Section 9.1 Video Guide Distance and Midpoint Formulas
|
|
- Jonathan Webster
- 5 years ago
- Views:
Transcription
1 Objectives: 1. Use the Distance Formula 2. Use the Midpoint Formula Section 9.1 Video Guide Distance and Midpoint Formulas Section 9.1 Objective 1: Use the Distance Formula Video Length 8:27 1. Eample: Determine the distance between 3, 5 and 3,3. Final answer: The Distance Formula The distance between two points P, and P, , is = P, P, Eample: Find the distance between 6, 6 and 5, 2. Final answer: Copright 2018 Pearson Education, Inc. 379
2 Section 9.1 Objective 2: Use the Midpoint Formula Video Length 2:43 Definition The of a line segment is the point located eactl halfwa between the two endpoints of the line segment. The Midpoint Formula M, The midpoint of the line segment from P, to P, M. is Eample: Find the midpoint of the line segment joining P1 0,8 and 2 4, 6 P. Final answer: 380 Copright 2018 Pearson Education, Inc.
3 Section 9.2 Video Guide Circles Objectives: 1. Write the Standard Form of the Equation of a Circle 2. Graph a Circle 3. Find the Center and Radius of a Circle Given an Equation in General Form Section 9.2 Objective 1: Write the Standard Form of the Equation of a Circle Video Length 5:44 Definition A is a set of all points in the Cartesian plane that are a fied distance r from a fied point hk,. The fied distance r is called the, and the fied point of the circle. hk, is called the Definition The of an equation of a circle with radius r and center hk, is =. 1. Eample: Determine the equation of the circle with radius 4 and center 5,2. Final answer: Copright 2018 Pearson Education, Inc. 381
4 Section 9.2 Objective 2: Graph a Circle Video Length 4: Eample: Graph the equation Copright 2018 Pearson Education, Inc.
5 Section 9.2 Objective 3: Find the Center and Radius of a Circle Given an Equation in General Form Video Length 7:28 Definition The of the equation of a circle is given b the equation when the graph eists. = 3. Eample: Graph the equation of the circle: Copright 2018 Pearson Education, Inc. 383
6 Section 9.3 Video Guide Parabolas Objectives: 1. Graph Parabolas in Which the Verte Is the Origin 2. Find the Equation of a Parabola 3. Graph a Parabola Whose Verte Is Not the Origin 4. Solve Applied Problems Involving Parabolas Section 9.3 Objective 1: Graph Parabolas in Which the Verte Is the Origin Video Length 5:54 Definition A is defined as the collection of all points P in the plane that are the same distance from a fied point F as the are from a fied line D. The point F is called the of the parabola, and the line D is its. As a result, a parabola is the set of points P for which = F V Now let's look at all four forms of a parabola whose verte is at the origin. Equation Verte Focus Directri 384 Copright 2018 Pearson Education, Inc.
7 Equation Verte Focus Directri Equation Verte Focus Directri Equation Verte Focus Directri 1. Eample: Graph the parabola Copright 2018 Pearson Education, Inc. 385
8 Section 9.3 Objective 2: Find the Equation of a Parabola Video Length 5:56 2. Eample: Find an equation of the parabola with verte at 0,0 and focus at equation. 5,0. Graph the Equation: 3. Eample: Find the equation of a parabola with verte at 0,0 if its ais of smmetr is the -ais and its graph contains the point 2,3. Graph the equation. Equation: 386 Copright 2018 Pearson Education, Inc.
9 Section 9.3 Objective 3: Graph a Parabola Whose Verte Is Not the Origin Video Length 6:32 Parabola with Ais of Smmetr Parallel to -Ais, Opens to the Right, a 0 Equation Verte Focus Directri Parabola with Ais of Smmetr Parallel to -Ais, Opens to the Left, a 0 Equation Verte Focus Directri Parabola with Ais of Smmetr Parallel to -Ais, Opens Up, a 0 Equation Verte Focus Directri Copright 2018 Pearson Education, Inc. 387
10 Parabola with Ais of Smmetr Parallel to -Ais, Opens Down, a 0 Equation Verte Focus Directri 4. Eample: Find an equation of the parabola with verte at 2,3 and focus at 0,3. Graph the equation. Equation: 388 Copright 2018 Pearson Education, Inc.
