# Roots and Coefficients of a Quadratic Equation Summary

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and + = (α +β) = α + β + αβ therefore α + β = (α+β) -αβ α 3 x β 3 = α 3 β 3 = (αβ) 3 and α 3 + β 3 = (α + β) 3-3αβ(α + β) Find the quadratic equation whose roots are α and β where α and β β α are the roots of the quadratic equation x 7x 3. From x 7x 3, α + β = 7 and αβ = 3. The sum of the new roots is: α β β α α β β α α 9 6 β (α β) αβ The new product of roots is: α β β α αβ αβ So the new equation is 6x 9x 9 9 6

2 Equations with related roots: If α and β are the roots of the equation, you can obtain an equation with roots α and β by substituting in y=x, thus. Find a quadratic equation with roots α- and β-, where α and β are the roots of the equation x 7x. x 7x Use the variable y where y=x-, so that x (y ). The required equation is therefore: (y ) 7 (y ) Which can be simplified to: y y

3 Series and Sums of Natural Numbers The sum of the first n natural numbers is given by: The sum of the squares of the first n natural numbers is given by: 6 The sum of the cubes of the first n natural numbers is given by: 3 Find: Splitting the expression gives: (r ) r Which gives: 00 (r ) r r ( ) ( ) (69 )

4 Find the sum of the squares of the odd numbers from to 9. The sum of the squares of the odd numbers from to 9 is equal to: Sum of all numbers from to 9 Sum of even numbers from to 9 The sum of the even numbers is: = ( ) r Therefore the sum of all odd numbers from to 9 is: 9 r r =0-9600=08 Find the value of n (r 3 3) r=n n (r 3 3) r=n n (r 3 3) r= n (r 3 3) r= Which gives: n n (r 3 3) r 3 n 3 n r 3 n 3 r= n (n ) 6n n (n ) 3n n n 3 3n 3n

5 Matrices The x identity matrix is called I, where: I When a matrix is multiplied by the identity matrix I, it stays the same. Find A B, where A 8 and B. A B 8 6 Find AB and BA where A 3 and B 6 7 AB BA

6 Transformations Every linear transformation can be represented by a x square matrix, M, of the form. To find the matrix M, representing a given transformation, you find the images of the two points (, 0) and (0, ). Find the matrix, M, representing an enlargement, scale factor, with the origin as the centre of enlargement. By drawing a graph, the images of (, 0) and (0, ) are (, 0) and (0, ). So M. Using the image (, ) to check: as expected. The general form for the matrix of rotation about the origin through angle Ѳ anticlockwise is. The general form for the matrix of a reflection in the line y=(tanѳ) is: Common Transformations is a stretch parallel to the y-axis with a scale factor a. is a stretch parallel to the x-axis with scale factor a. generally represents both a rotation and an enlargement. generally represents both a reflection and an enlargement.

7 Combining Two Transformations If the image of (x, y) under the transformation T is transformed by a second transformation T, represented by the matrix M, the image of (x, y) under the two transformations is given by M M. In general, MM M M. Find the matrix represented by the combined transformations: A stretch parallel to the x-axis of scale factor Followed by a stretch parallel to the y-axis of scale factor. The two matrices are: M and M. The combined transformation is given by M M. M M.

8 Graphs of Rational Functions Rational Functions with a Linear Numerator and Denominator: This is when it is in the form where a, b, c and d are constants. Consider the function sketch its graph.. To understand the function, it is useful to By observation, as x approaches the value -3, the numerator approaches -0 but the denominator approaches 0. As x keeps getting closer to -3, the y value will tend towards infinity without every touching x=-3. Hence the line x=-3 is an asymptote. To find the next asymptote, divide both the numerator and denominator by x. 8 This gives. As x gets very large, both and 3 3 approach the value zero. Therefore as x tends towards infinity, both fractions above tend towards zero. As a result, the curve approaches another asymptote of y=. After finding the x and y intercepts by using x=0 and y=0, you get: x=-3 as the vertical asymptote y= as the horizontal asymptote When x=0, y= When y=0, x= 8 3 3

9 Two Distinct Linear Factors in the Denominator These are where the denominator is a factorised quadratic, or one which can be factorised. If the denominator has two real solutions, there will be two vertical asymptotes. If the numerator is also a quadratic, the curve will usually cross the horizontal asymptote. To find the horizontal asymptote, divide through by x. To find the vertical asymptotes, equate the denominator to zero and then solve to find the asymptotes. To find where the curve crosses the x and y axes, substitute in x=0 and y=0. To find where the curve crosses the horizontal asymptote, substitute in the value of the horizontal asymptote (y=a). If the numerator is a linear expression and the denominator has two linear factors, the horizontal asymptote is always y=0, because if you expand the denominator; then divide through by x, y tends towards 0. Rational Functions with a Repeated Factor in the Denominator If the numerator is a quadratic expression, and the denominator is a quadratic expression with equal factors, e.g. ( 3)( 3) There is only one vertical asymptote. Considering the function above, as x approaches, the denominator approaches zero so the vertical asymptote is x=. To find the horizontal asymptote, divide through by x. To find the vertical asymptotes, equate the denominator to zero and then solve to find the asymptotes. To find where the curve crosses the x and y axes, substitute in x=0 and y=0. To find where the curve crosses the horizontal asymptote, substitute in the value of the horizontal asymptote (y=a). Rational Functions with an Irreducible Quadratic in the Denominator Not all curves in the form If the equation no vertical asymptote. have vertical asymptotes. has no real solutions, then the curve will have

