Course Outline for Pre-Calculus Mathematics Debra Cacchillo Saugerties High School Saugerties, NY 12477 845-247-6650 [email protected] College In The High School Course: Ulster County Community College – Full-Year Course Description: This course provides a background in the topics necessary for success in calculus. It is also designed for students who want to be better prepared for Advanced Placement Calculus. The course will include functions, graphing functions, solving higher order equations, system of equations, exponents, logarithms, trigonometry, conic sections, polar equations and their graphs, series, sequences, and limits. General Course Goals: 1. The student will explore functions 2. The student will be prepared to enter Calculus upon completion of this course Prerequisite: Algebra II and Trigonometry or College Algebra (with teacher recommendation) Text Book: Pre-Calculus with Limits, by Cynthia Young (Wiley) Grading: 1. Quarterly grades will be determined by using a total point evaluation that includes: Tests (weekly) Quizzes (1-4 per week) Homework (collected daily) Quarter Exams 2. Final Exam 3. Extra Help will be available every 9th period unless notified in advance (2:15 – 3:00pm) Student Requirements for Completion of the Course: Final grade will be determined by an average of the four quarter grades and the final exam (20% each). Course Outline and Objectives Section Description / Chapter 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Chapter 0 Linear Equation Solve linear equations in one variable Quadratic Equations Solve quadratic equations by factoring Solve quadratic equations by the square root method Solve quadratic equations by completing the square Solve quadratic equations by using the quadratic formula Other Types of Equations Solve rational equations Solve radical equations Solve equations that are in quadratic form Solve absolute equations Inequalities Graph inequalities and use interval notation Solve linear inequalities Solve polynomial inequalities Solve rational inequalities Solve absolute value inequalities Graphing Equations Graph equations in two variable in the Cartesian plane Use intercepts and symmetry as graphing aids Write the equation of a circle Derive and use the distance and midpoint formulas Lines Graph a line Find the slope of a line Write the equation of a line in general form, slope-intercept form, point-slope form or intercept form Find the slopes of parallel and perpendicular lines Modeling Variation Write an equation for a direct variation Write an equation for an indirect variation Find the constant of variation Time (in Days) 1 2 4 4 3 2 1 Section Description / Chapter A.7 Complex Numbers Understand that real numbers and imaginary numbers are subsets of the complex numbers Write radicals with negative radicals as imaginary numbers Add, subtract, multiply, and divide complex numbers Raise complex numbers to a power Time (in Days) 2 Chapter 1 1.1 1.2 1.3 1.4 Function Define a function Evaluate a function for any argument Determine if a relation is a function (vertical line test) Determine if an equation determines a function Use function notation Determine the domain and range of a function Graphs of Functions Recognize and graph common functions (linear, constant, identity, square, cube, square root, cube root, absolute value, and reciprocal functions) Classify functions as even, odd or neither Determine whether functions are increasing, decreasing, or neither Calculate the average rate of change of a function Evaluate the difference quotient for a function graph piecewise-defined functions Graphing Techniques: Transformations Sketch the graph of a function using vertical and horizontal shifts of common functions Sketch the graph of a function by reflecting a common function about the x-axis or y-axis Sketch the graph of a function by stretching or shrinking a common function Sketch the graph of a function using a sequence of transformations Combining Functions Add, subtract, multiply, and divide functions Evaluate composite functions Determine the domain of functions resulting from operations on and composition of functions 3 3 3 2 Section Description / Chapter 1.5 One-to-one Functions and Inverse Functions Determine algebraically and graphically whether a function is a one-to-one function Find the inverse of a function Graph the inverse function given the graph of the function Find the domain and range of inverse functions Time (in Days) 2 Chapter 2 2.1 2.2 2.3 2.4 2.5 2.6 Quadratic Functions Graph a quadratic function in standard form Graph a quadratic function in general form Find the equation of a parabola Solve application problems that involve the quadratic function Polynomial Functions of Higher Order Graph polynomial functions using transformations Identify real zeroes of polynomial functions and their multiplicities Determine the end behavior of a polynomial function Sketch graphs of polynomial functions using intercepts and end behavior Dividing Polynomials Divide polynomials with long division Divide polynomials with synthetic division The Real Zeroes of a Polynomial Function Use the remainder theorem to evaluate a polynomial function Use the factor theorem Use the rational roots theorem to list possible rational roots Use Descartes’ Rule of Signs to determine the possible combination