11 Section 9.3 Objective 4: Solve Applied Problems Involving Parabolas Video Length 2:37 5. Eample: A satellite dish is shaped like a paraboloid of revolution. The signals that emanate from a satellite strike the surface of the dish and are reflected to a single point, where the receiver is located. If the dish is 10 feet across at its opening and 4 feet deep at its center, at what position should the receiver be placed? Final answer: Note: Write our answer in a complete sentence. Copright 2018 Pearson Education, Inc. 389
12 Section 9.4 Video Guide Ellipses Objectives: 1. Graph an Ellipse Whose Center Is the Origin 2. Find the Equation of an Ellipse Whose Center Is the Origin 3. Graph an Ellipse Whose Center Is Not the Origin 4. Solve Applied Problems Involving Ellipses Section 9.4 Objective 1: Graph an Ellipse Whose Center Is the Origin Video Length 14:41 Definition An is the collection of all points in the plane the of whose distances from two fied points, called the, is a constant. P V 2 V 1 Equation of an Ellipse: Center at 0,0 ; Major Ais along the -Ais An equation of the ellipse with center 0,0, foci at c,0 and,0 a,0 is c, and vertices at a,0 and =, where and. The major ais is the -ais. 390 Copright 2018 Pearson Education, Inc.
13 1. Eample: Analze the equation: Major ais: Center: Foci: Vertices: Equation of an Ellipse: Center at 0,0 ; Major Ais along the -Ais An equation of the ellipse with center 0,0, foci at 0, c and 0,a is 0,c, and vertices at 0, a =, where and. The major ais is the -ais. and Copright 2018 Pearson Education, Inc. 391
14 2. Eample: Analze the equation: Major ais: Center: Foci: Vertices: 392 Copright 2018 Pearson Education, Inc.
15 Section 9.4 Objective 2: Find the Equation of an Ellipse Whose Center Is the Origin Video Length 6:12 3. Eample: Find an equation of the ellipse with center at the origin, one focus at 3,0 and a verte at 5,0. Graph the equation. Equation: Copright 2018 Pearson Education, Inc. 393
16 Section 9.4 Objective 3: Graph an Ellipse Whose Center Is Not the Origin Video Length 12:54 4. Eample: Analze the equation: Major ais: Center: Foci: Vertices: 394 Copright 2018 Pearson Education, Inc.
17 Section 9.4 Objective 4: Solve Applied Problems Involving Ellipses Video Length 3:25 Ellipses have an interesting reflection propert. If a source of sound or light is placed at one focus, the waves transmitted b the source reflect off the ellipse and concentrate at the other focus. This is the principle behind whispering galleries, which are rooms with elliptical ceilings. A person standing at one focus of the ellipse can whisper and be heard b a person standing at the other focus, because all the sound waves that reach the ceiling are reflected to the other person. 5. Eample: A whispering galler is 50 feet long. The distance from the center of the room to the foci is 15 feet. (a) Find an equation that describes the shape of the room. Equation: (b) How high is the room at its center? Final answer: Copright 2018 Pearson Education, Inc. 395
18 Section 9.5 Video Guide Hperbolas Objectives: 1. Graph a Hperbola Whose Center Is the Origin 2. Find the Equation of a Hperbola Whose Center Is the Origin 3. Find the Asmptotes of a Hperbola Whose Center Is the Origin Section 9.5 Objective 1: Graph a Hperbola Whose Center Is the Origin Video Length 19:37 Definition A is the collection of all points in the plane the of whose distances from two fied points, called the, is a constant. d F, P d F, P 2a 1 2 V 2 V 1 Equation of a Hperbola: Center at 0,0 ; Transverse Ais along the -Ais An equation of the hperbola with center 0,0, foci at c,0 and,0 a,0 is =, where. c, and vertices at a,0 and The transverse ais is the -ais. 396 Copright 2018 Pearson Education, Inc.
19 1. Eample: Analze the equation Transverse ais: Center: Foci: Vertices: Equation of a Hperbola: Center at 0,0 ; Transverse Ais along the -Ais An equation of the hperbola with center 0,0, foci at 0, c and 0,a is =, where. The transverse ais is the -ais. 0,c, and vertices at 0, a and Copright 2018 Pearson Education, Inc. 397
20 2. Eample: Analze the equation Transverse ais: Center: Foci: Vertices: 398 Copright 2018 Pearson Education, Inc.
