CCSS: N.CN.7: Solve quadratic equations with real coefficients that have complex solutions
|
|
- Georgina White
- 7 years ago
- Views:
Transcription
1 4.5 Completing The Square 1) Solve quadratic equations using the square root property 2) Generate perfect square trinomials by completing the square 3) Solve quadratic equations by completing the square CCSS: N.CN.7: Solve quadratic equations with real coefficients that have complex solutions F.IF.8.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 1. Look for and make use of structure 1. The square root property: a. Powers and roots are inverses, thus raising to a power and taking a root are inverse operations b. If the quadratic term in an equation can be isolated, it can be solved by taking the square root of each side c. Works best in an equation that does not have a linear term. d. Can be applied to perfect square trinomials because they can be written as a binomial raised to the second power 2. To generate a perfect square trinomial (complete the square): a. x 2 + bx + c; c = (0.5b) 2 b. so x 2 + bx + (b/2) 2 will always be a perfect square 3. To solve a quadratic equation by completing the square, the leading coefficient must be Example 1: Solve x x + 49 = 64 by using the square root property. 5. Example 2: Solve x 2-4x + 4 = 13 by using the square root property. 6. Example 3: Find the value of c that makes x x + c a perfect square. Then write the trinomial as a perfect square. 7. Example 4: Solve x 2 + 4x 12 = 0 by completing the square. 8. Example 5: Solve 3x 2-2x 1 = 0 by completing the square. 9. Example 6: Solve x 2 + 4x + 11 = 0 by completing the square
2 4.6 The Quadratic Formula and the Discriminant 1) solve quadratic equations using the quadratic formula 2) Use the discriminant to determine the number and type of roots of a quadratic equation CCSS: N.CN.7: Solve quadratic equations with real coefficients that have complex solutions A.SSE.1.b: Interpret complicated expressions by viewing one or more of their parts as a single entity 2. Look for and express regularity in repeated reasoning The Quadratic Formula uses the "a", "b", and "c" from "ax 2 + bx + c", where "a", "b", and "c" are just numbers; they are the "numerical coefficients". The Formula is derived from the process of completing the square, and is formally stated as: For ax 2 + bx + c = 0, the value of x is given by: Here are some examples of how the Quadratic Formula works: Solve x 2 + 3x 4 = 0 This quadratic happens to factor: x 2 + 3x 4 = (x + 4)(x 1) = 0...so I already know that the solutions are x = 4 and x = 1. How would my solution look in the Quadratic Formula? Using a = 1, b = 3, and c = 4, my solution looks like this: Then, as expected, the solution is x = 4, x = 1. Source:
3 1. All quadratic equations have two solutions. 2. If both solutions are real and different, the parabola has two x-intercepts. 3. If it has only one real solution, the quadratic is a perfect square. a. It still has two roots, they just happen to be the same number. b. The parabola will touch the x-axis only once. c. The double root will also be the vertex of the parabola. 4. The equation has no real solutions if a. It has a minimum that is above the x-axis or b. It has a maximum that is below the x-axis c. Both of its solutions are complex. 5. The discriminant: D=b 2 4ac (the radicand in the quadratic formula) can be used to determine the number of real roots. 6. If D>0 and is a perfect square, will have 2 real, rational roots. 7. If D>0 and is not a perfect square, will have 2 real, irrational roots. 8. If D=0, will have one real, rational root. ( b/2a,0) will be both roots and the x- intercept. 9. If D<0, will have no real (2 complex) roots. 10. Summary of methods for solving quadratic equations. a. Graphing (4.1, 4.2) b. Factoring (4.3) c. Square Root Property (4.4) d. Completing the Square (4.4, 4.5) e. Quadratic Formula (4.6) 11. Example 1: Solve x 2-8x = 33 by using the Quadratic Formula. 12. Example 2: Solve x 2-34x = 0 by using the Quadratic Formula. 13. Example 3: Solve x 2-6x + 2 = 0 by using the Quadratic Formula. 14. Example 4: Solve x = 6x by using the Quadratic Formula. 15. Example 5: Find the value of the discriminant for each quadratic equation. Then describe the number and type of roots for the equation. a. x 2 + 3x + 5 = 0 b. x 2-11x + 10 = 0