10 Stationary Points on the Graphs of Rational Functions Several of the functions in this chapter have one or more stationary points. Now consider the curve: 3 6 By sketching this graph you will see it has a minimum point. To find the minimum point, consider the line y=k intersecting the curve. Therefore, 3 6 Therefore (6 3) This gives 6 3. Since there is a minimum point, the roots must be equal, Thus b -ac=0. As there is no point on the curve for which y=, the minimum value is To find the value of x at the minimum point, sub the value of k (k=y) into equation. This gives x=- Inequalities With inequalities, normal rules apply unless you are multiplying or dividing both sides by a negative number. If you do this, the inequality symbol must be reversed however, This means that you cannot solve an inequality such as because you don t know whether cx+d is positive or negative, so you re unable to multiply both sides out by this. To solve an inequality like, there are two methods. The first method is to multiply out both sides by, which cannot be negative. Or sketch, solve and then, by comparing these two results, write down the solution to the inequality.

11 Conics The Parabola The standard equation for a parabola is. Translation: The translation on the parabola 8 to the parabola with equation 8 Reflection in the line y x: Reflecting 8 in the line y=x results in the new equation 8. Stretch parallel to the x-axis: A stretch, scale factor a, parallel to the x-axis transforms the parabola 8 8 into the new equation Stretch parallel to the y-axis: A stretch, scale factor a, parallel to the y-axis transforms the parabola to the equation 8 7 The Ellipse You can obtain an ellipse from a circle by applying a one-way stretch. Starting with a circle with the equation, then applying a stretch of scale factor a parallel to the x-axis followed by a stretch scale factor b parallel to the y-axis gives the standard equation of an ellipse: Translation: A translation of moves to Reflection in the line y x: To reflect an ellipse in the line y=x, replace the x s with y s and vice versa. Stretch parallel to the x-axis: A stretch parallel to the x-axis scale factor p, goes to Stretch parallel to the y-axis: A stretch, scale factor q, parallel to the y-axis goes to 3

12 The hyperbola The standard equation for a hyperbola is Translation: The translation of moves the hyperbola into a hyperbola with a new equation of: Reflection in the line y x: To reflect a hyperbola in the line y=x, swap around the x and y s. Stretch parallel to the x-axis: A stretch parallel to the x-axis scale factor p, goes to Stretch parallel to the y-axis: A stretch, scale factor q, parallel to the y-axis goes to The rectangular hyperbola The standard equation for a rectangular hyperbola is Translation: The translation of moves the rectangular hyperbola into a rectangular hyperbola with a new equation of. Reflection in the line y x: To reflect a rectangular hyperbola in the line y=x, swap around the x and y s, which will result in the same equation. Stretch parallel to the x-axis: A stretch parallel to the x-axis scale factor p, goes to Stretch parallel to the y-axis: A stretch, scale factor q, parallel to the y-axis goes to

13 Complex Numbers A complex number is a number in the form, where a and b are real numbers and. In the complex number, the real number a is called the real part, and the real number b is called the imaginary part of the complex number. - When a=0, the number is said to be wholly imaginary. - When b=0, the number is real. - If a complex number is equal to zero, both a and b are zero. If two complex numbers are equal, their real parts are equal and their imaginary parts are equal. If, then and. If z is a complex number, its complex conjugate is denoted by z*. If, then z* The quadratic formula can be used to find the roots to an equation. If is negative, you will have to introduce i. From chapter one, we have also learnt that the sum of roots is equal to the sum of the equations from the quadratic formula. The quadratic formula can be used to find the roots to an equation. If is negative, you will have to introduce i. From chapter one, we have also learnt that the sum of roots, equal to the sum of the equations from the quadratic formula., is Equations with z or z* can be solved by substituting in or z*. Then you can equate the real and imaginary coefficients.

15 Improper Integrals There are two types of improper integrals: The integral has as one of its limits. The integrand is undefined either at one of its limits, or somewhere between them limits. The function may have a denominator which equals 0 at some point. -Limits of integration between Replace with n, and once you have integrated, allow n to approach infinity. If the integral approaches a finite answer, then the integral can be found. If the integral doesn t approach a finite answer, the integral cannot be found. Determine Substitute n for : [ ] As n, the value of the integral approaches. Therefore the value of the improper integral is. Determine whether the integral value of the integral. has a value. If so, find the Substituting n for, then integrating gives: [ ] As n, the value of this integral does not approach a finite number, and so the integral cannot be found.