21 Section 9.5 Objective 2: Find the Equation of a Hperbola Whose Center Is the Origin Video Length 9:03 3. Eample: Find an equation of the hperbola with center at the origin, one focus at 5,0, and one verte at 2,0. Graph the equation. Equation: Copright 2018 Pearson Education, Inc. 399
22 Section 9.5 Objective 3: Find the Asmptotes of a Hperbola Whose Center Is the Origin Video Length 11:31 Hperbolas have a feature that circles and ellipses do not have; namel asmptotes. Asmptotes of a Hperbola 2 2 The hperbola 1 has two oblique asmptotes 2 2 a b = and = Asmptotes of a Hperbola 2 2 The hperbola 1 has two oblique asmptotes 2 2 a b = and = 4. Eample: Analze the equation Transverse ais: Center: Foci: Vertices: 400 Copright 2018 Pearson Education, Inc.
23 Section 9.6 Video Guide Sstems of Nonlinear Equations Objectives: 1. Solve a Sstem of Nonlinear Equations Using Substitution 2. Solve a Sstem of Nonlinear Equations Using Elimination Section 9.6 Objective 1: Solve a Sstem of Nonlinear Equations Using Substitution Video Length 3:06 We are now going to solve sstems of nonlinear equations. We can use the same methods that we learned when solving sstems of linear equations. That is, we can use the method of substitution or elimination Eample: Solve: Final answer: Note: You heard him! Verif our solutions. Copright 2018 Pearson Education, Inc. 401
24 Section 9.6 Objective 2: Solve a Sstem of Nonlinear Equations Using Elimination Video Length 2:24 2. Eample: Solve: Final answer: Note: You can quickl verif that there are four solutions b graphing the equations on the same set of aes. However, ou should alwas verif our work algebraicall to make sure ou didn't make an bonehead mistakes. 402 Copright 2018 Pearson Education, Inc.
Ready To Go On? Skills Intervention 10-1 Introduction to Conic Sections
Find this vocabular word in Lesson 10-1 and the Multilingual Glossar. Graphing Parabolas and Hperbolas on a Calculator A is a single curve, whereas a has two congruent branches. Identif and describe each
More informationChapter Summary. How does Chapter 10 fit into the BIGGER PICTURE of algebra?
Page of 5 0 Chapter Summar WHAT did ou learn? Find the distance between two points. (0.) Find the midpoint of the line segment connecting two points. (0.) Use distance and midpoint formulas in real-life
More informationChapter 7 Page 1 of 16. Lecture Guide. Math College Algebra Chapter 7. to accompany. College Algebra by Julie Miller
Chapter 7 Page 1 of 16 Lecture Guide Math 105 - College Algebra Chapter 7 to accompan College Algebra b Julie Miller Corresponding Lecture Videos can be found at Prepared b Stephen Toner & Nichole DuBal
More information-,- 2..J. EXAMPLE 9 Discussing the Equation of a Parabola. Solution
670 CHAPTER 9 Analtic Geometr Polnomial equations define parabolas whenever the involve two variables that are quadratic in one variable and linear in the other. To discuss this tpe of equation, we first
More informationThe telescopes at the W.M. Keck Observatory in Hawaii use hyperbolic mirrors.
UNIT 15 Conic Sections The telescopes at the W.M. Keck Observator in Hawaii use hperbolic mirrors. Copright 009, K1 Inc. All rights reserved. This material ma not be reproduced in whole or in part, including
More informationC H A P T E R 9 Topics in Analytic Geometry
C H A P T E R Topics in Analtic Geometr Section. Circles and Parabolas.................... 77 Section. Ellipses........................... 7 Section. Hperbolas......................... 7 Section. Rotation
More informationMath 101 chapter six practice exam MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Math 1 chapter si practice eam MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Which equation matches the given calculator-generated graph and description?
