5.2 Solving Quadratic Equations by Factoring
|
|
- Dominick Williams
- 5 years ago
- Views:
Transcription
1 Name. Solving Quadratic Equations b Factoring MATHPOWER TM, Ontario Edition, pp To solve a quadratic equation b factoring, a) write the equation in the form a + b + c = b) factor a + b + c c) use the zero product propert d) solve the two resulting equations to find the roots e) check our solutions. State the roots of each equation. a) ( )( + 7) = b) ( + )( ) = c) 7( ) = d) ( + )( + ) = Problem Solving. The hpotenuse of a right triangle is m longer than twice one of the other two sides. The third side of the triangle is m. Find the lengths of the unknown side and hpotenuse.. Write each equation in the form a + b + c =. a) = ( ) b) ( ) = + 7 c) = d) =. The length of a flag is twice the width. Find the width of a flag with an area of cm.. Solve and check. a) + 8 = b) + + =. Three consecutive even integers are such that the product of the first two is less than 9 times the third. Find the integers. c) q q 8 = d) m + m 7 = e) t 7t + = f) = g) t t = h) p p = 7. A rectangle is cm long and cm wide. When each dimension is increased b the same amount, the area is doubled. What are the new dimensions? i) ( ) = 7 + j) a a = k) ( ) = l) =. 8. Communication Write a quadratic equation whose roots are and Eplain our. thinking. Chapter Copright McGraw-Hill Rerson Limited
2 Name. Investigation: Graphing Quadratic Functions b Factoring MATHPOWER TM, Ontario Edition, p. 87 To sketch the graph of a quadratic function, a) find the -intercepts b factoring the equation in the form a + b + c = b) plot the points where the graph crosses the -ais, ( = ) c) use smmetr to find the -coordinate of the verte d) find the -coordinate of the verte b substituting the -coordinate of the verte into the equation in the form = a + b + c e) draw a smooth curve through the three points. Sketch the graphs of the following quadratic functions b locating the -intercepts, and then finding the coordinates of the verte. a) = ( )( ) b) = e) = + f) = 8. Use the -intercepts to determine the coordinates of the points on the -ais and the verte for the graph of each quadratic function. a) = ( + )( ) b) = 9 g) = + h) = + +. Sketch the graphs of the following quadratic functions b factoring to find the -intercepts, and then deducing the coordinates of the verte. a) = + b) = 7 +. Communication Eplain how to sketch a graph of each function using the intercepts and ais of smmetr. a) = ( ) c) = + d) =. b) = ( + ) Copright McGraw-Hill Rerson Limited Chapter
3 . The Quadratic Formula Name MATHPOWER TM, Ontario Edition, pp To solve a quadratic equation using the quadratic formula, write the equation in the form a + b + c =, a. b b ac The quadratic formula is = ±. a. Solve using the quadratic formula. a) 8 + = b) =. Application Use the Pthagorean Theorem to find the value of, to the nearest hundredth. + c) + 7 = d) =. Solve using the quadratic formula. Epress answers as eact roots. a) + = b) + + = 7 Problem Solving. The sum of the squares of three consecutive odd integers is 87. Find the integers. c) 8 = d) ( + 8) =. A plain mat is placed around a picture measuring 8 cm b cm so that the width of the mat is equal on all sides. The area of the mat is of the area of the picture. Find the width of the mat, to the nearest millimetre. e) b + b = f) + =. Solve using the quadratic formula. Round to the nearest hundredth, if necessar. a) = b) 9 = 7. A window is in the shape of a rectangle surmounted b a semicircle. The height of the rectangle is. m more than the width. The total area of the window is. m. Find the width and height of the window, to the nearest hundredth. c).( + ) =. d) = e) a = f) ( + ) + ( + ) = 8. Communication Is it possible to write two real numbers whose sum is and whose product is? Use the quadratic formula to help ou eplain. Chapter Copright McGraw-Hill Rerson Limited