16 Integral Undefined at a Particular Value If the integrand is undefined at one of the limits of integration, replace that limit of integration with p, and see what happens as p approaches the limit of integration. If the integral approaches a finite number, the integral can be found. If the integral doesn t approach a finite number, the integral cannot be found. If the integrand is undefined at some point between the two limits of integration, then you need to split the integral into two parts, one to the left, and one to the right of the point where the integrand is undefined and use a similar approach as before. Determine 0 0 When x=0, 0=p. [ ] [ ] [ ] [ 3 ] As p tends towards 0, no finite answer is reached. Therefore, the integral cannot be found.

17 Trigonometry Cosine: The general solution of the equation 36 or for any angle α is: Sine: The general solution of the equation 8 or for any angle α is: Tangent: The general solution of the equation 8 or for any angle α is: Find the general solution, in degrees, of the equation 3 3 is the cosine of 30. So the general solution for Ѳ is: 36 3 Therefore the general solution for Ѳ is: 7 6

18 Find the general solution, in radians to the equation 3 3 As 3 tan π 3, π 3x nπ π 3 3x x nπ π n 3 π π To simplify the equation further, allow m=-n. Therefore, x mπ 3 π 36 Find the general solution, in radians, of the equation π π sin x cos x Dividing both sides by the RHS gives: tan x π As tan () x x π nπ nπ π π

19 Numerical Solution of Equations Interval Bisection If you know there is a root of f(x)=0 between x=a and x=b, you can try. Whether this is positive or negative determines which side of the root lies. Show that the equation x 3 x 9 has a root between and. Use the method of interval bisection twice to obtain an interval of width 0. within which the root must lie. 3 9 () 9 3 () f() is negative and f() is positive, therefore a root lies between and. 87 f() is negative and f(.) is positive, therefore a root lies between and As f(.) is negative and f(.) is positive, there must be a root between. and. Linear Interpolation In solving the equation 3 9, you can deduce from () 3 and () 9 that the root of 3 9 is much more likely to be nearer to than, since () (). By plotting (, -3) and (, 9) on a graph, and connecting them by a straight line, the estimated root is +p. Using similar triangles, we can deduce: Thus the estimated root is +p which is +0.=.

20 Show that the equation has a root between 0 and. Use linear interpolation to find an approximate value of this root. Let () 3 The change of sign indicates that f(x)=0 is between x=0 and x=. Using similar triangles, with the root at x=p, we can deduce that 3 Therefore the required approximate value is 0. The Newton- Raphson Method If you are given an approximation to a root of f(x)=0, such as α, then is generally a better approximation. In its iterative form, the Newton- Raphson method for solving f(x)=0 gives:

21 Use the Newton-Raphson method once, with an initial value of x=3, to find an approximation for a root of the equation 8x x x. Give your answer to three significant figures. Let 8 Differentiation gives: 8 3. When x=3, 7 The required approximation is therefore: (3 ) 7 Step-by-Step Solution of Differential Equations The Euler formula is: This formula is used to find approximate solutions for a differential equation of the form, where f is a given function. In the equation, h is the step length, and (, ) are the points on a curve. The variables x and y satisfy the differential equation, and y= and x=. Use the Euler formula with step length 0. to find an approximation for the value of y when x=. and and 693 Thus, ( 693) 693 This is an approximation of y when x=. This time, and 7 Thus, 693 ( 7) 3 Therefore, when x=., the approximate value of y=.3

22 Linear Laws If you have results for x and y, and you need to prove that these results satisfy a formula, such as y 3 =ax +b, you will need to then calculate the values for y 3 and x, and then plot the graph of y 3 by x. The graph should be linear if it does satisfy the equation. Then, you can work out a and b by calculating the gradient and the y-intercept. : Two variables x and y, which are related by a law of the form y=ax n, can be turned into a linear relation if you use logarithms. From s Two variables x and y, which are related by a law of the form y=ab x, can be turned into a linear relation if you use logarithms.

### Math Review for AP Calculus

Math Review for AP Calculus This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet

### Algebra II Learning Targets

Chapter 0 Preparing for Advanced Algebra LT 0.1 Representing Functions Identify the domain and range of functions LT 0.2 FOIL Use the FOIL method to multiply binomials LT 0.3 Factoring Polynomials Use

### AS PURE MATHS REVISION NOTES

AS PURE MATHS REVISION NOTES 1 SURDS A root such as 3 that cannot be written exactly as a fraction is IRRATIONAL An expression that involves irrational roots is in SURD FORM e.g. 2 3 3 + 2 and 3-2 are

### Copyright 2018 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 2/10

Prep for Calculus This course covers the topics outlined below. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum (281 topics + 125 additional topics) Real

### PreCalculus. Curriculum (447 topics additional topics)

PreCalculus This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular needs.