More information+ 4 Ex: y = v = (1, 4) x = 1 Focus: (h, k + ) = (1, 6) L.R. = 8 units We can have parabolas that open sideways too (inverses) x = a (y k) 2 + h
Unit 7 Notes Parabolas: E: reflectors, microphones, (football game), (Davinci) satellites. Light placed where ras will reflect parallel. This point is the focus. Parabola set of all points in a plane that
More informationSummary, Review, and Test
944 Chapter 9 Conic Sections and Analtic Geometr 45. Use the polar equation for planetar orbits, to find the polar equation of the orbit for Mercur and Earth. Mercur: e = 0.056 and a = 36.0 * 10 6 miles
More informationCK- 12 Algebra II with Trigonometry Concepts 1
10.1 Parabolas with Verte at the Origin Answers 1. up. left 3. down 4.focus: (0, -0.5), directri: = 0.5 5.focus: (0.065, 0), directri: = -0.065 6.focus: (-1.5, 0), directri: = 1.5 7.focus: (0, ), directri:
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 information10.2 INTRODUCTION TO CONICS: PARABOLAS
Section 0.2 Introduction to Conics: Parabolas 733 0.2 INTRODUCTION TO CONICS: PARABOLAS What ou should learn Recognize a conic as the intersection of a plane a double-napped cone. Write equations of parabolas
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 informationThe details of the derivation of the equations of conics are com-
Part 6 Conic sections Introduction Consider the double cone shown in the diagram, joined at the verte. These cones are right circular cones in the sense that slicing the double cones with planes at right-angles
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 informationHooked on Conics. Chapter Introduction to Conics
Chapter 7 Hooked on Conics 7. Introduction to Conics In this chapter, we stud the Conic Sections - literall sections of a cone. Imagine a doublenapped cone as seen below being sliced b a plane. If we slice
More informationNot for reproduction
REVIEW OF CONIC SECTIONS In this section we give geometric definitions of parabolas, ellipses, and hperbolas and derive their standard equations. The are called conic sections, or conics, because the result
More informationEdexcel GCE A Level Maths. Further Maths 3 Coordinate Systems
Edecel GCE A Level Maths Further Maths 3 Coordinate Sstems Edited b: K V Kumaran kumarmaths.weebl.com 1 kumarmaths.weebl.com kumarmaths.weebl.com 3 kumarmaths.weebl.com 4 kumarmaths.weebl.com 5 1. An ellipse
More informationREVIEW OF CONIC SECTIONS
REVIEW OF CONIC SECTIONS In this section we give geometric definitions of parabolas, ellipses, and hperbolas and derive their standard equations. The are called conic sections, or conics, because the result
More informationSECOND-DEGREE INEQUALITIES
60 (-40) Chapter Nonlinear Sstems and the Conic Sections 0 0 4 FIGURE FOR EXERCISE GETTING MORE INVOLVED. Cooperative learning. Let (, ) be an arbitrar point on an ellipse with foci (c, 0) and ( c, 0)
More informationIn order to take a closer look at what I m talking about, grab a sheet of graph paper and graph: y = x 2 We ll come back to that graph in a minute.
Module 7: Conics Lesson Notes Part : Parabolas Parabola- The parabola is the net conic section we ll eamine. We talked about parabolas a little bit in our section on quadratics. Here, we eamine them more
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 informationMath 180 Chapter 10 Lecture Notes. Professor Miguel Ornelas
Math 180 Chapter 10 Lecture Notes Professor Miguel Ornelas 1 M. Ornelas Math 180 Lecture Notes Section 10.1 Section 10.1 Parabolas Definition of a Parabola A parabola is the set of all points in a plane
More informationMaintaining Mathematical Proficiency
Chapter Maintaining Mathematical Proficienc Find the -intercept of the graph of the linear equation. 1. = + 3. = 3 + 5 3. = 10 75. = ( 9) 5. 7( 10) = +. 5 + 15 = 0 Find the distance between the two points.
More informationNot for reproduction
ROTATION OF AES For a discussion of conic sections, see Review of Conic Sections In precalculus or calculus ou ma have studied conic sections with equations of the form A C D E F Here we show that the
More information5-4. Focus and Directrix of a Parabola. Key Concept Parabola VOCABULARY TEKS FOCUS ESSENTIAL UNDERSTANDING
5- Focus and Directri of a Parabola TEKS FOCUS VOCABULARY TEKS ()(B) Write the equation of a parabola using given attributes, including verte, focus, directri, ais of smmetr, and direction of opening.
More informationAnswers for the problems can be found at the end of this packet starting on Page 12.