4 Answers CHAPTER Quadratic Equations. Solving Quadratic Equations b Graphing. a) b) = 7 +. a), b), c), d), = + e). f)., g),. h) no solution i), j).,.. s. 8 m b m.,, or,, (, ) (, ) (, ) c) d) (, ) (, ) = + (, ) (, ) 7, (, 8). m b m b m =.. cm, 9 cm. Solving Quadratic Equations b Factoring. a), 7 b), c), d). a) + = b) 7 = c) = d) =. a), b), c), 7 d) 9, e) 7, f) g), h), 8 i) j), k),, l),. 8 m, 7 m. cm.,, 7. cm b cm 8. = and = so ( + ) Epand the =., equation to get + =. Then, multipl ever term b to eliminate the denominators, to get + = or + =.. Investigation: Graphing Quadratic Functions b Factoring. a) b) = ( )( ) (, ) (, ) (, ) (, ) (, ) = (, ) (, ) (, ) (, ) e) f) ( 7, ) (, ) = + (, ) g) h) (, ) (, ) (, ) = + (, ) (, ) = 8, = + + (, ) (, ),. a) The equation is in factored form, so there is onl one -intercept,. Plot (, ). To find the -intercept, let = and find. Plot (, ). A third point is smmetrical to (, ) about the ais of smmetr =. Plot (, ). Check b substituting (, ) into the equation.. a) (, ), (, ), (, ) b) ( 7, ), (7, ), (, 9) Copright McGraw-Hill Rerson Limited Chapter
5 b) There is one -intercept,. Plot (, ). To find the -intercept, let = and find. Plot (, ). A third point is smmetrical to (, ) about the ais of smmetr =. Plot (, ). Check b substituting (, ) into the equation.. The Quadratic Formula. a), b) c) 7,, d). a) ± b) ± c) d) ± e) f) no solution. a).7,.7 b).9,. c) 7.7,. d) no solution e) ±. f).9,.9. =.98., 7, 9, or 9, 7,.. cm or mm 7..9 m wide,.99 m high 8. If one number is and the other number is, the product of the two numbers, ( ), should equal. Tr to solve + =. There are no real solutions. ± 8± 7, 8 Chapter Copright McGraw-Hill Rerson Limited
Math 030 Review for Final Exam Revised Fall 2010 RH/ DM 1
Math 00 Review for Final Eam Revised Fall 010 RH/ DM 1 1. Solve the equations: (-1) (7) (-) (-1) () 1 1 1 1 f. 1 g. h. 1 11 i. 9. Solve the following equations for the given variable: 1 Solve for. D ab
More information9-1. The Function with Equation y = ax 2. Vocabulary. Graphing y = x 2. Lesson
Chapter 9 Lesson 9-1 The Function with Equation = a BIG IDEA The graph of an quadratic function with equation = a, with a 0, is a parabola with verte at the origin. Vocabular parabola refl ection-smmetric
More informationWrite each expression in terms of i : Add: (3 4i) (5 7i) (3 5) ( 4 7)i. 8 3i. Subtract: (3 4i) (5 7i) (3 4i) ( 5 7i) Find each product:
7_Ch09_online 7// 0:7 AM Page 9-0 9-0 CHAPTER 9 Quadratic Equations SECTION 9. Comple Numbers DEFINITIONS AND CONCEPTS EXAMPLES The imaginar number i is defined as Write each epression in terms of i :
More informationf(x) = 2x 2 + 2x - 4
4-1 Graphing Quadratic Functions What You ll Learn Scan the tet under the Now heading. List two things ou will learn about in the lesson. 1. Active Vocabular 2. New Vocabular Label each bo with the terms
More information17. f(x) = x 2 + 5x f(x) = x 2 + x f(x) = x 2 + 3x f(x) = x 2 + 3x f(x) = x 2 16x f(x) = x 2 + 4x 96
Section.3 Zeros of the Quadratic 473.3 Eercises In Eercises 1-8, factor the given quadratic polnomial. 1. 2 + 9 + 14 2. 2 + 6 + 3. 2 + + 9 4. 2 + 4 21. 2 4 6. 2 + 7 8 7. 2 7 + 12 8. 2 + 24 In Eercises
More informationMathematics 10 Page 1 of 7 The Quadratic Function (Vertex Form): Translations. and axis of symmetry is at x a.
