# Polynomials and Rational Functions. Quadratic Equations and Inequalities. Remainder and Factor Theorems. Rational Root Theorem

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1 Pre-Calculus Pre-AP Scope and Sequence - Year at a Glance Pre-Calculus Pre-AP - First Semester Pre-calculus with Limits; Larson/Hostetler Three Weeks 1 st 3 weeks 2 nd 3 weeks 3 rd 3 weeks 4 th 3 weeks 5 th 3 weeks 6 th 3 weeks Topics/ Concepts Rectangular Coordinates Graphs of Equations Linear Equations in Two Variables Functions Parent functions Transformations Inverse Functions Analyzing graphs Linear and Non- Linear Systems of Equation Two-Variable Linear Systems Multivariable Linear Systems Partial Fractions Systems of Inequalities Matrices and Systems of Equations Polynomials and Rational Functions Quadratic Equations and Inequalities Remainder and Factor Theorems Rational Root Theorem Locating Zeros of a Function Rational Equations and Partial Fractions Conics Circle Parabola Ellipse Hyperbola Transformations Systems of Second Degree Equations and Inequalities Tangents and Normals to Conic Sections Exponential and Logarithmic Functions Rational Exponents Number e Common Logarithms Natural Logarithms Exponential Growth Sequences and Series Arithmetic Geometric Infinite Convergent and Divergent Sigma Notation and the nth Term Binomial Theorem Permutations Probability and Odds Radical Equations and Inequalities Combinations Resource: Precalculus with Limits; Larson/Hostetler, Foerster (f) Chapter 1, Appendix A Chapter 16(f) Teks P.1A, P.1B, P.1C, P.1D, P.1E, P.2A, P.2B, P.2C, P.3A Chapter 7, Chapter 8 P.1B, P.2B, P.3A, P.3B, P.3C, Chapter 2 Chapter 16(f) Teks P.1C, P.1D, P.1E, P.3A Chapter 10 Chapter 12 & 13 (f) Teks P.1D, P.2A, P.5A, P.5B, P.5C Chapter 3 Teks P.1A, P.1B, P.1D, P.1E, P.2C, P.3A Chapter 9 Teks P.1B, P.3A, P.3B, P.3C, P.4A, P.4B, P.4C, P.4D

2 Pre-Calculus Pre-AP - Second Semester Pre-calculus with Limits; Larson/Hostetler Three Weeks 1 st 3 weeks 2 nd 3 weeks 3 rd 3 weeks 4 th 3 weeks 5 th 3 weeks 6 th 3 weeks Topics/ Concepts Trigonometric Functions Angles and Their Measure Central Angles and Arcs Circular Functions Special Angles Right Triangles Law of Sines Law of Cosines Graphs and Inverses of Trigonometric Functions Amplitude, Period and Phase Shift Principal Values of Inverses Simple Harmonic Motion Trigonometric Identities and Equations Sum and Difference Identities Double Angle and Half-Angle Identities Solving Trigonometric Equations Normal Form of a Linear Equation Vectors and Parametric Equations Geometric and Algebraic Vectors 3D Space Perpendicular Vectors Applications with Vectors Motion Modeling Polar Coordinates and Complex Numbers Polar Graphs Polar-Rectangular Coordinates Polar Form of Linear Function Polar Form of Complex Numbers Products and Quotients of Complex Numbers Applied Problems Connections to Calculus Limits Derivatives Resource: Precalculus with Limits; Larson/Hostetler Foerster (f) Area of Triangles Chapters 4, 5, 6 Teks P.1C, P.2B, P.3A, P.3C, P.3D, P.3E Chapters 4, 5, 6 Teks P.1A, P.1B, P.1C, P.1D, P.2A, P.2B, P.2C, P.3A, P.3B, P.3C Distance form a Point to a Line Chapters 4, 5, 6 Teks P.2C, P.3A, P.3C, P.3D Chapter 6 Teks P.5A, P.5C, P.5D, P.6A, P.6B Powers and Roots of Complex Numbers Chapter 10 Teks P.1A, P.1B, P.2A, P.3D Chapter 12 Chapter 16(f) *Teks b Calculus AB * These Teks, b Calculus AB, have been included for cross reference due to the lack of in-depth explanation with-in the Pre- Calculus Teks.