MAC 0 Review for Final Eam The eam will consists of problems similar to the ones below. When preparing, focus on understanding and general procedures (when available) rather than specific question. Answers
More information104Math. Find the equation of the parabola and sketch it in the exercises 10-18:
KING SAUD UNIVERSITY COLEGE OF SCIENCE DEPARTMENT OF MATHEMATICS Math Prof Messaoud Bounkhel List of Eercises: Chapter [Parabola] Find the elements of the parabola and sketch it in the eercises -9: ( )
More information3) Find the distance for each set of ordered pairs (remember to provide EXACT answers): 5) x 2 + y 2 + 6x 6y + 9 = 0 A) Ellipse (x 1) 2
Algebra Chapter Review 1) State the Midpoint Formula: State the Distance Formula: ID: 1 Name Date ) Find the midpoint for each set of ordered pairs: a) (1, ), (, ) b) (-, 0), (-, 3) Period c) (, ), (-,
More informationCircles. 1 Page Hannah Province Mathematics Department Southwest Tn Community College
Circles 1 Page To Graph a Circle; Graphing Calculator + y = 2 2 First Solve the equation for y: x 4 y = 4-x 2 2 y = ± 4 x 2 2 Graph as two separate equations y = 4 x y = 4 x 1 2 So that the circle doesn't
More information10.4 Nonlinear Inequalities and Systems of Inequalities. OBJECTIVES 1 Graph a Nonlinear Inequality. 2 Graph a System of Nonlinear Inequalities.
Section 0. Nonlinear Inequalities and Sstems of Inequalities 6 CONCEPT EXTENSIONS For the eercises below, see the Concept Check in this section.. Without graphing, how can ou tell that the graph of + =
More informationConic Sections. Pre-Calculus Unit Completing the Square. Solve each equation by completing the square x 2 + 8x 10 = 0
Pre-Calculus Unit 7 Conic Sections Name: 7.1 Completing the Square Solve each equation by completing the square. 1. x 2 + 4x = 21 6. x 2 5x 5 = 0 11. x 2 6x + 6 = 0 2. x 2 8x = 33 7. x 2 + 7x = 0 12. x
More informationUNCORRECTED. To recognise the rules of a number of common algebraic relations: y = x 1 y 2 = x
5A galler of graphs Objectives To recognise the rules of a number of common algebraic relations: = = = (rectangular hperbola) + = (circle). To be able to sketch the graphs of these relations. To be able
More informationEquations for Some Hyperbolas
Lesson 1-6 Lesson 1-6 BIG IDEA From the geometric defi nition of a hperbola, an equation for an hperbola smmetric to the - and -aes can be found. The edges of the silhouettes of each of the towers pictured
More informationDefinition of an Ellipse Drawing an Ellipse Standard Equations and Their Graphs Applications
616 9 Additional Topics in Analtic Geometr 53. Space Science. A designer of a 00-foot-diameter parabolic electromagnetic antenna for tracking space probes wants to place the focus 100 feet above the verte
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 informationUsing Intercept Form
8.5 Using Intercept Form Essential Question What are some of the characteristics of the graph of f () = a( p)( q)? Using Zeros to Write Functions Work with a partner. Each graph represents a function of
More informationAlgebra 2 Unit 9 (Chapter 9)
Algebra Unit 9 (Chapter 9) 0. Spiral Review Worksheet 0. Find verte, line of symmetry, focus and directri of a parabola. (Section 9.) Worksheet 5. Find the center and radius of a circle. (Section 9.3)
More informationAnalytic Geometry in Two and Three Dimensions
CHAPTER 8 Analtic Geometr in Two and Three Dimensions 8.1 Conic Sections and Parabolas 8.2 Ellipses 8.3 Hperbolas 8.4 Translation and Rotation of Aes 8.5 Polar Equations of Conics 8.6 Three-Dimensional
More informationSection 7.1 Objective 1: Solve Quadratic Equations Using the Square Root Property Video Length 12:12
Section 7.1 Video Guide Solving Quadratic Equations by Completing the Square Objectives: 1. Solve Quadratic Equations Using the Square Root Property. Complete the Square in One Variable 3. Solve Quadratic
More informationDistance and Midpoint Formula 7.1
Distance and Midpoint Formula 7.1 Distance Formula d ( x - x ) ( y - y ) 1 1 Example 1 Find the distance between the points (4, 4) and (-6, -). Example Find the value of a to make the distance = 10 units
More informationThe second type of conic is called an ellipse, and is defined as follows. Definition of Ellipse
72 Chapter 10 Topics in Analtic Geometr 10.3 ELLIPSES What ou should learn Write equations of ellipses in standard form and graph ellipses. Use properties of ellipses to model and solve real-life problems.