Mathematics 10 Page 1 of 7 Verte form of Quadratic Relations The epression a p q defines a quadratic relation called the verte form with a horizontal translation of p units and vertical translation of
More informationAnswers Investigation 2
Applications 1. a. Square Side Area 4 16 2 6 36 7 49 64 9 1 10 100 11 121 Rectangle Length Width Area 0 0 9 1 9 10 2 20 11 3 33 12 4 4 13 6 14 6 4 1 7 10 b. (See Figure 1.) c. The graph and the table both
More information2 nd Semester Final Exam Review Block Date
Algebra 1B Name nd Semester Final Eam Review Block Date Calculator NOT Allowed Graph each function. 1 (10-1) 1. (10-1). (10-1) 3. (10-1) 4. 3 Graph each function. Identif the verte, ais of smmetr, and
More informationReview of Essential Skills and Knowledge
Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope
More informationLESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II
LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,
More informationVertex. March 23, Ch 9 Guided Notes.notebook
March, 07 9 Quadratic Graphs and Their Properties A quadratic function is a function that can be written in the form: Verte Its graph looks like... which we call a parabola. The simplest quadratic function
More informationNational 5 Mathematics
St Andrew s Academ Mathematics Department National 5 Mathematics UNIT 4 ASSESSMENT PREPARATION St Andrew's Academ Maths Dept 016-17 1 Practice Unit Assessment 4A for National 5 1. Simplif, giving our answer
More informationMATH 115: Final Exam Review. Can you find the distance between two points and the midpoint of a line segment? (1.1)
MATH : Final Eam Review Can ou find the distance between two points and the midpoint of a line segment? (.) () Consider the points A (,) and ( 6, ) B. (a) Find the distance between A and B. (b) Find the
More informationPre-Calculus B Semester 1 Review Packet December 2015
Pre-Calculus B Semester Review Packet December 05 Name DISCLAIMER The memor on all calculators will be cleared the da of the final. If ou have programs on our calculator that ou would like to keep, please
More informationFinal Exam Review Part 2 #1 Page 1 / 21
Final Eam Review Part #1 Intermediate Algebra / MAT 135 Spring 017 Master ( Master Templates) Student Name/ID: v 1. Solve for, where is a real number. v v + 1 + =. Solve for, where is a real number. +
More informationAlgebra II Notes Unit Five: Quadratic Functions. Syllabus Objectives: 5.1 The student will graph quadratic functions with and without technology.
Sllabus Objectives:.1 The student will graph quadratic functions with and without technolog. Quadratic Function: a function that can be written in the form are real numbers Parabola: the U-shaped graph
More informationSection 2.5: Graphs of Functions
Section.5: Graphs of Functions Objectives Upon completion of this lesson, ou will be able to: Sketch the graph of a piecewise function containing an of the librar functions. o Polnomial functions of degree
More informationFinal Exam Review Part 2 #4
Final Eam Review Part # Intermediate Algebra / MAT 135 Fall 01 Master (Prof. Fleischner) Student Name/ID: 1. Solve for, where is a real number. + = 8. Solve for, where is a real number. 9 1 = 3. Solve
More informationFinal Exam Review Part 2 #4
Final Eam Review Part # Intermediate Algebra / MAT 135 Fall 01 Master (Prof. Fleischner) Student Name/ID: 1. Solve for, where is a real number. + =. Solve for, where is a real number. 9 1 = 3. Solve for,
More informationmath FALL developmental mathematics sullivan 1e
TSIpractice eam review 1 131 180 plus 34 TSI questions for elementar and intermediate algebra m0300004301 aaa Name www.alvarezmathhelp.com math0300004301 FALL 01 100 interactmath developmental mathematics
More informationLesson 9.1 Using the Distance Formula
Lesson. Using the Distance Formula. Find the eact distance between each pair of points. a. (0, 0) and (, ) b. (0, 0) and (7, ) c. (, 8) and (, ) d. (, ) and (, 7) e. (, 7) and (8, ) f. (8, ) and (, 0)
More informationMath 0210 Common Final Review Questions (2 5 i)(2 5 i )
Math 0 Common Final Review Questions In problems 1 6, perform the indicated operations and simplif if necessar. 1. ( 8)(4) ( )(9) 4 7 4 6( ). 18 6 8. ( i) ( 1 4 i ) 4. (8 i ). ( 9 i)( 7 i) 6. ( i)( i )
More informationChapter 9 Prerequisite Skills
Name: Date: Chapter 9 Prerequisite Skills BLM 9. Consider the function f() 3. a) Show that 3 is a factor of f(). If f() ( 3)g(), what is g()?. Factor each epression fully. a) 30g 4g 6fg 8g c) 6 5 d) 5
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polnomial Degree and Finite Differences 1. Identif the degree of each polnomial. a. 1 b. 0. 1. 3. 3 c. 0 16 0. Determine which of the epressions are polnomials. For each polnomial, state its
More informationreview math0410 (1-174) and math 0320 ( ) aafinm mg
Eam Name review math04 (1-174) and math 0320 (17-243) 03201700aafinm0424300 mg MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Simplif. 1) 7 2-3 A)
More informationPolynomial and Rational Functions
Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define
More informationWriting Quadratic Functions in Standard Form
Chapter Summar Ke Terms standard form (general form) of a quadratic function (.1) parabola (.1) leading coefficient (.) second differences (.) vertical motion model (.3) zeros (.3) interval (.3) open interval
More informationLast modified Spring 2016
Math 00 Final Review Questions In problems 6, perform the indicated operations and simplif if necessar.. 8 6 8. 7 6. ( i) ( 4 i) 4. (8 i). ( 9 i)( 7 i) 6. ( i)( i) In problems 7-, solve the following applications.
