# Precalculus Conic Sections Unit 6. Parabolas. Label the parts: Focus Vertex Axis of symmetry Focal Diameter Directrix

Size: px
Start display at page:

Download "Precalculus Conic Sections Unit 6. Parabolas. Label the parts: Focus Vertex Axis of symmetry Focal Diameter Directrix"

Transcription

1 PICTURE: Parabolas Name Hr Label the parts: Focus Vertex Axis of symmetry Focal Diameter Directrix Using what you know about transformations, label the purpose of each constant: y a x h 2 k It is common to rewrite the equation of the parabola in standard form: 2 y k 4p x h 2 x h 4p y k Vertex: Focus: Vertex: Focus: p > 0 p < 0 p > 0 p < 0

2 Examples: Describe the graph (y 1) 2 = 4(x 1). Write the equation for the parabola. Write the equation for the parabola. Find the equation with F( 3, 4), directrix y = 2.

3 Graph the parabola: y y = x + 1 Find the equation with V( 2, 3) and F(0, 3). Graph the parabola: x 2 + 6x 4y + 1 = 0. Engineers often design roads with parabolic surfaces to allow rain to drain off. A particular road that is 32 feet wide is 0.4 foot higher in the center than on the sides. a) Find the equation of the parabola. (Place the vertex at the origin.) b) How far from the center of the road is the road surface 0.1 foot lower than in the middle?

4 Ellipses PICTURE: x h y k a 1 b Label the parts: Center Major Axis Minor Axis Vertices Foci Eccenticity: EXAMPLES: Find the equation of the ellipse.

5 Graph the ellipse: (x 2) 2 4 (y + 3)2 + = 1 8 Graph the ellipse: x 1 y Find an equation with C(2, 3), one focus (3, 3) and one vertex (5, 3) Find the equation with V(±4, 0) and F(±2, 0)

6 Graph the ellipse: 4x 2 + y 2 8x + 4y + 4 = 0 Graph the ellipse: 3x 2 + 5y 2 12x + 30y + 42 = 0 Where is this applied? Newton s work showed that objects in closed orbits must have circular or elliptical paths. However, if the velocity of an orbiting body is increased, its orbital path changes to a parabola or hyperbola, and it escapes the gravitational pull of the Sun and leaves the solar system. APPLICATION: Halley s Comet passed by Earth in has an elliptical orbit with the sun at one focus. The eccentricity of the orbit is The length of the major axis of the orbit is approximately astronomical units. (An astronomical unit is about 93 million miles.) Find the standard form of the equation of the orbit. Place the center of the orbit at the origin and place the major axis on the x-axis. (NOTE: You will have a chance to view Halley s Comet in 2061 when is passes by Earth again.)

7 Hyperbolas PICTURE: x h y k a 1 b y k x h b 1 a Label the parts: Center Foci Transverse Axis Minor Axis Vertices Asymptotes EXAMPLES: Find the equation of the hyperbola.

8 Graph the hyperbola: (x + 2) 2 9 (y 5)2 = 1 49 Determine the equation of the hyperbola having center C( 1, 1), a vertex on the y-axis, and a focus at F( 3, 1). Graph the hyperbola: x 2 4x 4y 2 8y = 1 Graph the hyperbola: 4x 2 3y 2 + 8x + 16 = 0 Find the equation for the hyperbola with F (0, ±3), transverse axis length 4.

9 CLASSIFYING CONICS: Ax 2 + Cy 2 + Dx + Ey + F = 0 START Is A or C equal to 0? YES PARABOLA NO Is A = C? YES CIRCLE NO Do A and C have the same sign? ELLIPSE HYPERBOLA

10 PRACTICE: Write an equation in standard form for each conic section Focus Focus Find the equation of the conic section in standard form with the given characteristics: 5. HYPERBOLA with vertices (0, ±2) and asymptotes y = ±2x 6. PARABOLA with vertex (3, 3) and focus (3, 9 4 ) 7. ELLIPSE with foci (±2, 0) and major axis length of 10