More informationSection 7.3: Parabolas, from College Algebra: Corrected Edition by Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a Creative Commons
Section 7.: Parabolas, from College Algebra: Corrected Edition b Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a Creative Commons Attribution-NonCommercial-ShareAlike.0 license. 0, Carl Stitz.
More informationRELATIONS AND FUNCTIONS through
RELATIONS AND FUNCTIONS 11.1.2 through 11.1. Relations and Functions establish a correspondence between the input values (usuall ) and the output values (usuall ) according to the particular relation or
More informationName Class Date. Deriving the Standard-Form Equation of a Parabola
Name Class Date 1. Parabolas Essential Question: How is the distance formula connected with deriving equations for both vertical and horizontal parabolas? Eplore Deriving the Standard-Form Equation of
More information1 is equal to. 1 (B) a. (C) a (B) (D) 4. (C) P lies inside both C & E (D) P lies inside C but outside E. (B) 1 (D) 1
Single Correct Q. Two mutuall perpendicular tangents of the parabola = a meet the ais in P and P. If S is the focus of the parabola then l a (SP ) is equal to (SP ) l (B) a (C) a Q. ABCD and EFGC are squares
More informationAPPENDIX D Rotation and the General Second-Degree Equation
APPENDIX D Rotation and the General Second-Degree Equation Rotation of Aes Invariants Under Rotation After rotation of the - and -aes counterclockwise through an angle, the rotated aes are denoted as the
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 informationName Please print your name as it appears on the class roster.
Berkele Cit College Practice Problems Math 1 Precalculus - Final Eam Preparation Name Please print our name as it appears on the class roster. SHORT ANSWER. Write the word or phrase that best completes
More informationChapter 9. Conic Sections and Analytic Geometry. 9.2 The Hyperbola. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 9 Conic Sections and Analytic Geometry 9. The Hyperbola Copyright 014, 010, 007 Pearson Education, Inc. 1 Objectives: Locate a hyperbola s vertices and foci. Write equations of hyperbolas in standard
More informationIAS 3.1 Conic Sections
Year 13 Mathematics IAS 3.1 Conic Sections Robert Lakeland & Carl Nugent Contents Achievement Standard.................................................. The Straight Line.......................................................
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 information4.2 Parabolas. Explore Deriving the Standard-Form Equation. Houghton Mifflin Harcourt Publishing Company. (x - p) 2 + y 2 = (x + p) 2
COMMON CORE. d Locker d LESSON Parabolas Common Core Math Standards The student is epected to: COMMON CORE A-CED.A. Create equations in two or more variables to represent relationships between quantities;
More informationCircles. 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.
More informationFair Game Review. Chapter 8. Graph the linear equation. Big Ideas Math Algebra Record and Practice Journal
Name Date Chapter Graph the linear equation. Fair Game Review. =. = +. =. =. = +. = + Copright Big Ideas Learning, LLC Big Ideas Math Algebra Name Date Chapter Fair Game Review (continued) Evaluate the
More informationCoordinate geometry. + bx + c. Vertical asymptote. Sketch graphs of hyperbolas (including asymptotic behaviour) from the general
A Sketch graphs of = a m b n c where m = or and n = or B Reciprocal graphs C Graphs of circles and ellipses D Graphs of hperbolas E Partial fractions F Sketch graphs using partial fractions Coordinate
More informationThe Distance Formula. The Midpoint Formula
Math 120 Intermediate Algebra Sec 9.1: Distance Midpoint Formulas The Distance Formula The distance between two points P 1 = (x 1, y 1 ) P 2 = (x 1, y 1 ), denoted by d(p 1, P 2 ), is d(p 1, P 2 ) = (x
More informationComplete Solutions Manual. Technical Calculus with Analytic Geometry FIFTH EDITION. Peter Kuhfittig Milwaukee School of Engineering.
Complete Solutions Manual Technical Calculus with Analtic Geometr FIFTH EDITION Peter Kuhfittig Milwaukee School of Engineering Australia Brazil Meico Singapore United Kingdom United States 213 Cengage
More informationItems with a symbol next to the item number indicate that a student should be prepared to complete items like these with or without a calculator.
HNRS ALGEBRA B Semester Eam Review The semester B eamination for Honors Algebra will consist of two parts. Part is selected response on which a calculator will NT be allowed. Part is short answer on which
More informationIdentifying second degree equations
Chapter 7 Identifing second degree equations 71 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +
More informationGuided Practice. Application. Practice and Apply. Homework Help. Extra Practice.