More informationGeometry and Honors Geometry Summer Review Packet 2014
Geometr and Honors Geometr Summer Review Packet 04 This will not be graded. It is for our benefit onl. The problems in this packet are designed to help ou review topics from previous mathematics courses
More informationLESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II
1 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The
More informationALGEBRA 1 CP FINAL EXAM REVIEW
ALGEBRA CP FINAL EXAM REVIEW Alg CP Sem Eam Review 0 () Page of 8 Chapter 8: Eponents. Write in rational eponent notation. 7. Write in radical notation. Simplif the epression.. 00.. 6 6. 7 7. 6 6 8. 8
More informationThe Coordinate Plane. Circles and Polygons on the Coordinate Plane. LESSON 13.1 Skills Practice. Problem Set
LESSON.1 Skills Practice Name Date The Coordinate Plane Circles and Polgons on the Coordinate Plane Problem Set Use the given information to show that each statement is true. Justif our answers b using
More informationGlossary. Also available at BigIdeasMath.com: multi-language glossary vocabulary flash cards. An equation that contains an absolute value expression
Glossar This student friendl glossar is designed to be a reference for ke vocabular, properties, and mathematical terms. Several of the entries include a short eample to aid our understanding of important
More information+ = + + = x = + = + = 36x
Ch 5 Alg L Homework Worksheets Computation Worksheet #1: You should be able to do these without a calculator! A) Addition (Subtraction = add the opposite of) B) Multiplication (Division = multipl b the
More informationUNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS
Answer Ke Name: Date: UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS Part I Questions. For the quadratic function shown below, the coordinates of its verte are, (), 7 6,, 6 The verte is
More informationUNIT #9 ROOTS AND IRRATIONAL NUMBERS REVIEW QUESTIONS
Answer Key Name: Date: UNIT #9 ROOTS AND IRRATIONAL NUMBERS REVIEW QUESTIONS Part I Questions. Which of the following is the value of 6? () 6 () 4 () (4). The epression is equivalent to 6 6 6 6 () () 6
More informationRev Name Date. Solve each of the following equations for y by isolating the square and using the square root property.
