Summer 2016 Bellmore-Merrick Central High School District
Bellmore-Merrick Central High School District BOARD OF EDUCATION Skip Haile President Janet Goller Vice President Trustees Marion Blane JoAnn DeLauter Wendy Gargiulo Dr. Nancy Kaplan Nina Lanci Gina Piskin ADMINISTRATION John DeTommaso Superintendent of Schools Cynthia Strait Régal Deputy Superintendent, Business Dr. Mara Bollettieri Assistant Superintendent, Personnel David Seinfeld Assistant Superintendent, Curriculum and Instruction 2
Written by Sara Gant Christine Guglielmo Summer 2016 Supervised by James Morris Calhoun High School This is one-semester course intended to bridge the gap for r motivated students who wish to advance from Algebra 2 Trigonometry into Calculus. Students will primarily focus on functions that include linear, polynomial, rational, exponential and logarithmic functions. This course will also cover select topics from advanced trigonometry including trigonometric and circular functions, their graphs, and trigonometric identities and equations. Students will also be introduced to limits, derivatives, and integrals. * Due of the abundance of no-calculator questions on the Advanced Placement exam, Jumpstart should be predominantly taught without a calculator. Table of Contents Topic Page 1. Nature of Graphs... 4 2. Functions and Relations... 5 3. Rational Functions... 6 4. Polynomials... 7 5. Trigonometry and Analytic Trigonometry... 8 6. Exponents and Logarithms... 9 7. Limits... 10 8. Derivatives and their Applications... 11 9. Integrals... 12 10. Supplemental Materials... 13 3
Unit #1: Nature of Graphs (approximately 9 days) Overview This unit studies functions and their graphs. Students will be able to determine domain and range graphically, determine symmetry, graph parent functions and their transformations, and graph piecewise functions. * The students should be able to complete all the tasks without the use of a calculator * Topic Description Days Needed Introduction of Domain and Range Even and Odd Functions Parent Graphs Transformations: Reflections and Vertical and Horizontal shift Transformations: Dilations Transformations: (Include examples with multiple transformations) Ex. y 4x 6 7 Piecewise Functions The students will be able to determine the domain and range of a function from the graph. The students should be able to express the domain/range in set notation and interval notation. The students will be able to prove/show if a function is even, odd or neither. The students will be able to test for symmetry with the y-axis and the origin. The students will be able to graph the parent graphs without the use of a calculator. The students should know the domain and range of the parent graphs. The students should know if the parent graph has any vertical/horizontal asymptotes and/or x- and y-intercepts. The students will be able to describe and graph the parent graph and the transformations without a calculator. The students should be able to identify the domain and range of the parent graph and the transformed graph. The students will be able to describe and graph the parent graph and the transformations without a calculator. The students should be able to identify the domain and range of the parent graph and the transformed graph. The students will be able to describe and graph the parent graph and the transformations without a calculator. The students should be able to identify the domain and range of the parent graph and the transformed graph. The students will be able to evaluate piecewise functions. The students will be able to graph piecewise functions without a calculator. The students should be able to determine the domain/range of the Textbook Reference Pages 1 44-49 1 60-63 1 19-42 See the list of parent graphs in the supplemental material 1 66-82 1 66-82 1 66-82 1-2 42, 49 70, 71 piecewise function. Review 1 N/A Test 1 N/A 4
Unit #2: Relations and Functions (approximately 8 days) Overview This unit studies functions, inverse functions and reviews basic factoring such as greatest common factor, different of two square, trinomial factoring as well as factoring a trinomials with the leading coefficient greater than one, perfect cubes and factor by grouping. Students will be able to determine the domain of a function algebraically, find the inverse of a function. Students will also be able to factor the sum and difference of cubes * The students should be able to complete most of the tasks without the use of a calculator * Topic Description Days Needed Relations and Functions The students will be able to define relations and functions. The students will be able to identify if a relation is a function using the vertical line test. 1 Textbook Reference Pages 39-71 Restricted Domains The student will be able to determine the domain of a function algebraically in all 4 cases: 1 1 y, y x, y, y logx x x 1 39-71 Range of functions The students will be able to determine the range of a function using the graphs and transformations from unit #1. The students will be able to determine the range of harder functions using a graphing calculator. The students will be able to identify max/mins using the calculator. 2 39-71 Factoring Expressions The students will be able to factor using greatest common factor, difference of two squares, trinomial factoring as well as factoring a trinomial with the leading coefficient greater than one, perfect cubes, and factor by grouping. 1 A31 A34, A37 Inverse Functions The students will be able to prove algebraically that f(x) and g(x) are functions. The students will be able to find the inverse graphically and algebraically and apply the horizontal line test. 1 92 101 Example 2 on page 93 Review 1 N/A Test 1 N/A 5