4 4.7 Transformations of Quadratic Graphs 1) Write quadratic equations in vertex form 2) Transform graphs of quadratic functions 3) Write equations of quadratic functions by interpreting the graph CCSS: F.IF.8.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 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. 7. Look for and make use of structure 1. Vertex form: f(x) = a(x - h) 2 + k Transformations review (from section 2.7): 2. Translation a. Slide b. Addition 3. Reflection a. Flip b. Multiplication by Dilation a. Stretch/compress (enlarge/shrink) b. Multiplication 5. Horizontal/Vertical inside/outside TRANSFORMATION CHANGE TO PARENT GRAPH Translation f(x + h) Horizontal translation h units left f(x - h) Horizontal translation h units right f(x) - h Vertical translation down h units f(x) + h Vertical translation up h units Reflection -f(x) Vertical reflection (across the x axis) f(-x) Horizontal reflection (across the y-axis) Dilation af(x), a > 1 Vertical stretch by a factor of a
5 6. Example 1: Write each equation in vertex form. a. f(x) = x 2 2x + 4 b. f(x) = -3x 2 18x Example 2: Graph f(x) = -2x 2 + 4x Quadratic Inequalities 1) Graph quadratic inequalities in two variables 2) Solve quadratic inequalities in one variable graphically and algebraically CCSS: 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 1. Make sense of problems and persevere in solving them 1. Is the parabola solid or dotted? 2. Do you shade the inside or the outside (use a test point)? 3. For which x-values will the parabola lie above (or below) the x-axis? 4. Example 1: Graph f(x) > x 2 3x Example 2: Solve x 2 4x + 3 < 0 by graphing. 6. Example 3: Solve 0-2x 2 6x + 1 by graphing. 7. Example 4: Solve x 2 + x 2 algebraically.
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationAlgebra 2/Trig Unit 2 Notes Packet Period: Quadratic Equations
Algebra 2/Trig Unit 2 Notes Packet Name: Date: Period: # Quadratic Equations (1) Page 253 #4 6 **Check on Graphing Calculator (GC)** (2) Page 253 254 #20, 26, 32**Check on GC** (3) Page 253 254 #10 12,
More informationBEST METHODS FOR SOLVING QUADRATIC INEQUALITIES.
BEST METHODS FOR SOLVING QUADRATIC INEQUALITIES. I. GENERALITIES There are 3 common methods to solve quadratic inequalities. Therefore, students sometimes are confused to select the fastest and the best
More information7.1 Graphs of Quadratic Functions in Vertex Form
7.1 Graphs of Quadratic Functions in Vertex Form Quadratic Function in Vertex Form A quadratic function in vertex form is a function that can be written in the form f (x) = a(x! h) 2 + k where a is called
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationPolynomial Operations and Factoring
Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.
More informationManhattan Center for Science and Math High School Mathematics Department Curriculum
Content/Discipline Algebra 1 Semester 2: Marking Period 1 - Unit 8 Polynomials and Factoring Topic and Essential Question How do perform operations on polynomial functions How to factor different types
More information2.5 Transformations of Functions
2.5 Transformations of Functions Section 2.5 Notes Page 1 We will first look at the major graphs you should know how to sketch: Square Root Function Absolute Value Function Identity Function Domain: [
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More informationAlgebra II A Final Exam
Algebra II A Final Exam Multiple Choice Identify the choice that best completes the statement or answers the question. Evaluate the expression for the given value of the variable(s). 1. ; x = 4 a. 34 b.
More informationALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section
ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 5-3.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 6-4.2 Solving Equations by
More informationThis unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide.