Rev 8-8-3 Name Date TI-8 GC 3 Using GC to Graph Parabolae that are Not Functions of Objectives: Recall the square root propert Practice solving a quadratic equation f Graph the two parts of a hizontal
More informationVertex form of a quadratic equation
Verte form of a quadratic equation Nikos Apostolakis Spring 017 Recall 1. Last time we looked at the graphs of quadratic equations in two variables. The upshot was that the graph of the equation: k = a(
More informationChapter 8 Vocabulary Check
28 CHAPTER 8 Quadratic Equations and Functions d. What is the level of methane emissions for that ear? (Use our rounded answer from part (c).) (Round this answer to 2 decimals places.) Use a graphing calculator
More informationMth 95 Module 4 Chapter 8 Spring Review - Solving quadratic equations using the quadratic formula
Mth 95 Module 4 Chapter 8 Spring 04 Review - Solving quadratic equations using the quadratic formula Write the quadratic formula. The NUMBER of REAL and COMPLEX SOLUTIONS to a quadratic equation ( a b
More informationMaintaining Mathematical Proficiency
Name Date Chapter 8 Maintaining Mathematical Proficienc Graph the linear equation. 1. = 5. = + 3 3. 1 = + 3. = + Evaluate the epression when =. 5. + 8. + 3 7. 3 8. 5 + 8 9. 8 10. 5 + 3 11. + + 1. 3 + +
More informationMath Intermediate Algebra
Math 095 - Intermediate Algebra Final Eam Review Objective 1: Determine whether a relation is a function. Given a graphical, tabular, or algebraic representation for a function, evaluate the function and
More informationLesson 4.1 Exercises, pages
Lesson 4.1 Eercises, pages 57 61 When approimating answers, round to the nearest tenth. A 4. Identify the y-intercept of the graph of each quadratic function. a) y = - 1 + 5-1 b) y = 3-14 + 5 Use mental
More informationSolutions to the Math 1051 Sample Final Exam (from Spring 2003) Page 1
Solutions to the Math 0 Sample Final Eam (from Spring 00) Page Part : Multiple Choice Questions. Here ou work out the problems and then select the answer that matches our answer. No partial credit is given
More informationREVIEW. cos 4. x x x on (0, x y x y. 1, if x 2
Math ` Part I: Problems REVIEW Simplif (without the use of calculators). log. ln e. cos. sin (cos ). sin arccos( ). k 7. k log (sec ) 8. cos( )cos 9. ( ) 0. log (log) Solve the following equations/inequalities.
More informationMath 121. Practice Questions Chapters 2 and 3 Fall Find the other endpoint of the line segment that has the given endpoint and midpoint.
Math 11. Practice Questions Chapters and 3 Fall 01 1. Find the other endpoint of the line segment that has the given endpoint and midpoint. Endpoint ( 7, ), Midpoint (, ). Solution: Let (, ) denote the
More informationLesson 10.1 Solving Quadratic Equations
Lesson 10.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with each set of conditions. a. One -intercept and all nonnegative y-values b. The verte in the third quadrant and no
More informationExam 2 Review F15 O Brien. Exam 2 Review:
Eam Review:.. Directions: Completely rework Eam and then work the following problems with your book notes and homework closed. You may have your graphing calculator and some blank paper. The idea is to
More informationQUADRATIC FUNCTIONS AND COMPLEX NUMBERS
CHAPTER 86 5 CHAPTER TABLE F CNTENTS 5- Real Roots of a Quadratic Equation 5-2 The Quadratic Formula 5-3 The Discriminant 5-4 The Comple Numbers 5-5 perations with Comple Numbers 5-6 Comple Roots of a
More informationMay 27, QUADRATICS.notebook. Apr 26 17:43. Apr 26 18:27. Apr 26 18:40. Apr 28 10:22. Apr 28 10:34. Apr 28 10:33. Starter
1. Factorise: 2 - - 6 2. Solve for : 2( + 1) = - 1 3. Factorise: 2-25 To solve quadratic equations.. Factorise: 2 2-8 5. State the gradient of the line: + 12 = 2 Apr 26 17:3 Apr 26 18:27 Solving Quadratic
More informationAlgebra 2 Honors Summer Packet 2018
Algebra Honors Summer Packet 018 Solving Linear Equations with Fractional Coefficients For these problems, ou should be able to: A) determine the LCD when given two or more fractions B) solve a linear
More information2 nd Semester Final Exam Review Block Date
Algebra 1B Name nd Semester Final Eam Review Block Date Calculator NOT Allowed Graph each function. Identif the verte and ais of smmetr. 1 (10-1) 1. (10-1). 3 (10-) 3. 4 7 (10-) 4. 3 6 4 (10-1) 5. Predict
More informationLESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II
1 LESSON #4 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The
More informationStudy Guide and Intervention
6- NAME DATE PERID Stud Guide and Intervention Graphing Quadratic Functions Graph Quadratic Functions Quadratic Function A function defined b an equation of the form f () a b c, where a 0 b Graph of a
More informationQUADRATIC GRAPHS ALGEBRA 2. Dr Adrian Jannetta MIMA CMath FRAS INU0114/514 (MATHS 1) Quadratic Graphs 1/ 16 Adrian Jannetta
QUADRATIC GRAPHS ALGEBRA 2 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Quadratic Graphs 1/ 16 Adrian Jannetta Objectives Be able to sketch the graph of a quadratic function Recognise the shape
More informationBRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE. MTH 05 Review Sheet
BRONX COMMUNITY COLLEGE of the Cit Universit of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE MTH 05 Review Sheet Go to http://www.cun.edu/testing for more information on the CUNY Elementar Algebra
More informationUnit 2 Notes Packet on Quadratic Functions and Factoring
Name: Period: Unit Notes Packet on Quadratic Functions and Factoring Notes #: Graphing quadratic equations in standard form, verte form, and intercept form. A. Intro to Graphs of Quadratic Equations: a
More information4 B. 4 D. 4 F. 3. How can you use the graph of a quadratic equation to determine the number of real solutions of the equation?