Unit #3: Rational Functions (approximately 6 days) Overview This unit studies what a rational function is and how the graph behaves. The students will know how to determine the domain and range of a rational function as well as a better understanding of a vertical asymptote. The students will be able to find the horizontal asymptote and understand that this is the behavior of the function at infinity. Topic Description Days Needed Textbook Reference Pages Introduction to Rational Functions and Discontinuities The students will be able to identify a rational function and the functions asymptotes. The students will be able to identify each of the discontinuities (point, jump and infinite discontinuity) on the given graphs. 1 181-193 The x- and y-intercepts of a Function, Points of Discontinuity, and Vertical Asymptotes The students will be able to find the x- and y-intercepts algebraically. The students will review even/odd/symmetry. The students will be able to determine the coordinates of any holes. The students will be able to find the equation of any vertical asymptotes. 1 181-193 Horizontal and Slant Asymptotes The students will be able to determine the horizontal/slant asymptotes of a function by comparing the growth rates of the quotient. 1 181-193 Graphing Rational Functions The students will be able to sketch the graph of rational functions. The sketch should include any holes, asymptotes and intercepts. 1 181-193 Review 1 N/A Test 1 N/A 6
Unit #4: Polynomial Functions (approximately 7 days) Overview This unit studies the zeros of higher order polynomial equations. The students will be able to solve for all roots of a higher order polynomial equation as well as write the polynomial equation given the roots. Students will also understand how to draw a possible rational function given the nature of the roots. Topic Description Days Needed Textbook Reference Pages The Nature of Roots The students will be able to describe the different types of roots and identify the roots from the graphs. The students will be able to sketch the graph when given a list of the types of roots. 1 126-180 Descartes Rule of Signs, The Rational Roots Theorem, and Synthetic Division The students will review the rational root theorem, the fundamental theorem of algebra, Descartes rule of signs and removing roots using synthetic division. The students will be able to sketch the graph using the information from the above theorems. 1 126-180 Solving for the Zeros of a Polynomial Function The students will be able to solve for the roots of a higher order functions. They may use the calculator to check the real roots. 1 126-180 Building Polynomial Functions (Part 1) The students will be able to build the polynomial from the given roots. (algebraically) 1 126-180 Building Polynomial Functions (Part II) The students will be able to build the polynomial from the given roots. The students should be able to identify the roots from the graph. 1 126-180 Review 1 N/A Test 1 N/A 7
Unit #5: Trigonometry and Analytic Trigonometry (approximately 10 days) Overview Students will have a better understanding of the unit circle, the special right triangles, the inverse trigonometric functions and solving trigonometric equations with and without a calculator. Topic Description Days Needed Trigonometric Identities (Part 1) Trigonometric Identities (Part II) The student should memorize the trigonometric identities. The students should be able to prove trigonometric identities. The student will be able to evaluate/simplify basic trigonometric expressions using the trigonometric identities. Textbook Reference Pages 1 372-379 1 377-379 Proving Trigonometric Identities The Unit Circle and Special Right Triangles The students will to prove trigonometric identities. The students will know how to evaluate the trigonometric functions without a calculator. The students must know the exact values of the trigonometric functions. The students will evaluate the special angles for the trig functions 1-2 380 386 1 279-318 Exact Vales of Trigonometric Functions, and Inverse Trigonometric Functions The students will be able to determine the exact value of all six trigonometric functions without a calculator. The students will be able to evaluate inverse trig functions without a calculator. 1 279 318 319 350 365 366 Solving Trigonometric Equations Algebraically (Part 1) The students will be able to solve first degree trigonometric equations without the use of a calculator. 1 387 415 Solving Trigonometric Equations Algebraically (Part II) Solving Trigonometric Equations Algebraically (Part III) Solving Trigonometric Equations Graphically The students will be able to solve second degree trigonometric equations without the use of a calculator. The students will be able to solve second degree trigonometric equations that require substitution of a trig identity with and without the use of a calculator. The students will be able to solve trigonometric equations by graphing with the use of a calculator. Examples: 1 1 cos x, cos x sin x, 3sin x 1 cos x 2 2 1 387 415 1 387 415 1 387 415 Review 1 N/A Test 1 N/A 8
Unit #6: Exponential and Logarithmic Functions (approximately 6 days) Overview This unit reviews the properties of exponents and logarithms, the properties of logarithms and solving exponential and logarithmic equations. Topic Description Days Needed Textbook Reference Pages Properties of Exponents The students will be able to evaluate numerical expressions with negative and rational exponents without the use of a calculator. The students will be able to write expressions with rational exponents in radical form and rewrite expressions that have negative exponents with only positive exponents. 1 A14 A26 Properties of Logarithms The students will be able to convert from log form to exponential form and from exponential to log form. The students will be able to evaluate logs without a calculator. Examples: 3 log10, log100, log10, ln e, ln e 1 2 1 227-254 Solving Exponential and Logarithmic Equations Algebraically Solving Exponential and Logarithmic Equations Graphically The students will be able to solve logarithmic equations without a calculator. The students will be able to solve exponential equations with logs without a calculator. The students will be able to solve exponential equations graphically with the use of a calculator. Example: 3e 1 16 2x 1 1 227-254 1 227 254 Review 1 N/A Test 1 N/A 9