COLLEGE ALGEBRA UNIT 2 WRITING ASSIGNMENT This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. 1) What is the
More informationMA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
More informationAlgebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only
Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationMSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions
MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions The goal of this workshop is to familiarize you with similarities and differences in both the graphing and expression of polynomial
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationWarm-Up Oct. 22. Daily Agenda:
Evaluate y = 2x 3x + 5 when x = 1, 0, and 2. Daily Agenda: Grade Assignment Go over Ch 3 Test; Retakes must be done by next Tuesday 5.1 notes / assignment Graphing Quadratic Functions 5.2 notes / assignment
More informationBookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line
College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina - Beaufort Lisa S. Yocco, Georgia Southern University
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationAlgebra 2: Q1 & Q2 Review
Name: Class: Date: ID: A Algebra 2: Q1 & Q2 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which is the graph of y = 2(x 2) 2 4? a. c. b. d. Short
More informationPolynomial and Rational Functions
Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving
More informationQUADRATIC EQUATIONS AND FUNCTIONS
Douglas College Learning Centre QUADRATIC EQUATIONS AND FUNCTIONS Quadratic equations and functions are very important in Business Math. Questions related to quadratic equations and functions cover a wide
More informationMathematics. Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 1: Quadratic Functions
Georgia Standards of Excellence Frameworks Mathematics Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 1: Quadratic Functions These materials are for nonprofit educational purposes only. Any
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More informationFACTORING QUADRATICS 8.1.1 and 8.1.2
FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.
More informationMathematics Curriculum
Common Core Mathematics Curriculum Table of Contents 1 Polynomial and Quadratic Expressions, Equations, and Functions MODULE 4 Module Overview... 3 Topic A: Quadratic Expressions, Equations, Functions,
More informationUnit 7 Quadratic Relations of the Form y = ax 2 + bx + c
Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c Lesson Outline BIG PICTURE Students will: manipulate algebraic expressions, as needed to understand quadratic relations; identify characteristics
More informationPARABOLAS AND THEIR FEATURES
STANDARD FORM PARABOLAS AND THEIR FEATURES If a! 0, the equation y = ax 2 + bx + c is the standard form of a quadratic function and its graph is a parabola. If a > 0, the parabola opens upward and the
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationList the elements of the given set that are natural numbers, integers, rational numbers, and irrational numbers. (Enter your answers as commaseparated
MATH 142 Review #1 (4717995) Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Description This is the review for Exam #1. Please work as many problems as possible
More informationLinear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)}
Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in
More information1 Shapes of Cubic Functions
MA 1165 - Lecture 05 1 1/26/09 1 Shapes of Cubic Functions A cubic function (a.k.a. a third-degree polynomial function) is one that can be written in the form f(x) = ax 3 + bx 2 + cx + d. (1) Quadratic
More informationSouth Carolina College- and Career-Ready (SCCCR) Algebra 1
South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR) Mathematical Process
More informationBrunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year.
Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Goal The goal of the summer math program is to help students
More informationIntroduction to Quadratic Functions
Introduction to Quadratic Functions The St. Louis Gateway Arch was constructed from 1963 to 1965. It cost 13 million dollars to build..1 Up and Down or Down and Up Exploring Quadratic Functions...617.2
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte
More informationSection 3.1 Quadratic Functions and Models
Section 3.1 Quadratic Functions and Models DEFINITION: A quadratic function is a function f of the form fx) = ax 2 +bx+c where a,b, and c are real numbers and a 0. Graphing Quadratic Functions Using the
More informationStudents Currently in Algebra 2 Maine East Math Placement Exam Review Problems
Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write
More informationThis unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
More informationWeek 1: Functions and Equations
Week 1: Functions and Equations Goals: Review functions Introduce modeling using linear and quadratic functions Solving equations and systems Suggested Textbook Readings: Chapter 2: 2.1-2.2, and Chapter
More informationPrentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)
Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify
More information1.2 GRAPHS OF EQUATIONS. Copyright Cengage Learning. All rights reserved.
1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs
More informationAlgebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.
More informationSection 5.0A Factoring Part 1
Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (
More informationCreating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities
Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned
More informationMATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationCONVERT QUADRATIC FUNCTIONS FROM ONE FORM TO ANOTHER (Standard Form <==> Intercept Form <==> Vertex Form) (By Nghi H Nguyen Dec 08, 2014)
CONVERT QUADRATIC FUNCTIONS FROM ONE FORM TO ANOTHER (Standard Form Intercept Form Vertex Form) (By Nghi H Nguyen Dec 08, 2014) 1. THE QUADRATIC FUNCTION IN INTERCEPT FORM The graph of the quadratic
More informationExamples of Tasks from CCSS Edition Course 3, Unit 5
Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can
More informationFlorida Math for College Readiness
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourth-year math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
More informationDRAFT. Algebra 1 EOC Item Specifications
DRAFT Algebra 1 EOC Item Specifications The draft Florida Standards Assessment (FSA) Test Item Specifications (Specifications) are based upon the Florida Standards and the Florida Course Descriptions as
More informationSimplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.