3 Functions and Their Graphs (1st 3 weeks) Essential Learning Outcomes Identify functions by definition and graph and state their domain and range Discuss continuity; determine intervals of increase, decrease or constant; describe end behavior Find the sum, difference, product, and quotient of two functions Find the composition of two functions given their equations and be able to determine functions given their composition. Be able to reflect graphs of functions over x-axis, y-axis and line y = x. Be able to test a function for symmetry and determine if it is odd, even or neither algebraically and from the graph. Given the graph of y = f(x), be able to graph a f(x), f(x) + d, f(x - c), f(bx), f( x ), and f( x ). Given a graph that has a transformation, write the equation for the graph. (Optional to address this in this unit) Determine if a function is periodic and, if so, give its period and amplitude and graph it using shifts and stretches. Find the inverse of a function and, given two functions, determine if they are inverses of each other algebraically and graphically. Solve application problems using composite functions. TEKS P.1A, P.1B, P.1C, P.1D, P.1E, P.2A, P.2B, P.2C, P.3A Topics (not in sequential order) Polynomials and Rational Functions Quadratic Equations and Inequalities Remainder and Factor Theorems Rational Root Theorem Locating Zeros of a Function Rational Equations and Partial Fractions Radical Equations and Inequalities Suggested Resources Chapter 1, Appendix A Assessments TAKS 1,2,3,4

4 Conics (2 nd 3 weeks) Essential Learning Outcomes solve 2 X 2 systems linear combination substitution graphing (with a graphing calculator) intersecting lines parallel lines lines that coincide table (graphing calculator) matrices define a matrix and use a matrix to represent data inverse matrices by hand (2X2 only) inverse matrices with calculator use operations of matrices to solve problems systems of inequalities graphing calculator graph by hand Test a point Solve 3 X 3 systems Matrices (calculator) Perform matrix multiplication to show multiplication is not commutative By hand (demo only) With technology determine what a solution to a system of equations/inequalities means in relationship to the problem determine if the solution to a system of equations/inequalities is reasonable for given contexts connect algebraic solutions to graphical and tabular solutions TEKS P.1B, P.2B, P.3A, P.3B, P.3C, Topics (not in sequential order) Circle Parabola Ellipse Hyperbola Transformations Systems of Second Degree Equations and Inequalities Tangents and Normals to Conic Sections Suggested Assessments TAKS Resources Chapter 7, Chapter 8 1,2,3,4,5,6,7

5 Polynomials and Rational Functions (3 rd 3 weeks) Essential Learning Outcomes TEKS Topics (not in sequential order) Identify and classify polynomials by degree Find zeros (roots) and factors of polynomial functions (equations) Use synthetic substitution/division Apply Theorems: Remainder, Factor, Rational Root, Complex Conjugate, and Fundamental Theorem of Algebra Graph a polynomial function given a factorable equation Write the equation for a polynomial function given its graph or a set of data Find maximum and minimum values with or without a graphing calculator and apply to situations (area, volume, cost, distance, etc.) Solve and graph (on a number line) polynomial inequalities with answer in interval notation Sketch the graph of a polynomial function using properties of end behavior, zeros, and multiplicity of zeros P.1C, P.1D, P.1E, P.3A Polynomials and Rational Functions Quadratic Equations and Inequalities Remainder and Factor Theorems Rational Root Theorem Locating Zeros of a Function Rational Equations and Partial Fractions Suggested Assessments TAKS Resources Chapter 2 1,2,3,4,5 Graph rational functions, including those containing more than one vertical asymptote, an oblique asymptote, removable discontinuities, and a crossover of the horizontal or oblique asymptote by locating the asymptotes, holes, and intercepts as transformations of one of the 1 1 parent functions f ( x) = or f ( x) =. 2 x x Manipulate the function rule algebraically to obtain graphing form. x (Example: Use long division to rewrite the as 1+ x + 2 x + 2 Describe the continuity and limits of the function. State the domain and range of the function and write the equations of its asymptotes. Write a rational function given the graph, or description of characteristics such as a verbal description of the transformations of the parent function, the asymptote equations, or function notation showing the transformations. Make connections between the graphical, tabular, verbal, and symbolic representations of the function. Describe the symmetry of the function by identifying it as odd or even. Radical Equations and Inequalities