3.1 Solving Quadratic Equations COMMON CORE Learning Standards HSA-SSE.A. HSA-REI.B.b HSF-IF.C.8a Essential Question Essential Question How can ou use the graph of a quadratic equation to determine the
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Math 2 Stud Guide-Chapters 8 and 9 Name Date: Time: MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find all square roots of the number. ) 600 9,
More informationAnswers Investigation 3
Answers Investigation Applications. a., b. s = n c. The numbers seem to be increasing b a greater amount each time. The square number increases b consecutive odd integers:,, 7,, c X X=. a.,,, b., X 7 X=
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 1 b. 0.2 1. 2 3.2 3 c. 20 16 2 20 2. Determine which of the epressions are polynomials. For each polynomial,
More informationUNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives
Chapter 3 3Quadratics Objectives To recognise and sketch the graphs of quadratic polnomials. To find the ke features of the graph of a quadratic polnomial: ais intercepts, turning point and ais of smmetr.
More informationNAME DATE PERIOD. Study Guide and Intervention. Solving Quadratic Equations by Graphing. 2a = -
NAME DATE PERID - Study Guide and Intervention Solving Quadratic Equations by Graphing Solve Quadratic Equations Quadratic Equation A quadratic equation has the form a + b + c = 0, where a 0. Roots of
More informationBasic Fraction and Integer Operations (No calculators please!)
P1 Summer Math Review Packet For Students entering Geometry The problems in this packet are designed to help you review topics from previous mathematics courses that are important to your success in Geometry.
More informationReteaching (continued)
Quadratic Functions and Transformations If a, the graph is a stretch or compression of the parent function b a factor of 0 a 0. 0 0 0 0 0 a a 7 The graph is a vertical The graph is a vertical compression
More informationPrecalculus Honors - AP Calculus A Information and Summer Assignment
Precalculus Honors - AP Calculus A Information and Summer Assignment General Information: Competenc in Algebra and Trigonometr is absolutel essential. The calculator will not alwas be available for ou
More informationx Radical Sign: Radicand: the number beneath the radical sign
Sllabus Objective: 9.4 The student will solve quadratic equations using graphic and algebraic techniques to include the quadratic formula, square roots, factoring, completing the square, and graphing.
More informationAdd or subtract. Write the answer in lowest terms. 1) 15 13x 6. 13x A) 9 13x. 9 26x D) 9. B) 13x 9. 2) 2 r + 5 r 9 A) 18r 7 r(9 r) 7r 18 r(9 r)
Fall 2018 Preview Eam 4 Math 1 TTh This is intended onl as a resource to assist in our studing. This preview does not constitute acontract of inclusion and eclusion. MULTIPLE CHOICE. Choose the one alternative
More informationFind the dimensions of rectangle ABCD of maximum area.
QURTIS (hapter 1) 47 7 Infinitel man rectangles ma e inscried within the right angled triangle shown alongside. One of them is illustrated. a Let = cm and = cm. Use similar triangles to find in terms of.
More informationN x. You should know how to decompose a rational function into partial fractions.