MAC 1105 Final Review Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. 1) 8x 2-49x + 6 x - 6 A) 1, x 6 B) 8x - 1, x 6 x -
More informationWentzville School District Algebra 1: Unit 8 Stage 1 Desired Results
Wentzville School District Algebra 1: Unit 8 Stage 1 Desired Results Unit Title: Quadratic Expressions & Equations Course: Algebra I Unit 8 - Quadratic Expressions & Equations Brief Summary of Unit: At
More information3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes
Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general
More information6.1 Add & Subtract Polynomial Expression & Functions
6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More informationGraphing Linear Equations
Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope
More informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction
More informationexpression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.
A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are
More informationPolynomial Expressions and Equations
Polynomial Expressions and Equations This is a really close-up picture of rain. Really. The picture represents falling water broken down into molecules, each with two hydrogen atoms connected to one oxygen
More informationUnit 1 Equations, Inequalities, Functions
Unit 1 Equations, Inequalities, Functions Algebra 2, Pages 1-100 Overview: This unit models real-world situations by using one- and two-variable linear equations. This unit will further expand upon pervious
More informationEL-9650/9600c/9450/9400 Handbook Vol. 1
Graphing Calculator EL-9650/9600c/9450/9400 Handbook Vol. Algebra EL-9650 EL-9450 Contents. Linear Equations - Slope and Intercept of Linear Equations -2 Parallel and Perpendicular Lines 2. Quadratic Equations
More informationFactoring Polynomials and Solving Quadratic Equations
Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3
More informationALGEBRA I (Created 2014) Amherst County Public Schools
ALGEBRA I (Created 2014) Amherst County Public Schools The 2009 Mathematics Standards of Learning Curriculum Framework is a companion document to the 2009 Mathematics Standards of Learning and amplifies
More informationAlgebra II Unit Number 4
Title Polynomial Functions, Expressions, and Equations Big Ideas/Enduring Understandings Applying the processes of solving equations and simplifying expressions to problems with variables of varying degrees.
More informationPRE-CALCULUS GRADE 12
PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
More informationAlgebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks
Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks
More informationThis is a square root. The number under the radical is 9. (An asterisk * means multiply.)
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize
More informationAlgebraic Concepts Algebraic Concepts Writing
Curriculum Guide: Algebra 2/Trig (AR) 2 nd Quarter 8/7/2013 2 nd Quarter, Grade 9-12 GRADE 9-12 Unit of Study: Matrices Resources: Textbook: Algebra 2 (Holt, Rinehart & Winston), Ch. 4 Length of Study:
More informationEquations. #1-10 Solve for the variable. Inequalities. 1. Solve the inequality: 2 5 7. 2. Solve the inequality: 4 0
College Algebra Review Problems for Final Exam Equations #1-10 Solve for the variable 1. 2 1 4 = 0 6. 2 8 7 2. 2 5 3 7. = 3. 3 9 4 21 8. 3 6 9 18 4. 6 27 0 9. 1 + log 3 4 5. 10. 19 0 Inequalities 1. Solve
More informationis the degree of the polynomial and is the leading coefficient.
Property: T. Hrubik-Vulanovic e-mail: thrubik@kent.edu Content (in order sections were covered from the book): Chapter 6 Higher-Degree Polynomial Functions... 1 Section 6.1 Higher-Degree Polynomial Functions...
More informationUnit 7: Radical Functions & Rational Exponents
Date Period Unit 7: Radical Functions & Rational Exponents DAY 0 TOPIC Roots and Radical Expressions Multiplying and Dividing Radical Expressions Binomial Radical Expressions Rational Exponents 4 Solving
More informationAlgebra 2 Chapter 5 Practice Test (Review)
Name: Class: Date: Algebra 2 Chapter 5 Practice Test (Review) Multiple Choice Identify the choice that best completes the statement or answers the question. Determine whether the function is linear or
More informationwww.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More informationNorth Carolina Math 2
Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4.