### Pre-Calculus and Trigonometry Capacity Matrix

Review Polynomials A1.1.4 A1.2.5 Add, subtract, multiply and simplify polynomials and rational expressions Solve polynomial equations and equations involving rational expressions Review Chapter 1 and their

### ALGEBRA & TRIGONOMETRY FOR CALCULUS MATH 1340

ALGEBRA & TRIGONOMETRY FOR CALCULUS Course Description: MATH 1340 A combined algebra and trigonometry course for science and engineering students planning to enroll in Calculus I, MATH 1950. Topics include:

### Polynomial and Rational Functions. Chapter 3

Polynomial and Rational Functions Chapter 3 Quadratic Functions and Models Section 3.1 Quadratic Functions Quadratic function: Function of the form f(x) = ax 2 + bx + c (a, b and c real numbers, a 0) -30

### Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers

Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers CLASSIFICATIONS OF NUMBERS NATURAL NUMBERS = N = {1,2,3,4,...}

### Pre-Calculus and Trigonometry Capacity Matrix

Pre-Calculus and Capacity Matri Review Polynomials A1.1.4 A1.2.5 Add, subtract, multiply and simplify polynomials and rational epressions Solve polynomial equations and equations involving rational epressions

### Secondary Honors Algebra II Objectives

Secondary Honors Algebra II Objectives Chapter 1 Equations and Inequalities Students will learn to evaluate and simplify numerical and algebraic expressions, to solve linear and absolute value equations

### PreCalculus. Curriculum (637 topics additional topics)

PreCalculus This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular needs.

### WAYNESBORO AREA SCHOOL DISTRICT CURRICULUM ALGEBRA II

UNIT: Review of Basic Algebra Skills as Needed SR1 and any Supplemental Materials UNIT : What skills from Algebra I are used in Algebra II? Review Algebra I Skills as Needed SR1 and any additional resources

### UNIT 3 MATHEMATICAL METHODS ALGEBRA

UNIT 3 MATHEMATICAL METHODS ALGEBRA Substitution of Values Rearrangement and Substitution Polynomial Expressions Expanding Expressions Expanding Expressions by Rule Perfect Squares The Difference of Two

### Algebra 2 with Trigonometry Correlation of the ALEKS course Algebra 2 with Trigonometry to the Tennessee Algebra II Standards

Algebra 2 with Trigonometry Correlation of the ALEKS course Algebra 2 with Trigonometry to the Tennessee Algebra II Standards Standard 2 : Number & Operations CLE 3103.2.1: CLE 3103.2.2: CLE 3103.2.3:

### Test Codes : MIA (Objective Type) and MIB (Short Answer Type) 2007

Test Codes : MIA (Objective Type) and MIB (Short Answer Type) 007 Questions will be set on the following and related topics. Algebra: Sets, operations on sets. Prime numbers, factorisation of integers

### STEM-Prep Pathway SLOs

STEM-Prep Pathway SLOs Background: The STEM-Prep subgroup of the MMPT adopts a variation of the student learning outcomes for STEM from the courses Reasoning with Functions I and Reasoning with Functions

### Precalculus. Precalculus Higher Mathematics Courses 85

Precalculus Precalculus combines the trigonometric, geometric, and algebraic techniques needed to prepare students for the study of calculus, and strengthens students conceptual understanding of problems

### Sophomore Year: Algebra II Textbook: Algebra II, Common Core Edition Larson, Boswell, Kanold, Stiff Holt McDougal 2012

Sophomore Year: Algebra II Tetbook: Algebra II, Common Core Edition Larson, Boswell, Kanold, Stiff Holt McDougal 2012 Course Description: The purpose of this course is to give students a strong foundation

### Homework. Basic properties of real numbers. Adding, subtracting, multiplying and dividing real numbers. Solve one step inequalities with integers.

Morgan County School District Re-3 A.P. Calculus August What is the language of algebra? Graphing real numbers. Comparing and ordering real numbers. Finding absolute value. September How do you solve one

### Precalculus. Course Text. Course Description. Course Objectives. StraighterLine MAT201: Precalculus

Precalculus Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. Precalculus, 6th edition, McGraw-Hill, 2008. ISBN: 9780073312637 [This text is available as an etextbook at purchase

### STATE COUNCIL OF EDUCATIONAL RESEARCH AND TRAINING TNCF DRAFT SYLLABUS.

STATE COUNCIL OF EDUCATIONAL RESEARCH AND TRAINING TNCF 2017 - DRAFT SYLLABUS Subject :Mathematics Class : XI TOPIC CONTENT Unit 1 : Real Numbers - Revision : Rational, Irrational Numbers, Basic Algebra

### Tennessee s State Mathematics Standards Precalculus

Tennessee s State Mathematics Standards Precalculus Domain Cluster Standard Number Expressions (N-NE) Represent, interpret, compare, and simplify number expressions 1. Use the laws of exponents and logarithms

### New Jersey Quality Single Accountability Continuum (NJQSAC) A-CED 1-4; A-REI 1-2,5-7; F-IF 1-2, 4-5; N-Q 1-3; N-RN

New Jersey Quality Single Accountability Continuum (NJQSAC) (former ICM) Date: Unit 1, September 4-30 How do we use functions to solve real world problems? How can we model a real-life situation with a

### Mathematics, Algebra, and Geometry

Mathematics, Algebra, and Geometry by Satya http://www.thesatya.com/ Contents 1 Algebra 1 1.1 Logarithms............................................ 1. Complex numbers........................................

### V. Graph Sketching and Max-Min Problems

V. Graph Sketching and Max-Min Problems The signs of the first and second derivatives of a function tell us something about the shape of its graph. In this chapter we learn how to find that information.