Section 7. Partial Fractions 0. 0 7 0 0 0 0 Solution:, 0 Equation Equation Eq. Eq. 07. nswers will var. Section 7. Partial Fractions N You should know how to decompose a rational function into partial
More information12x y (4) 2x y (4) 5x y is the same as
Name: Unit #6 Review Quadratic Algebra Date: 1. When 6 is multiplied b the result is 0 1 () 9 1 () 9 1 () 1 0. When is multiplied b the result is 10 6 1 () 7 1 () 7 () 10 6. Written without negative eponents
More informationVocabulary. The Pythagorean Identity. Lesson 4-3. Pythagorean Identity Theorem. Mental Math
Lesson 4-3 Basic Basic Trigonometric Identities Identities Vocabular identit BIG IDEA If ou know cos, ou can easil fi nd cos( ), cos(90º - ), cos(180º - ), and cos(180º + ) without a calculator, and similarl
More informationToolkit of Basic Function Families Solutions
I. Linear Functions a. Domain and range: All real numbers unless a = 0, then the range is onl b. b. Graph patterns: Straight lines with slope a and -intercept (0, b) c. Table patterns: Constant additive
More informationMini-Lecture 8.1 Solving Quadratic Equations by Completing the Square
Mini-Lecture 8.1 Solving Quadratic Equations b Completing the Square Learning Objectives: 1. Use the square root propert to solve quadratic equations.. Solve quadratic equations b completing the square.
More informationCollege Algebra Final, 7/2/10
NAME College Algebra Final, 7//10 1. Factor the polnomial p() = 3 5 13 4 + 13 3 + 9 16 + 4 completel, then sketch a graph of it. Make sure to plot the - and -intercepts. (10 points) Solution: B the rational
More informationAlgebra 2 CPA Summer Assignment 2018
Algebra CPA Summer Assignment 018 This assignment is designed for ou to practice topics learned in Algebra 1 that will be relevant in the Algebra CPA curriculum. This review is especiall important as ou
More informationNorthwest High School s Algebra 2/Honors Algebra 2
Northwest High School s Algebra /Honors Algebra Summer Review Packet 0 DUE Frida, September, 0 Student Name This packet has been designed to help ou review various mathematical topics that will be necessar
More informationAlgebra I Quadratics Practice Questions
1. Which is equivalent to 64 100? 10 50 8 10 8 100. Which is equivalent to 6 8? 4 8 1 4. Which is equivalent to 7 6? 4 4 4. Which is equivalent to 4? 8 6 From CCSD CSE S Page 1 of 6 1 5. Which is equivalent
More informationThe standard form of the equation of a circle is based on the distance formula. The distance formula, in turn, is based on the Pythagorean Theorem.
Unit, Lesson. Deriving the Equation of a Circle The graph of an equation in and is the set of all points (, ) in a coordinate plane that satisf the equation. Some equations have graphs with precise geometric
More informationLearning Goals. College of Charleston Department of Mathematics Math 101: College Algebra Final Exam Review Problems 1
College of Charleston Department of Mathematics Math 0: College Algebra Final Eam Review Problems Learning Goals (AL-) Arithmetic of Real and Comple Numbers: I can classif numbers as natural, integer,
More informationChapter 9 Notes Alg. 1H 9-A1 (Lesson 9-3) Solving Quadratic Equations by Finding the Square Root and Completing the Square
Chapter Notes Alg. H -A (Lesson -) Solving Quadratic Equations b Finding the Square Root and Completing the Square p. *Calculator Find the Square Root: take the square root of. E: Solve b finding square
More information9.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED LESSON
CONDENSED LESSON 9.1 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 solve
More informationNEXT-GENERATION Advanced Algebra and Functions
NEXT-GENERATIN Advanced Algebra and Functions Sample Questions The College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunit.
More informationQuadratic Functions Objective: To be able to graph a quadratic function and identify the vertex and the roots.