of positive and negative real zeroes (optional) Find the real zeroes of a polynomial function Factor a polynomial function Use the intermediate value theorem to approximate a real zero Complex Zeros: Fundamental Theorem of Algebra Understand the fundamental theorem of algebra guarantees at least one zero Understand why complex zeros occur in conjugate pairs Understand the n zeros theorem Rational Functions Find domain of a rational function Determine the vertical, horizontal, and slant asymptotes of rational functions Graph rational functions 3 3 2 4 1 3 Section Description / Chapter 3.1 3.2 3.3 3.4 3.5 4.1 Chapter 3 Exponential Functions and Their Graphs Evaluate exponential functions Graph exponential functions Find the domain and range of exponential functions Define the number e Solve problems involving applications of the exponential function Logarithmic Functions and Their Graphs Convert exponential expressions to logarithmic expressions Convert logarithmic expressions to exponential expressions Evaluate logarithmic expressions Approximate common and natural logarithms using a calculator Graph logarithmic functions Determine domain restrictions on logarithmic functions Solve problems involving applications of the logarithmic function Properties of Logarithms Understand that logarithmic functions are inverses of exponential functions and derive the properties of logarithms Write a single logarithm as a sum or difference of logarithms Write a logarithmic expression as a single logarithm Evaluate logarithms of a general base (change of base formula) Exponential and Logarithmic Equations Solve exponential equations Solve logarithmic equations Exponential and Logarithmic Models Use exponential growth, exponential decay, logarithmic, logistic growth, and Guassin distribution models to represent real-world problems Chapter 4 Angle Measure Convert angle measures between degrees and radians Find the complement or supplement of an angle Identify coterminal angles Calculate the length of an arc along a circle Calculate the area of a circular sector Time (in Days) 3 3 3 2 2 2 Section Description / Chapter 4.2 4.3 4.4 4.5 5.1 5.2 Right Triangle Trigonometry Learn the trigonometric functions as ratios of sides of a right triangle Evaluate trigonometric functions exactly for special angles Evaluate trigonometric functions using a calculator Solve right triangles Trigonometric Functions of Angles Define trigonometric functions for angles in the Cartesian plane Understand the ranges for trigonometric angles Calculate the trigonometric function values for quadrantal angles Determine the reference angle of an nonacute angle Calculate trigonometric function values for all angles The Law of Sines Classify an oblique triangle as one of four cases Derive the law of sines Understand the ambiguous case can yield no triangle, one triangle, or two triangles Solve triangles using the law of sines The Law of Cosines Solve oblique triangles using the law of cosines Derive the law of cosines Solve triangles using the law of sines and the law of cosines Derive the formula for the area of a triangle Find the area of a triangle Chapter 5 Trigonometric Functions: The unit Circle Approach Draw the unit circle showing the special angles, and label cosine and sine values Determine the domain and range of circular functions Classify trigonometric functions as even or odd Graphs of Sine and Cosine Functions Graph the sine and cosine functions Determine the domain and range of the sine and cosine functions Determine the amplitude and period of sinusoidal functions and graph Determine the phase shift of a sinusoidal function and graph Solve harmonic motion problems Graph sums of functions: addition of ordinates (optional) Time (in Days) 2 2 2 3 2 3 Section Description / Chapter 5.3 6.1 6.2 6.3 6.4 6.5 6.6 Graphs of Other Trigonometric Functions Determine the domain and range of the tangent, cotangent, secant, and cosecant functions Determine the period and graph basic tangent, cotangent, secant, and cosecant functions Graph translated tangent functions Chapter 6 Verifying Trigonometric Identities Verify fundamental identities (reciprocal, quotient, even-odd, and cofunction identities) Simplify trigonometric expressions using identities Verify trigonometric identities Sum and Difference Identities Derive the sum and difference identities for the cosine, sine, and tangent functions Find exact values of trigonometric functions using the sum and difference identities Double-Angle and Half-Angle Identities Derive double-angle identities using the sum of two angle identity Derive half-angle identities using double-angle identities Use double-angle and half-angle identities to find exact values of certain trigonometric functions Use double-angle and half-angle identities to help in verifying identities Product-To-Sum and Sum-To-Product Identities Derive the product-to-sum identities using the sum and difference identities Derive the sum-to-product identities using the product-to-sum identities Express product of trigonometric functions as sums of trigonometric functions and sums of trigonometric functions as products of trigonometric functions Inverse Trigonometric Functions Develop inverse trigonometric functions Find values of inverse trigonometric functions Graph inverse trigonometric functions