6 Conics (4th 3 weeks) Essential Learning Outcomes Know the geometric and algebraic descriptions of circle, parabola, hyperbola, and ellipse Represent conic sections in standard form (graphing form) and in parametric form and graph Determine the equation of a conic given the graph Determine the equation and graph of a conic from given information Apply properties of conic sections to solve real-life problems (both as parametric and rectangular) orbits of planets LORAN Convert from general to standard form Describe how properties of conics are used in the physical world, such as reflective properties (flashlight, whispering room, lithotripsy. etc.) elliptipool cooling towers for nuclear reactors (optional) graphs of rotated conics TEKS P.5A, P.5C, P.5D, P.6A, P.6B Topics (not in sequential order) Conics Circle Parabola Ellipse Hyperbola Transformations Systems of Second Degree Equations and Inequalities Tangents and Normals to Conic Sections Suggested Resources Chapter 10, Chapter 12 & 13(f) Assessments TAKS 1,2,3,4,5,6,10

7 Exponential and Logarithmic Functions (5th 3 weeks) Essential Learning Outcomes TEKS Topics (not in sequential order) Suggested Resources Assessments TAKS Know the definition of an exponential function, identify restrictions and graph Define the logarithmic function as the inverse of an exponential function and graph Know and apply laws of integral and rational exponents Apply transformations to exponential and logarithmic functions Know restrictions on the log function Identify intervals where graph of an exponential function is increasing or decreasing, as well as its domain, range, and zeros Use exponential growth and decay formulas to solve application problems: 0 ( 1 t A= A + r) A = A b k 0 t nt 0 1 r A= A + n A= A e 0 rt Use rational exponents to simplify expressions and solve equations Write an exponential function from a set of data Know the definition of e and the function f(x) = e x and use it in application problem. P.1A, P.1B, P.1D, P.1E, P.2C, P.3A Exponential and Logarithmic Functions Rational Exponents Number e Common Logarithms Natural Logarithms Exponential Growth Chapter 3 1,2,3,4,5,8

8 Sequences and Series (6th 3 weeks) Essential Learning Outcomes TEKS Topics (not in sequential order) Suggested Resources Assessments TAKS Identify a sequence as arithmetic (linear), geometric (exponential when r > 0), or other Define a sequence both explicitly and recursively Explore representations of a sequence on a graph and table Use sequences defined recursively to solve problems Find the sum of the first n terms and the value of a specific term of an arithmetic and geometric series Find or estimate the limit of an infinite sequence or determine that the limit does not exist Find the sum of an infinite geometric series when possible Use sigma notation to represent series Expand a binomial of the form ( a b) n + and find the nth term of a binomial expansion Use sequences and series to solve real-life problems P.1B, P.3A, P.3B, P.3C, P.4A, P.4B, P.4C, P.4D Sequences and Series Arithmetic Geometric Infinite Convergent and Divergent Sigma Notation and the nth Term Binomial Theorem Permutations Probability and Odds Combinations Chapter 9 1,2,3,4,5,8

9 Trigonometric Functions (1 st 3 weeks, 2 nd Semester) Essential Learning Outcomes Unit Circle: know definition of radians, degree measure, standard position, coterminal angles, quadrantal angles, coordinates, and the six trig functions Understand the concept of radian measure as a location on the Unit Circle) Define sine, cosine and tangent in terms of x, y and r Convert between radian measure and degree measure of special angles (multiples π π π π,,, ) Know the sine, cosine and tangent values of special angles (multiples π, π, π, π ) Find the values of cotangent, secant, and cosecant from sine, cosine, and tangent Find the reference angle of any given angle Recognize equivalent fractions created by rationalized denominators Find arc length and area of a sector (may be taught in another unit) Solve right triangles using sin, cos, and tan Find area of triangles Use Law of Sines and Law of Cosines Solve application problems involving navigation and surveying, including angles of elevation and depression, area of parallelogram using area of triangle. TEKS P.1C, P.2B, P.3A, P.3C, P.3D, P.3E Topics (not in sequential order) Suggested Resources Assessments TAKS Trigonometric Functions Chapter 4, 5, 6 1,2,3,4 Angles and Their Measure Central Angles and Arcs Circular Functions Special Angles Right Triangles Law of Sines Law of Cosines Area of Triangles