### Π xdx cos 2 x

Π 5 3 xdx 5 4 6 3 8 cos x Help Your Child with Higher Maths Introduction We ve designed this booklet so that you can use it with your child throughout the session, as he/she moves through the Higher course,

1 Quadratic Functions and Factoring 1.1 Graph Quadratic Functions in Standard Form 1.2 Graph Quadratic Functions in Vertex or Intercept Form 1.3 Solve by Factoring 1.4 Solve by Factoring 1.5 Solve Quadratic

### Curriculum Catalog

2017-2018 Curriculum Catalog 2017 Glynlyon, Inc. Table of Contents PRE-CALCULUS COURSE OVERVIEW...1 UNIT 1: RELATIONS AND FUNCTIONS... 1 UNIT 2: FUNCTIONS... 1 UNIT 3: TRIGONOMETRIC FUNCTIONS... 2 UNIT

### sketching Jan 22, 2015 Calculus with Algebra and Trigonometry II Lecture 2Maxima andjan minima, 22, 2015 convexity/concav 1 / 15

Calculus with Algebra and Trigonometry II Lecture 2 Maxima and minima, convexity/concavity, and curve sketching Jan 22, 2015 Calculus with Algebra and Trigonometry II Lecture 2Maxima andjan minima, 22,

### A-Level Maths Revision notes 2014

A-Level Maths Revision notes 2014 Contents Coordinate Geometry... 2 Trigonometry... 4 Basic Algebra... 7 Advanced Algebra... 9 Sequences and Series... 11 Functions... 12 Differentiation... 14 Integration...

### College Algebra with Trigonometry

College Algebra with Trigonometry This course covers the topics outlined below. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum (556 topics + 614 additional

### Saxon Advanced Mathematics Scope and Sequence

hmhco.com Saxon Advanced Mathematics Scope and Sequence Foundations Calculator Perform two-variable analysis Use graphing calculators Find roots of equations Solve systems of equations Exponentials and

### GAT-UGTP-2018 Page 1 of 5

SECTION A: MATHEMATICS UNIT 1 SETS, RELATIONS AND FUNCTIONS: Sets and their representation, Union, Intersection and compliment of sets, and their algebraic properties, power set, Relation, Types of relation,

### Prentice Hall Mathematics, Algebra Correlated to: Achieve American Diploma Project Algebra II End-of-Course Exam Content Standards

Core: Operations on Numbers and Expressions Priority: 15% Successful students will be able to perform operations with rational, real, and complex numbers, using both numeric and algebraic expressions,

### Maths Higher Prelim Content

Maths Higher Prelim Content Straight Line Gradient of a line A(x 1, y 1 ), B(x 2, y 2 ), Gradient of AB m AB = y 2 y1 x 2 x 1 m = tanθ where θ is the angle the line makes with the positive direction of

### THEIR GRAPHS, AND THEIR

St. Michael Albertville High School Teacher: Kim Benson Algebra 2 (Master) August 2015 CEQs: WHAT RELATIONSHIP S EXIST BETWEEN VARIOUS FUNCTIONS, THEIR GRAPHS, AND THEIR SOLUTION(S)? HOW DO WE SIMPLIFY

### To get horizontal and slant asymptotes algebraically we need to know about end behaviour for rational functions.

Concepts: Horizontal Asymptotes, Vertical Asymptotes, Slant (Oblique) Asymptotes, Transforming Reciprocal Function, Sketching Rational Functions, Solving Inequalities using Sign Charts. Rational Function

### 2016 Notes from the Marking Centre - Mathematics

2016 Notes from the Marking Centre - Mathematics Question 11 (a) This part was generally done well. Most candidates indicated either the radius or the centre. Common sketching a circle with the correct

### Solutionbank Edexcel AS and A Level Modular Mathematics

Page of Exercise A, Question The curve C, with equation y = x ln x, x > 0, has a stationary point P. Find, in terms of e, the coordinates of P. (7) y = x ln x, x > 0 Differentiate as a product: = x + x

### Algebra 2 (2006) Correlation of the ALEKS Course Algebra 2 to the California Content Standards for Algebra 2

Algebra 2 (2006) Correlation of the ALEKS Course Algebra 2 to the California Content Standards for Algebra 2 Algebra II - This discipline complements and expands the mathematical content and concepts of

### Fundamentals of Engineering (FE) Exam Mathematics Review

Fundamentals of Engineering (FE) Exam Mathematics Review Dr. Garey Fox Professor and Buchanan Endowed Chair Biosystems and Agricultural Engineering October 16, 2014 Reference Material from FE Review Instructor

### correlated to the Indiana Academic Standards for Precalculus CC2

correlated to the Indiana Academic Standards for Precalculus CC2 6/2003 2003 Introduction to Advanced Mathematics 2003 by Richard G. Brown Advanced Mathematics offers comprehensive coverage of precalculus

### y d y b x a x b Fundamentals of Engineering Review Fundamentals of Engineering Review 1 d x y Introduction - Algebra Cartesian Coordinates

Fundamentals of Engineering Review RICHARD L. JONES FE MATH REVIEW ALGEBRA AND TRIG 8//00 Introduction - Algebra Cartesian Coordinates Lines and Linear Equations Quadratics Logs and exponents Inequalities