Trigonometric Equations Solve trigonometric equations by inspection, using algebraic techniques, using the inverse functions, and using trigonometric identities Time (in Days) 2 3 2 2 1 2 3 Section Description / Chapter 7.5 7.3 7.4 8.1 8.2 8.4 Chapter 7 Polar Coordinates and Graphs of Polar Equations Plot point in a polar coordinate plane Convert between rectangular and polar coordinates Convert equations between rectangular and polar form Graph polar equations Polar Form of a Complex Number Graph a point in the complex plane Convert complex numbers from rectangular to polar form and from polar to rectangular form Products, Quotients, Powers, and Roots of Complex Numbers Find the product of two complex roots Find the quotient of two complex numbers Raise a complex number to an integer power(DeMoivre’s Theorem) Find the nth root of a complex number Chapter 8 Systems of Linear Equations in Two Variables Solve a linear system of equations by using the substitution method, elimination method, and graphing Solve applications involving systems of linear equations Systems of Linear Equations in Three Variables Solve a system if linear equations in three variables using a combination of both the elimination method and the substitution method Identify three types of solutions: one solution(point), no solution, or infinitely many solutions Solve application problems involving systems of linear equations in three variables Matrix Algebra Add and subtract matrices Perform scalar multiplication Multiply two matrices , understand why multiplication of some matrices is undefined, and realize matrix multiplication is not commutative Write a system of linear equations as a matrix equation Find the inverse of a square matrix and understand that only a square matrix can have an inverse Solve systems of linear equations using inverse matrices Time (in Days) 3 2 3 1 3 4 Section Description / Chapter 8.5 8.6 8.7 9.1 9.2 9.3 9.4 9.5 9.6 The Determinant of a Square Matrix and Cramer’s Rule Find the determinant of a 2 x 2 and n x n matrix Understand that if a determinant of a matrix is equal to zero, then the matrix does not have an inverse Use Cramer’s rule to solve a square system of linear equations Partial Fractions Decompose rational expressions into sums of partial fractions when the denominators contain: distinct linear factors, repeated factors, distinct irreducible quadratic factors, or repeated irreducible quadratic factors Systems of Linear Inequalities in Two Variables Graph a linear inequality in two variables Graph a system of linear inequalities in two variables Chapter 9 Conic Basics Learn the name of each conic section and understand each conic as an intersection of a plane and a cone Define conics and recognize the algebraic equation associated with each conic The Parabola Find the equation of a parabola whose vertex is at the origin Find the equation of a parabola whose vertex is at the point (h, k) The Ellipse Graph an ellipse given the equation and name the center, vertices, endpoints of the minor axis, and eccentricity number Write the equation of an ellipse centered at the origin Write the equation of an ellipse centered at the point (h, k) The Hyperbola Given the equation of a hyperbola, find the foci, vertices, and asymptotes Graph a hyperbola using asymptotes as aids Write the equation of a hyperbola centered at the origin Write the equation of a hyperbola centered at the point (h, k) Systems of Nonlinear Equations Solve a system of nonlinear equations with elimination Solve a system of nonlinear equations with substitution Systems of Nonlinear Inequalities Graph a nonlinear inequality in two variables Graph a system of nonlinear inequalities in two variables Chapter 11 Time (in Days) 3 3 2 1 2 3 2 3 1 Section Description / Chapter 11.1 11.2 11.4 8.3 11.3 Introduction to Limits Understand the meaning of a limit and be able to estimate limits numerically and graphically Understand that the limit of a function at a point may exist even if the function is not defined at the point Estimate limits of functions by inspecting graphs and determine whether limits of functions exist or fail to exist Understand the difference between a limit of a function and a onesided limit of a function Techniques for Finding Limits Find limits of functions using limit laws, direct substitution, and algebra Use left-hand and right-hand limits to find the limit of a function Limits at Infinity Evaluate limits of functions at infinity Optional Topics (time permitting) Systems of Linear Equations and Matrices Write a system of linear equations as an augmented matrix Perform row operations on an augmented matrix Write a matrix in row-echelon form Solve systems of linear equations using Guassian elimination with back substitution Tangent Lines and Derivatives Define the slope of a tangent line to a graph by using limits Find the equation of a tangent line to a curve by using the difference quotient Use limits to define the derivative of a function Find the instantaneous rate of change of a function Time (in Days) 4 3 2 (3) (3) Total Instruction Days: 137 (145) Total Test Days: 35 Total 172 (180) This schedule may not be possible. It is very tight. A few more topics may become optional. A final evaluation of the topics and sequence will be determined at the end of the first session.