Name: Quadratic Functions Objective: To be able to graph a quadratic function and identif the verte and the roots. Period: Quadratic Function Function of degree. Usuall in the form: We are now going to
More informationMath 1050 REVIEW for Exam 1. Use synthetic division to find the quotient and the remainder. 1) x3 - x2 + 6 is divided by x + 2
Math 0 REVIEW for Eam 1 Use snthetic division to find the quotient and the remainder. 1) 3-2 + 6 is divided b + 2 Use snthetic division to determine whether - c is a factor of the given polnomial. 2) 3-32
More information3. TRANSLATED PARABOLAS
3. TRANSLATED PARABOLAS The Parabola with Verte V(h, k) and Aes Parallel to the ais Consider the concave up parabola with verte V(h, k) shown below. This parabola is obtained b translating the parabola
More informationMathematics. Mathematics 2. hsn.uk.net. Higher HSN22000
hsn.uk.net Higher Mathematics UNIT Mathematics HSN000 This document was produced speciall for the HSN.uk.net website, and we require that an copies or derivative works attribute the work to Higher Still
More informationCh. 9.3 Vertex to General Form. of a Parabola
Ch. 9.3 Verte to General Form Learning Intentions: of a Parabola Change a quadratic equation from verte to general form. Learn to square a binomial & factor perfectsquare epressions using rectangle diagrams.
More informationChapter 9 Vocabulary Check
9 CHAPTER 9 Eponential and Logarithmic Functions Find the inverse function of each one-to-one function. See Section 9.. 67. f = + 68. f = - CONCEPT EXTENSIONS The formula = 0 e kt gives the population
More informationReady To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions
Read To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Find these vocabular words in Lesson 5-1 and the Multilingual Glossar. Vocabular quadratic function parabola verte
More informationSystems of Linear and Quadratic Equations. Check Skills You ll Need. y x. Solve by Graphing. Solve the following system by graphing.
NY- Learning Standards for Mathematics A.A. Solve a sstem of one linear and one quadratic equation in two variables, where onl factoring is required. A.G.9 Solve sstems of linear and quadratic equations
More informationLesson Goals. Unit 4 Polynomial/Rational Functions Quadratic Functions (Chap 0.3) Family of Quadratic Functions. Parabolas
Unit 4 Polnomial/Rational Functions Quadratic Functions (Chap 0.3) William (Bill) Finch Lesson Goals When ou have completed this lesson ou will: Graph and analze the graphs of quadratic functions. Solve
More informationCALCULUS BASIC SUMMER REVIEW
NAME CALCULUS BASIC SUMMER REVIEW Slope of a non vertical line: rise y y y m run Point Slope Equation: y y m( ) The slope is m and a point on your line is, ). ( y Slope-Intercept Equation: y m b slope=
More informationPolynomial and Rational Functions
Polnomial and Rational Functions Figure -mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of work b Horia Varlan;
More informationAlgebra 1 Unit 9 Quadratic Equations
Algebra 1 Unit 9 Quadratic Equations Part 1 Name: Period: Date Name of Lesson Notes Tuesda 4/4 Wednesda 4/5 Thursda 4/6 Frida 4/7 Monda 4/10 Tuesda 4/11 Wednesda 4/12 Thursda 4/13 Frida 4/14 Da 1- Quadratic
More information2.1 The Rectangular Coordinate System
. The Rectangular Coordinate Sstem In this section ou will learn to: plot points in a rectangular coordinate sstem understand basic functions of the graphing calculator graph equations b generating a table
More information12. Quadratics NOTES.notebook September 21, 2017
1) Fully factorise 4y 2-5y - 6 Today's Learning: To find the equation of quadratic graphs using substitution of a point. 2) Epand the brackets and simplify: (m + 4)(2m - 3) 3) Calculate 20% of 340 without
More information206 Calculus and Structures
06 Calculus and Structures CHAPTER 4 CURVE SKETCHING AND MAX-MIN II Calculus and Structures 07 Copright Chapter 4 CURVE SKETCHING AND MAX-MIN II 4. INTRODUCTION In Chapter, we developed a procedure for
More informationAlgebra I Practice Questions ? 1. Which is equivalent to (A) (B) (C) (D) 2. Which is equivalent to 6 8? (A) 4 3
1. Which is equivalent to 64 100? 10 50 8 10 8 100. Which is equivalent to 6 8? 4 8 1 4. Which is equivalent to 7 6? 4 4 4. Which is equivalent to 4? 8 6 Page 1 of 0 11 Practice Questions 6 1 5. Which
More information( ) 2 + 2x 3! ( x x ) 2
Review for The Final Math 195 1. Rewrite as a single simplified fraction: 1. Rewrite as a single simplified fraction:. + 1 + + 1! 3. Rewrite as a single simplified fraction:! 4! 4 + 3 3 + + 5! 3 3! 4!
More information