### Mathematics Precalculus: Academic Unit 7: Conics

Understandings Questions Knowledge Vocabulary Skills Conics are models of real-life situations. Conics have many reflective properties that are used in every day situations Conics work can be simplified

### Northampton County Schools Pre-Calculus Mathematics Pacing Guide

Northampton County Schools Pre-Calculus Mathematics 2008-2009 Pacing Guide Grading Periods Included 1 st Six Weeks 2 nd Six Weeks 3 rd Six Weeks Northampton County Schools Department of Curriculum & Instruction

### Circles. Example 2: Write an equation for a circle if the enpoints of a diameter are at ( 4,5) and (6, 3).

Conics Unit Ch. 8 Circles Equations of Circles The equation of a circle with center ( hk, ) and radius r units is ( x h) ( y k) r. Example 1: Write an equation of circle with center (8, 3) and radius 6.

### Pearson Georgia High School Mathematics Advanced Algebra

A Correlation of Pearson Georgia High School Mathematics Advanced Algebra 2014 to the Gwinnett County Academic Knowledge and Skills (AKS) Mathematics Table of Contents A - Process Skills... 1 C - Geometry...

### Algebra 2A Unit 1 Week 1 Day Activity Unit 1 Week 2 Day Activity Unit 1 Week 3 Day Activity Unit 2 Week 1 Day Activity

Algebra 2A Unit 1 Week 1 1 Pretest Unit 1 2 Evaluating Rational Expressions 3 Restrictions on Rational Expressions 4 Equivalent Forms of Rational Expressions 5 Simplifying Rational Expressions Unit 1 Week

### WA State Common Core Standards - Mathematics

Number & Quantity The Real Number System Extend the properties of exponents to rational exponents. 1. Explain how the definition of the meaning of rational exponents follows from extending the properties

### G r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s ( 3 0 S ) Final Practice Exam

G r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s ( 3 0 S ) Final Practice Exam G r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s Final Practice Exam Name: Student Number: For Marker

### Key Facts and Methods

Intermediate Maths Key Facts and Methods Use this (as well as trying questions) to revise by: 1. Testing yourself. Asking a friend or family member to test you by reading the questions (on the lefthand

### PURE MATHEMATICS Unit 1

PURE MATHEMATICS Unit 1 FOR CAPE EXAMINATIONS DIPCHAND BAHALL CAPE is a registered trade mark of the Caribbean Examinations Council (CXC). Pure Mathematics for CAPE Examinations Unit 1 is an independent

### , Precalculus, Quarter 1

2017.18, Precalculus, Quarter 1 The following Practice Standards and Literacy Skills will be used throughout the course: Standards for Mathematical Practice Literacy Skills for Mathematical Proficiency

### MATHEMATICS SYLLABUS SECONDARY 4th YEAR

European Schools Office of the Secretary-General Pedagogical Development Unit Ref.:010-D-591-en- Orig.: EN MATHEMATICS SYLLABUS SECONDARY 4th YEAR 6 period/week course APPROVED BY THE JOINT TEACHING COMMITTEE

### 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

### Pre-Calculus and Trigonometry Capacity Matrix

Information Pre-Calculus and Capacity Matri Review Polynomials A1.1.4 A1.2.5 Add, subtract, multiply and simplify polynomials and rational epressions Solve polynomial equations and equations involving

### FP1 practice papers A to G

FP1 practice papers A to G Paper Reference(s) 6667/01 Edexcel GCE Further Pure Mathematics FP1 Advanced Subsidiary Practice Paper A Time: 1 hour 30 minutes Materials required for examination Mathematical

### Honors Algebra II/Trigonometry

Honors Algebra II/Trigonometry Scranton School District Scranton, PA Honors Algebra II/Trigonometry Prerequisite: Successful completion of Geometry or Honors Geometry Be in compliance with the SSD Honors

### Summer 2017 Review For Students Entering AP Calculus AB/BC

Summer 2017 Review For Students Entering AP Calculus AB/BC Holy Name High School AP Calculus Summer Homework 1 A.M.D.G. AP Calculus AB Summer Review Packet Holy Name High School Welcome to AP Calculus

### MAP 2302 MAP 4103 MAE 3920 MAE 4360 MAS 4301 MAS Introduction to Abstract Algebra I. Introduction to Abstract Algebra

B.S. In Mathematics Florida A&M University MAC 2311 MAD 2120 MAC 2312 MAE 1920 MAC 2313 STA 2023 MHF 4202 MAE 2920 MAS 3105 MAP 2302 MAP 4103 MAS 4301 MAE 3920 MAE 4360 MTG 4212 MAS 4203 FTCE Skills &

### correlated to the Virginia Standards of Learning Algebra II with Trigonometry

correlated to the Virginia Standards of Learning Algebra II with Trigonometry AII/T.1 The student will identify field properties, axioms of equality and inequality, and properties of order that are valid

### OCR Maths FP1. Topic Questions from Papers. Complex Numbers. Answers

OCR Maths FP1 Topic Questions from Papers Complex Numbers Answers PhysicsAndMathsTutor.com . 1 (i) i Correct real and imaginary parts z* = i 1i Correct conjugate seen or implied Correct real and imaginary

### SUBJECT: ADDITIONAL MATHEMATICS CURRICULUM OUTLINE LEVEL: 3 TOPIC OBJECTIVES ASSIGNMENTS / ASSESSMENT WEB-BASED RESOURCES. Online worksheet.