Unit #7: Limits (approximately 12 days) Overview This unit studies limits and an introduction to Calculus. The students will be able to evaluate limits graphically and algebraically. The students will also be able to discuss continuity at a point. Topic Description Days Needed Introduction to Limits Evaluating Limits Graphically Continuity at a Point Graphically Evaluating Limits Algebraically Continuity at a Point Algebraically Types of Discontinuity Limits at Infinity The students will be able to discuss the definition of limit and where we see limits in nature. The students will be able to evaluate one-sided limits The students will be able to evaluate limits from a graph. The students will be able to determine if a function is continuous at a point given a graph. The students should be able to evaluate a limit using direct substitution. The students will evaluate limits algebraically. The teacher should use examples involving factoring, synthetic division, trig identities, rationalizing and complex fractions. The students will be able to determine continuity at a point of a piecewise function as well as standard functions algebraically. The students will be able to identify the four types of discontinuity (jump, oscillation, infinite, point). The students should understand how the discontinuities relate to limits. The students will be able to evaluate limits as x approaches infinity. The students will look at the growth of graphs to determine the limit. The students will learn how to determine vertical & horizontal asymptotes using limits. 1 1 Textbook Reference Pages 866 900 # 1 10 852 Example 5 1 See Supplemental Material 2 1 1-2 1-2 853 863 868 869 See Supplemental Material 859 # 13 44 See Supplemental Material 182 882 883 887 # 1 35 Special Trig. Limits The students will be able to determine the following limits using the graphing calculator. The algebraic solutions to these problems will be covered in AP Calculus. These limits should be memorized for use in the derivatives unit. 1 sin x cos x lim,lim, x 0 x x 0 x h tan x e 1 lim,lim, x 0 x x 0 h cos x 1 lim x 0 x Review 1 N/A Test 1 N/A 10
Unit #8: Derivatives (approximately 15 days) Overview This unit introduces the students to the definition of the derivate. The students will know how to use the product, quotient and chain rule to evaluate and find derivatives of complex functions. The students will also learn about applications of derivatives. You may want to have two assessments for this unit. Topic Description Days Needed Textbook Reference Pages Average Rate of Change and Instantaneous Rate of Change The students will be able to determine the average rate of change. Using the average rate of change and limits, the students will derive the difference quotient. 1 See Supplemental Material The Difference Quotient The students will use the difference quotient to find derivatives, such as x 1 1 y x 2, y sin x, y e, y x 6 See Supplemental Material Proofs of Derivatives The students will use the difference quotient to prove the power rule, product rule and quotient rule. 1 See Supplemental Material The Power Rule, Product Rule, and Quotient Rule Derivatives of Trig Functions and the Natural Log Function The students will be able to determine the derivative using the product, quotient and power rule. The students will be able to determine the derivative of y tan x, y ln x, y cos x 1 1 See Supplemental Material See Supplemental Material The Chain Rule The students will be able to determine the derivatives of functions using the chain rule. 1 See Supplemental Material (Quiz or Test) The students should memorize the derivative rules. 1 N/A Slopes of Tangent Lines and Normal Lines The students will be able to determine the slope of the line tangent to the curve using the derivative. The students will be able to write the equations of the tangent and normal lines. 1 878 # 43 50 879 # 55 70 Curve Sketching (Part 1) The students will be able to use the first and second derivative to determine max/min points and points of inflection. The students will be able to find the intervals where the function is increasing/decreasing/concave up/concave down. The students ill check their work using the calculator. 2 See Supplemental Material The students will be able to determine the graph of f(x) given the graph of the first and second derivative. Curve Sketching The students will be able to determine the x-value of See Supplemental 2 (Part II) the max/min/point of inflection. The students will be Material to determine the intervals where the function is increasing/decreasing/concave up/concave down. Review 1 N/A Test 1 N/A 11
Unit #9: Integrals (approximately 7 days) Overview This unit studies the relationship between the derivative and the integral. The students will be able to find the area under one curve and between two curves. Topic Description Days Needed Textbook Reference Pages Introduction to Integrals and Antiderivatives The students will be able to use the formula to find an antiderivative. The students will understand the need for + C. 1 See Supplemental Material Integrals The students will use the power rule and algebraic simplification to integrate. The students should be able to use the basic integral properties. 1 2 See Supplemental Material Definite Integrals The students will understand the difference between definite and indefinite integrals. The students will be able to evaluate definite integrals without a calculator. 1 See Supplemental Material Area Under the Curve The students will be able to use integrals to find the area of basic regions. The students should be able to evaluate the area with and without the calculator. 1 See Supplemental Material Area Between Two Curves The students will be able to use integrals to find the area between two curves. The students should be able to evaluate the area with and without the calculator. 1 See Supplemental Material Review 1 N/A Test 1 N/A 12
Supplemental Materials 13
14 Parent Graphs: (The parent graphs are also located on the last two pages of the textbook) The students should be able to recognize each of these parent graphs: 2 3 1 log ln y x y x y x y x y x y x 3 [[ ]] x x y x y x y x y x y e y a sin cos tan sec csc cot y x y x y x y x y x y x