10 Graphs and Inverses of Trigonometric Functions (2nd 3 weeks, 2 nd Semester) Essential Learning Outcomes TEKS Topics (not in sequential order) Graphs and Inverses of Graph all six trig functions and identify their Trigonometric Functions period, amplitude, domain, range and zeros Graph all six trig functions with parameter changes (i.e. y = a sin b(x c) + d; f( x ), and f( x ) are optional) Write/identify the equation of a trig function from its graph State whether a trig function graph is odd or even Investigate continuity, end behavior, vertical asymptotes, increasing/decreasing (especially tan, cot, sec, and csc) Model real-life data using sine and cosine functions with and without a regression equation Define and graph principle trig functions (1 to 1 functions) and their inverses (sine, cosine, and tangent). Make connections between graphs of inverse trig functions and the Unit Circle). Find values of inverse trig functions with and without the calculator (sine, cosine, and tangent) Sketch the graphs of the functions Sin -1, Cos - 1, Tan -1 and state the domain and range. (Emphasize the range of inverse sine and tangent) P.1A, P.1B, P.1C, P.1D, P.2A, P.2B, P.2C, P.3A, P.3B, P.3C Amplitude, Period and Phase Shift Principal Values of Inverses Simple Harmonic Motion Suggested Resources Assessments TAKS Chapter 4, 5, 6 1,2,3,4,5

11 Trigonometric Identities and Equations (3 rd 3 weeks, 2 nd Semester) Essential Learning Outcomes TEKS Topics (not in sequential order) Suggested Resources Assessments TAKS Know and apply the following identities: P.2C, P.3A, P.3C, Trigonometric Identities and Equations Sum and Difference Identities Chapter 4, 5, 6 1,2,3,4,5,6 Pythagorean P.3D Reciprocal Quotient Negative angle Cofunctions Sum and difference Double angle Double Angle and Half-Angle Identities Solving Trigonometric Equations Normal Form of a Linear Equation Distance form a Point to a Line Half angle (optional) Prove trig identities Simplify and expand trig expressions Solve trig equations

12 Vectors and Parametric Equations (4th 3 weeks, 2 nd Semester) Essential Learning Outcomes TEKS Topics (not in sequential order) Suggested Resources Assessments TAKS Using geometric and algebraic representation of vectors, add, subtract, and multiply by a scalar and find the magnitude (length) and direction Show, analyze, and solve navigation and force problems using vectors and trig Use vectors, vector equations, and parametric equations to show an object s motion. Use polar coordinates to plot points on a graph Determine if two vectors are parallel or perpendicular using dot product Calculate the angle between two vectors Work with 3-dimensional vectors to find their magnitude and midpoint Find equations of planes parallel and perpendicular to another plane Write complex numbers and find products in polar form Find powers and roots of complex numbers Convert polar coordinates to rectangular coordinates and vice versa P.5A, P.5C, P.5D, P.6A, P.6B Vectors and Parametric Equations Geometric and Algebraic Vectors 3D Space Perpendicular Vectors Applications with Vectors Motion Modeling Chapter 6 1,2,3,4,5,6

13 Polar Coordinates and Complex Numbers (5th 3 weeks, 2 nd Semester) Essential Learning Outcomes TEKS Topics (not in sequential order) Suggested Resources Assessments TAKS Determine if two vectors are parallel or perpendicular using dot product Calculate the angle between two vectors Work with 3-dimensional vectors to find their magnitude and midpoint Find equations of planes parallel and perpendicular to another plane Write complex numbers and find products in polar form Find powers and roots of complex numbers Convert polar coordinates to rectangular coordinates and vice versa P.1A, P.1B, P.2A, P.3D Polar Coordinates and Complex Numbers Polar Graphs Polar-Rectangular Coordinates Polar Form of Linear Function Polar Form of Complex Numbers Products and Quotients of Complex Numbers Powers and Roots of Complex Numbers Chapter 10 2, 5

14 Limits and an introduction to Calculus (16th 3 weeks, 2 nd Semester) Essential Learning Outcomes TEKS Topics (not in sequential order) Suggested Resources Assessments TAKS The student will Estimate limits using graphical and numerical approaches Learn different ways a limit can fail to exist Evaluate limits using properties of limits Develop and use a strategy for finding limits Evaluate a limit using division and rationalization techniques Use properties of continuity Determine infinite limits from the left and right Find and sketch the vertical asymptotes of the graph of a function Use limits to justify vertical asymptotes Find limits as x approaches infinity Find and sketch the horizontal asymptotes of the graph of a function Find the slope of the tangent line to a curve at a point Use the limit definition to find the derivative of a function Understand the relationship between differentiability and continuity Find the derivative of a function using the basic differentiation rules All *111.54b Calculus AB Applied Problems Connections to Calculus Limits Derivatives Chapter 12 Chapter 16(f) N/A

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