More information2.1 Increasing, Decreasing, and Piecewise Functions; Applications
2.1 Increasing, Decreasing, and Piecewise Functions; Applications Graph functions, looking for intervals on which the function is increasing, decreasing, or constant, and estimate relative maxima and minima.
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationPolynomials and Quadratics
Polynomials and Quadratics Want to be an environmental scientist? Better be ready to get your hands dirty!.1 Controlling the Population Adding and Subtracting Polynomials............703.2 They re Multiplying
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationNSM100 Introduction to Algebra Chapter 5 Notes Factoring
Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the
More informationPearson Algebra 1 Common Core 2015
A Correlation of Pearson Algebra 1 Common Core 2015 To the Common Core State Standards for Mathematics Traditional Pathways, Algebra 1 High School Copyright 2015 Pearson Education, Inc. or its affiliate(s).
More informationAssessment Schedule 2013
NCEA Level Mathematics (9161) 013 page 1 of 5 Assessment Schedule 013 Mathematics with Statistics: Apply algebraic methods in solving problems (9161) Evidence Statement ONE Expected Coverage Merit Excellence
More information5. Factoring by the QF method
5. Factoring by the QF method 5.0 Preliminaries 5.1 The QF view of factorability 5.2 Illustration of the QF view of factorability 5.3 The QF approach to factorization 5.4 Alternative factorization by the
More informationAlgebra 1. Curriculum Map
Algebra 1 Curriculum Map Table of Contents Unit 1: Expressions and Unit 2: Linear Unit 3: Representing Linear Unit 4: Linear Inequalities Unit 5: Systems of Linear Unit 6: Polynomials Unit 7: Factoring
More informationCRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide
Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are
More informationMATH 108 REVIEW TOPIC 10 Quadratic Equations. B. Solving Quadratics by Completing the Square
Math 108 T10-Review Topic 10 Page 1 MATH 108 REVIEW TOPIC 10 Quadratic Equations I. Finding Roots of a Quadratic Equation A. Factoring B. Quadratic Formula C. Taking Roots II. III. Guidelines for Finding
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationPractice Test Answer and Alignment Document Mathematics: Algebra II Performance Based Assessment - Paper
The following pages include the answer key for all machine-scored items, followed by the rubrics for the hand-scored items. - The rubrics show sample student responses. Other valid methods for solving
More informationGouvernement du Québec Ministère de l Éducation, 2004 04-00815 ISBN 2-550-43543-5
Gouvernement du Québec Ministère de l Éducation, 004 04-00815 ISBN -550-43543-5 Legal deposit Bibliothèque nationale du Québec, 004 1. INTRODUCTION This Definition of the Domain for Summative Evaluation
More informationGraphic Designing with Transformed Functions
Math Objectives Students will be able to identify a restricted domain interval and use function translations and dilations to choose and position a portion of the graph accurately in the plane to match
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More informationFactoring Polynomials
Factoring Polynomials Factoring Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of 6x 2 x 2 are 2x + 1 and 3x 2. In this section, we will be factoring
More information4.4 Transforming Circles
Specific Curriculum Outcomes. Transforming Circles E13 E1 E11 E3 E1 E E15 analyze and translate between symbolic, graphic, and written representation of circles and ellipses translate between different
More informationMath 1. Month Essential Questions Concepts/Skills/Standards Content Assessment Areas of Interaction
Binghamton High School Rev.9/21/05 Math 1 September What is the unknown? Model relationships by using Fundamental skills of 2005 variables as a shorthand way Algebra Why do we use variables? What is a
More informationa. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F
FINAL REVIEW WORKSHEET COLLEGE ALGEBRA Chapter 1. 1. Given the following equations, which are functions? (A) y 2 = 1 x 2 (B) y = 9 (C) y = x 3 5x (D) 5x + 2y = 10 (E) y = ± 1 2x (F) y = 3 x + 5 a. all
More informationIn this section, you will develop a method to change a quadratic equation written as a sum into its product form (also called its factored form).
CHAPTER 8 In Chapter 4, you used a web to organize the connections you found between each of the different representations of lines. These connections enabled you to use any representation (such as a graph,
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More information