Circles Vocabular circle center tangent Write equations of circles. Graph circles. are circles important in air traffic control? Radar equipment can be used to detect and locate objects that are too far
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 informationThe formulas below will be provided in the examination booklet. Compound Interest: r n. Continuously: n times per year: 1
HONORS ALGEBRA B Semester Eam Review The semester B eamination for Honors Algebra will consist of two parts. Part will be selected response on which a calculator will not be allowe Part will be short answer
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 informationConic Sections CHAPTER OUTLINE. The Circle Ellipses and Hyperbolas Second-Degree Inequalities and Nonlinear Systems FIGURE 1
088_0_p676-7 /7/0 :5 PM Page 676 (FPG International / Telegraph Colour Librar) Conic Sections CHAPTER OUTLINE. The Circle. Ellipses and Hperbolas.3 Second-Degree Inequalities and Nonlinear Sstems O ne
More informationChapter 12 and 13 Math 125 Practice set Note: the actual test differs. Given f(x) and g(x), find the indicated composition and
Chapter 1 and 13 Math 1 Practice set Note: the actual test differs. Given f() and g(), find the indicated composition. 1) f() = - ; g() = 3 + Find (f g)(). Determine whether the function is one-to-one.
More informationPreCalculus. American Heritage Upper School Summer Math Packet
! PreCalculus American Heritage Upper School Summer Math Packet All Upper School American Heritage math students are required to complete a summer math packet. This packet is intended for all students
More informationGraph and Write Equations of Parabolas
TEKS 9.2 a.5, 2A.5.B, 2A.5.C Graph and Write Equations of Parabolas Before You graphed and wrote equations of parabolas that open up or down. Now You will graph and write equations of parabolas that open
More informationAnswers. Chapter Start Thinking Sample answer: y-intercept: 8 5. x x
. ( 7, ) 9. (, 9 ) 0. (, 7). no solution. (, 7). no solution. no solution. ( 7, ). infinitel man solutions 7. (, 7 ). infinitel man solutions 9. (, 9) 70. 9a + a + 7. b b + 9 7. c + 90c + 7. 9d d + 7.
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question
Midterm Review 0 Precalculu Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question ) A graph of a function g is shown below. Find g(0). (-, ) (-, 0) - -
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 information4 The Cartesian Coordinate System- Pictures of Equations
The Cartesian Coordinate Sstem- Pictures of Equations Concepts: The Cartesian Coordinate Sstem Graphs of Equations in Two Variables -intercepts and -intercepts Distance in Two Dimensions and the Pthagorean
More informationChapter 2 Polynomial, Power, and Rational Functions
Section. Linear and Quadratic Functions and Modeling 6 Chapter Polnomial, Power, and Rational Functions Section. Linear and Quadratic Functions and Modeling Eploration. $000 per ear.. The equation will
More informationMath Review Packet #5 Algebra II (Part 2) Notes
SCIE 0, Spring 0 Miller Math Review Packet #5 Algebra II (Part ) Notes Quadratic Functions (cont.) So far, we have onl looked at quadratic functions in which the term is squared. A more general form of
More informationSkills Practice Skills Practice for Lesson 12.1
Skills Practice Skills Practice for Lesson.1 Name Date Try to Stay Focused Ellipses Centered at the Origin Vocabulary Match each definition to its corresponding term. 1. an equation of the form a. ellipse
More informationto cross cancel to swap (the direction of the inequality symbol) to substitute to find to prove, to show, to verify to plot, to graph
SHORT GLOSSARY OF MATHEMATICS Numbers natural, whole, integer rational, irrational fraction, numerator, denominator real imaginar comple positive, negative even, odd Operations sum, to add, plus difference,
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 informationMathematics 309 Conic sections and their applicationsn. Chapter 2. Quadric figures. ai,j x i x j + b i x i + c =0. 1. Coordinate changes
Mathematics 309 Conic sections and their applicationsn Chapter 2. Quadric figures In this chapter want to outline quickl how to decide what figure associated in 2D and 3D to quadratic equations look like.