TERM 1 Simultaneous Online worksheet. Week 1 Equations in two Solve two simultaneous equations where unknowns at least one is a linear equation, by http://www.tutorvista.com/mat substitution. Understand

### Integrated Mathematics I, II, III 2016 Scope and Sequence

Mathematics I, II, III 2016 Scope and Sequence I Big Ideas Math 2016 Mathematics I, II, and III Scope and Sequence Number and Quantity The Real Number System (N-RN) Properties of exponents to rational

### Muskogee Public Schools Curriculum Map

Muskogee Public Schools Curriculum Map 2009-20010 Course: Algebra II Grade Level: 9-12 Nine- 1 st Nine Standard 1: Number Systems and Algebraic Operations - The student will perform operations with rational,

### Algebra 2 with Trigonometry

Algebra 2 with Trigonometry This course covers the topics shown below; new topics have been highlighted. Students navigate learning paths based on their level of readiness. Institutional users may customize

### TRIGONOMETRIC RATIOS AND GRAPHS

Mathematics Revision Guides Trigonometric Ratios and Graphs Page 1 of 15 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 TRIGONOMETRIC RATIOS

### Aldine I.S.D. Benchmark Targets/ Algebra 2 SUMMER 2004

ASSURANCES: By the end of Algebra 2, the student will be able to: 1. Solve systems of equations or inequalities in two or more variables. 2. Graph rational functions and solve rational equations and inequalities.

### Centerville High School Curriculum Mapping Algebra II 1 st Nine Weeks

Centerville High School Curriculum Mapping Algebra II 1 st Nine Weeks Chapter/ Lesson Common Core Standard(s) 1-1 SMP1 1. How do you use a number line to graph and order real numbers? 2. How do you identify

### abc Mathematics Further Pure General Certificate of Education SPECIMEN UNITS AND MARK SCHEMES

abc General Certificate of Education Mathematics Further Pure SPECIMEN UNITS AND MARK SCHEMES ADVANCED SUBSIDIARY MATHEMATICS (56) ADVANCED SUBSIDIARY PURE MATHEMATICS (566) ADVANCED SUBSIDIARY FURTHER

### Indiana Academic Standards for Precalculus

PRECALCULUS correlated to the Indiana Academic Standards for Precalculus CC2 6/2003 2004 Introduction to Precalculus 2004 by Roland E. Larson and Robert P. Hostetler Precalculus thoroughly explores topics

### Algebra 2. Chapter 4 Exponential and Logarithmic Functions. Chapter 1 Foundations for Functions. Chapter 3 Polynomial Functions

Algebra 2 Chapter 1 Foundations for Chapter 2 Quadratic Chapter 3 Polynomial Chapter 4 Exponential and Logarithmic Chapter 5 Rational and Radical Chapter 6 Properties and Attributes of Chapter 7 Probability

### Fundamental Theorem of Algebra (NEW): A polynomial function of degree n > 0 has n complex zeros. Some of these zeros may be repeated.

.5 and.6 Comple Numbers, Comple Zeros and the Fundamental Theorem of Algebra Pre Calculus.5 COMPLEX NUMBERS 1. Understand that - 1 is an imaginary number denoted by the letter i.. Evaluate the square root

### Curriculum Guide Algebra 2 Advanced

Unit 1: Equations and Inequalities Biblical Worldview Essential Questions: Is your life balanced as a believer? Are you a real Christian? 13 Lessons A2#1, A2#2 1. Use a number line to graph and order real

### Accuplacer College Level Math Study Guide

Testing Center Student Success Center Accuplacer Study Guide The following sample questions are similar to the format and content of questions on the Accuplacer College Level Math test. Reviewing these

### 1 a) Remember, the negative in the front and the negative in the exponent have nothing to do w/ 1 each other. Answer: 3/ 2 3/ 4. 8x y.

AP Calculus Summer Packer Key a) Remember, the negative in the front and the negative in the eponent have nothing to do w/ each other. Answer: b) Answer: c) Answer: ( ) 4 5 = 5 or 0 /. 9 8 d) The 6,, and

### Mathematics Three Points

Subject Pre- Mathematics Three Points and Statistics Probability Topics Unit 1 Basic operations, Fractions, Factors, Arithmetic Progression, Quadratic Equations, Reading Graphs, Profit and Loss Percentages,

### Algebra II. Key Resources: Page 3

Algebra II Course This course includes the study of a variety of functions (linear, quadratic higher order polynomials, exponential, absolute value, logarithmic and rational) learning to graph, compare,

### Pure Core 2. Revision Notes

Pure Core Revision Notes June 06 Pure Core Algebra... Polynomials... Factorising... Standard results... Long division... Remainder theorem... 4 Factor theorem... 5 Choosing a suitable factor... 6 Cubic

