from a table). F-LE 3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. F-LE 5: Interpret the parameters in a linear or exponential function in terms of a context. A-CED 1: Create equations and inequalities in one variable and use them to solve problems. F-LE 4: For exponential Solving Exponential Equations and Inequalities: -solve exponential equations -write exponential 7-2 L: Solve the exponential equation. M: Solve word problems using exponential equations. H: Solve complex exponential equations. Ex: Solve for x: =4 x

28 Pre-Calculus Curriculum Guide Page 28 models, express as a logarithm the solution to ab ct =d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. F-IF 7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude F-BF 3: Identify the effect on the graph of replacing f(x) by f(x)+k, kf(x), f(kx), and f(x+k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. A-REI 10: Understand that the graph of an equation in two functions -compound interest -solve exponential inequalities Logarithms and Logarithmic Functions: -logarithmic and exponential form -evaluate logarithmic expressions -parent function -graph logarithmic functions -find inverses of exponential functions 7-3 L: Write each equation in exponential or logarithmic form. Graph functions. M: Evaluate logarithmic expressions. H: Consider y=logbx in which b, x, and y are real numbers. Zero can be in the domain sometimes, always, or never. Justify your answer.

29 Pre-Calculus Curriculum Guide Page 29 variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). A-SSE 2: Use the structure of an expression to identify ways to rewrite it. A-CED 1: Create equations and inequalities in one variable and use them to solve problems. A-CED 1: Create equations and inequalities in one variable and use them to solve problems. A-CED 1: Create equations and inequalities in one variable and use them to solve problems. A-REI 11: Explain why the x- coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately, ex: using Solving Logarithmic Equations and Inequalities: -solve various types of logarithmic equations Properties of Logarithms: -use the product, quotient, and power property to approximate logarithms -use properties to solve equations Common Logarithms -find common logarithms -solve logarithmic equations -solve exponential equations and inequalities using logarithms -change of base formula L: Solve logarithmic equations with the variable on one side. M/H: Solve equations with variables on each side. L/M: Use given logarithmic approximations to evaluate logarithmic expressions. H: Solve logarithmic equations. L: Use a calculator to evaluate the log expression. M: Use the change of base formula. H: Solve log equations and inequalities.

30 Pre-Calculus Curriculum Guide Page 30 technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. A-SSE 2: Use the structure of an expression to identify ways to rewrite it. Base e and Natural Logarithms: -exponential and natural logarithmic form -simply expressions -solve equations and inequalities -continuously compounded interest 7-7 L: Write an equivalent exponential or log functions. M: Write expressions as a single logarithm. Solve equations. H: Solve word problems using logarithmic equations or inequalities, and base e equations. F-IF 8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions and classify them as representing exponential growth or decay. Using Exponential and Logarithmic Functions: -exponential growth and decay -carbon dating -continuous growth -logistic growth 7-8 L/M/H: Solve word problems of varying difficulty using exponential and logarithmic functions. F-LE 4: For exponential models, express as a logarithm

31 Pre-Calculus Curriculum Guide Page 31 the solution to ab ct =d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. A-SSE 3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. c. Use the properties of exponents to transform expressions for exponential functions F-BF 1: Write a function that describes a relationship between two quantities. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. F-BF 5: Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and

32 Pre-Calculus Curriculum Guide Page 32 exponents 2 nd 9 weeks Unit 6, Ch 8: Rational Functions and Relations A-APR 7: Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. A-APR 7: Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. A-CED 2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and Key Understanding Resources, Materials, and Activities Multiplying and Dividing Rational Expressions: -simplify rational expressions -determine when expressions are undefined -simplify complex fractions Adding and Subtracting Rational Expressions: -find the LCM -add and subtract with monomial and polynomial denominators -simplify complex fractions Graphing Reciprocal Functions: -parent function -limitations on domain -graph functions and Assessment Stem L/M/H: Simplify various levels of rational expressions. L: Find the LCM for the set of polynomials. M: Simplify the complex rational expression. H: Solve rational equations. Ex: Find the expression that make s the statement true: (x-6)/(x+3) x?/(x-6) = x-2 L: Identify the asymptotes, domain, and range given the graph. M: Graph the function. H: Solve word problems.

33 Pre-Calculus Curriculum Guide Page 33 scales. F-BF 3: Identify the effect on the graph of replacing f(x) by f(x)+k, kf(x), f(kx), and f(x+k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. transformations -write equations A-REI 10: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). A-CED 2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. F-IF 9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by Graphing Rational Functions: -vertical and horizontal asymptotes -oblique asymptotes -point discontinuity 8-4 L/M/H: Graph the function. Determine the asymptotes and any points of discontinuity.

34 Pre-Calculus Curriculum Guide Page 34 verbal descriptions). Ex: given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. A-REI 10: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). A-REI 11: Explain why the x- coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately, ex: using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. F-IF 7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more

35 Pre-Calculus Curriculum Guide Page 35 complicated cases. d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. A-CED 2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Variation Functions: -direct, joint, and inverse variation -combined variation 8-5 L: Solve variation expressions. M: Determine the type of variation shown in the relation. H: Solve word problems involving variations. A-REI 10: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). A-CED 1: Create equations and inequalities in one variable and use them to solve problems. A-REI 2: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Solving Rational Equations and Inequalities: -solve rational equations -mixture, distance, and work problems -solve rational inequalities 8-6 L: Solve the rational equation. M: Solve rational inequalities. H: Solve distance and work problems. 3 rd 9 weeks Unit 7, Ch 1: Functions from Key Understanding Resources, Materials, Assessment Stem

36 Pre-Calculus Curriculum Guide Page 36 a Calculus Perspective F-IF 1: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y=f(x). F-IF 2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. -interval notation -identify functions -find function values -find domains algebraically -evaluate piecewisedefined functions and Activities ch 1-1 L: Determine if a graph or relation is a function. Find function values. M: Determine if real-world situations are functions (Ex: Bank account numbers and the account balance). H: Identify functions by the equation. F-IF 5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Ex: if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. A-REI 10: Understand that the graph of an equation in two -estimate function values ch 1-2 L: State the domain, range, intercepts, and symmetry of a graph. State if the

37 Pre-Calculus Curriculum Guide Page 37 variables is the set of all its solutions plotted in the coordinate plane, often forming a curve. F-IF 1: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y=f(x). -find domain and range -find y-intercepts -find zeros (xintercepts, solutions, roots) -symmetry; even and odd functions function is even or odd. M/H: Solve word problems by graphing the function. Determine domain, range, intercepts and symmetry. State the meaning behind each feature. F-IF 4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F-IF 5: Relate the domain of a

38 Pre-Calculus Curriculum Guide Page 38 function to its graph and, where applicable, to the quantitative relationship it describes. Ex: if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. F-IF 9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Ex: given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. F-IF 4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums -continuity -types of discontinuity -identify point of continuity or discontinuity -Intermediate Value Theorem -approximate zeros -end behavior for graphs that approach infinity and approach a specific value ch 1-3 L: Determine if a function is continuous or discontinuous at a given x-value. Describe the end behavior of a graph. M: Given a function, determine the limit as x approaches infinity. H: Solve word problems and determine the relevance of the end behavior.

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