More informationUnit 12 Study Notes 1 Systems of Equations
You should learn to: Unit Stud Notes Sstems of Equations. Solve sstems of equations b substitution.. Solve sstems of equations b graphing (calculator). 3. Solve sstems of equations b elimination. 4. Solve
More informationInstructor: Imelda Valencia Course: A3 Honors Pre Calculus
Student: Date: Instructor: Imelda Valencia Course: A3 Honors Pre Calculus 01 017 Assignment: Summer Homework for those who will be taking FOCA 017 01 onl available until Sept. 15 1. Write the epression
More informationabsolute value The distance of a number from zero on a real number line.
G L O S S A R Y A absolute value The distance of a number from zero on a real number line. acute angle An angle whose measure is less than 90. acute triangle A triangle in which each of the three interior
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 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 informationPrecalculus Spring Final Review Name
Precalculus Spring Final Review Name Solve the equation on the interval 0 θ < ". ) sin θ + sin θ = 0 A) 0, ", " 3, 5" 3 B) 0, ", 3" C) 0, ", " 3, " 3 D) 0, ", " 3, 5" 3 SHORT ANSWER. Answer the question.
More informationInclination of a Line. Definition of Inclination
76 Chapter 0 Topics in Analtic Geometr 0. LINES What ou should learn Find the inclination of a line. Find the angle between two lines. Find the distance between a point and a line. Wh ou should learn it
More informationMath125 Exam 5 Review Name. Do the following as indicated.
Math Eam Review Name Do the following as indicated. For the given functions f and g, find the requested function. ) f() = - 6; g() = 9 Find (f - g)(). ) ) f() = 33 + ; g() = - Find (f g)(). 3) f() = ;
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 informationFunctions and Graphs TERMINOLOGY
5 Functions and Graphs TERMINOLOGY Arc of a curve: Part or a section of a curve between two points Asmptote: A line towards which a curve approaches but never touches Cartesian coordinates: Named after
More information(6, 4, 0) = (3, 2, 0). Find the equation of the sphere that has the line segment from P to Q as a diameter.
Solutions Review for Eam #1 Math 1260 1. Consider the points P = (2, 5, 1) and Q = (4, 1, 1). (a) Find the distance from P to Q. Solution. dist(p, Q) = (4 2) 2 + (1 + 5) 2 + (1 + 1) 2 = 4 + 36 + 4 = 44
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 informationSTUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs
STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE Functions & Graphs Contents Functions and Relations... 1 Interval Notation... 3 Graphs: Linear Functions... 5 Lines and Gradients... 7 Graphs: Quadratic
More informationAnalytic Geometry in Three Dimensions
Analtic Geometr in Three Dimensions. The Three-Dimensional Coordinate Sstem. Vectors in Space. The Cross Product of Two Vectors. Lines and Planes in Space The three-dimensional coordinate sstem is used
More informationAttributes and Transformations of Quadratic Functions VOCABULARY. Maximum value the greatest. Minimum value the least. Parabola the set of points in a
- Attributes and Transformations of Quadratic Functions TEKS FCUS VCABULARY TEKS ()(B) Write the equation of a parabola using given attributes, including verte, focus, directri, ais of smmetr, and direction
More informationChapter 8 Analytic Geometry in Two and Three Dimensions
Section 8 Conic Sections and a New Look at Parabolas 09 Chapter 8 Analtic Geometr in Two and Three Dimensions Section 8 Conic Sections and a New Look at Parabolas 0 Eploration From Figure 8, we see that
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 informationMath 2412 Pre Calculus TEST 2 Prep Fall 2011
Math 41 Pre Calculus TEST Prep Fall 011 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the eact value under the given conditions. 1) sin α
More informationSolutions to the Exercises of Chapter 4
Solutions to the Eercises of Chapter 4 4A. Basic Analtic Geometr. The distance between (, ) and (4, 5) is ( 4) +( 5) = 9+6 = 5 and that from (, 6) to (, ) is ( ( )) +( 6 ( )) = ( + )=.. i. AB = (6 ) +(
More informationAnswers. Chapter Warm Up. Sample answer: The graph of h is a translation. 3 units right of the parent linear function.
Chapter. Start Thinking As the string V gets wider, the points on the string move closer to the -ais. This activit mimics a vertical shrink of a parabola... Warm Up.. Sample answer: The graph of f is a
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 information8.7 The Parabola. PF = PD The fixed point F is called the focus. The fixed line l is called the directrix.
8.7 The Parabola The Hubble Space Telescope orbits the Earth at an altitude of approimatel 600 km. The telescope takes about ninet minutes to complete one orbit. Since it orbits above the Earth s atmosphere,
More information