### Pre-Calculus Mathematics Curriculum

Pre-Calculus Mathematics Curriculum First day introductions, materials, policies, procedures and Summer Exam (2 days) Unit 1 Estimated time frame for unit 1 Big Ideas Essential Question Competencies Lesson

### MODULE 1: FOUNDATIONS OF MATHEMATICS

MODULE 1: FOUNDATIONS OF MATHEMATICS GENERAL OBJECTIVES On completion of this Module, students should: 1. acquire competency in the application of algebraic techniques; 2. appreciate the role of exponential

### ADVANCED PROGRAMME MATHEMATICS: PAPER I MODULE 1: CALCULUS AND ALGEBRA

GRADE 1 EXAMINATION NOVEMBER 017 ADVANCED PROGRAMME MATHEMATICS: PAPER I MODULE 1: CALCULUS AND ALGEBRA Time: hours 00 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists

### Algebra II CP Final Exam Review Packet. Calculator Questions

Name: Algebra II CP Final Exam Review Packet Calculator Questions 1. Solve the equation. Check for extraneous solutions. (Sec. 1.6) 2 8 37 2. Graph the inequality 31. (Sec. 2.8) 3. If y varies directly

### ALGEBRA II (COMMON CORE) FACTS YOU MUST KNOW COLD FOR THE REGENTS EXAM

ALGEBRA II (COMMON CORE) FACTS YOU MUST KNOW COLD FOR THE REGENTS EXAM NYS Mathematics Regents Preparation Created by Trevor Clark Algebra II [Common Core] Regents Exam Study Guide ALGEBRA & FUNCTIONS

### Benchmark Computation The student models, performs, and explains computation with complex numbers and polynomials in a variety of situations.

Standard Number and Computation The student uses numerical and computational concepts and procedures in a variety of situations. Indicator/Objective Critical Vocabulary Knowledge Indicators The student

### Pure Core 1. Revision Notes

Pure Core Revision Notes Ma 06 Pure Core Algebra... Indices... Rules of indices... Surds... 4 Simplifing surds... 4 Rationalising the denominator... 4 Quadratic functions... 5 Completing the square....

### Lone Star College-CyFair Formula Sheet

Lone Star College-CyFair Formula Sheet The following formulas are critical for success in the indicated course. Student CANNOT bring these formulas on a formula sheet or card to tests and instructors MUST

### TARGET : JEE 2013 SCORE. JEE (Advanced) Home Assignment # 03. Kota Chandigarh Ahmedabad

TARGT : J 01 SCOR J (Advanced) Home Assignment # 0 Kota Chandigarh Ahmedabad J-Mathematics HOM ASSIGNMNT # 0 STRAIGHT OBJCTIV TYP 1. If x + y = 0 is a tangent at the vertex of a parabola and x + y 7 =

### Circle Theorems. Angles at the circumference are equal. The angle in a semi-circle is x The angle at the centre. Cyclic Quadrilateral

The angle in a semi-circle is 90 0 Angles at the circumference are equal. A B They must come from the same arc. Look out for a diameter. 2x Cyclic Quadrilateral Opposite angles add up to 180 0 A They must

### 1. 4 2y 1 2 = x = x 1 2 x + 1 = x x + 1 = x = 6. w = 2. 5 x

.... VII x + x + = x x x 8 x x = x + a = a + x x = x + x x Solve the absolute value equations.. z = 8. x + 7 =. x =. x =. y = 7 + y VIII Solve the exponential equations.. 0 x = 000. 0 x+ = 00. x+ = 8.

### Composition of and the Transformation of Functions

1 3 Specific Outcome Demonstrate an understanding of operations on, and compositions of, functions. Demonstrate an understanding of the effects of horizontal and vertical translations on the graphs of

### + 2gx + 2fy + c = 0 if S

CIRCLE DEFINITIONS A circle is the locus of a point which moves in such a way that its distance from a fixed point, called the centre, is always a constant. The distance r from the centre is called the

### Algebra II. A2.1.1 Recognize and graph various types of functions, including polynomial, rational, and algebraic functions.

Standard 1: Relations and Functions Students graph relations and functions and find zeros. They use function notation and combine functions by composition. They interpret functions in given situations.

### Mathematics syllabus for Grade 11 and 12 For Bilingual Schools in the Sultanate of Oman

03 04 Mathematics syllabus for Grade and For Bilingual Schools in the Sultanate of Oman Prepared By: A Stevens (Qurum Private School) M Katira (Qurum Private School) M Hawthorn (Al Sahwa Schools) In Conjunction

### Grade 11 or 12 Pre-Calculus

Grade 11 or 12 Pre-Calculus Strands 1. Polynomial, Rational, and Radical Relationships 2. Trigonometric Functions 3. Modeling with Functions Strand 1: Polynomial, Rational, and Radical Relationships Standard

### MATHia Unit MATHia Workspace Overview TEKS

1 Function Overview Searching for Patterns Exploring and Analyzing Patterns Comparing Familiar Function Representations Students watch a video about a well-known mathematician creating an expression for