11 Classify each conic section, write in standard form, and graph (including foci) x 10x y x y x y y x 20y x y x 2 8 0

12 5. 2 y x y x 25y 24x 250 y x 4y 54x 8y x 25y 100 y

13 9. Stein Glass Co. makes parabolic headlights for a variety of automobiles. If one of its headlights has a parabolic surface generated by the parabola x 2 = 12y where should its light bulb be placed? 10. A cannon fires a cannonball. The path of the cannonball is parabolic with vertex at the highest point of the path. If the cannonball lands 1600 feet from the cannon and the highest point it reaches is 3200 feet above the ground, find an equation for the path of the cannonball. (HINT: Place the origin at the location of the cannon.) 11. A lithotripter machine uses an elliptical reflector to break up kidney stones. A spark plug in the reflector generates energy waves at one focus of an ellipse. The reflector directs these waves toward the kidney stone positioned at the other focus of the ellipse with enough energy to break up the stone. The lengths of the major and minor axes of the ellipse are 280 mm and 160 mm. How far is the spark from the kidney stone? Spark Plug Kidney Stone

14 x h y k 12. The equation a describes a degenerate ellipse. What does b 0 this mean and how is it different from a regular ellipse? If you need a hint, go to the website and click on the Degenerate Conics link. TRUE OR FALSE??? Explain your answers. 13. It is easier to distinguish the graph of an ellipse from the graph of a circle when the eccentricity of the ellipse is close to The area of a circle with diameter d = 2r = 8 is greater than the area of an ellipse with major axis 2a = If the vertex and focus of a parabola are on a horizontal line, then the directrix of the parabola is a vertical line. 16. If the asymptotes of the hyperbola a = b. x a y b 1 intersect at right angles, then

15 Complete the square (Part 1) 1. y = x 2 + 2x y = x 2 20x y = 2x x 4 4. y = 3x x x = 3y 2 9y x = 5y y + 8 Complete the square (Part 2) 1. x 2 + 4y 2 + 6x 8y + 9 = x 2 + y 2 8x + 4y 8 = x 2 4y 2 54x + 40y + 37 = x 2 9y x 36y + 43 = 0

16 Trig Tuesday: Simplify the following expressions and match them with their equivalent term. (You will not use all the letters in the right-hand column.) You must show work to receive credit. 1. sec 2 x 1 A sin x 3. sin x cot x 4. 2 sin x 1 cos x tan 2 x 6. csc 2 x cot 2 x B. 2 C. 1 D. csc x E. sec x + csc x F. cot 2 x G. sin x cos x 1 sin x sin x cos x sin xcos x 2 cot x csc x 1 H. tan 2 x I. sec x + 1 J. cos x K. sec 2 x L cos x cot x 11. sin 2 x + cos 2 x + 1 M. 1 sin x N. csc x + 1 O. 1 cos x

### Honors Precalculus Chapter 8 Summary Conic Sections- Parabola

Honors Precalculus Chapter 8 Summary Conic Sections- Parabola Definition: Focal length: y- axis P(x, y) Focal chord: focus Vertex x-axis directrix Focal width/ Latus Rectum: Derivation of equation of parabola:

### 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.

### Distance 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

### Conic Sections. Geometry - Conics ~1~ NJCTL.org. Write the following equations in standard form.

Conic Sections Midpoint and Distance Formula M is the midpoint of A and B. Use the given information to find the missing point. 1. A(, 2) and B(3, -), find M 2. A(5, 7) and B( -2, -), find M 3. A( 2,0)

### 3. A( 2,0) and B(6, -2), find M 4. A( 3, 7) and M(4,-3), find B. 5. M(4, -9) and B( -10, 11) find A 6. B(4, 8) and M(-2, 5), find A

Midpoint and Distance Formula Class Work M is the midpoint of A and B. Use the given information to find the missing point. 1. A(4, 2) and B(3, -8), find M 2. A(5, 7) and B( -2, -9), find M 3. A( 2,0)

### CONIC SECTIONS TEST FRIDAY, JANUARY 5 TH

CONIC SECTIONS TEST FRIDAY, JANUARY 5 TH DAY 1 - CLASSIFYING CONICS 4 Conics Parabola Circle Ellipse Hyperbola DAY 1 - CLASSIFYING CONICS GRAPHICALLY Parabola Ellipse Circle Hyperbola DAY 1 - CLASSIFYING

### 3. A( 2,0) and B(6, -2), find M 4. A( 3, 7) and M(4,-3), find B. 5. M(4, -9) and B( -10, 11) find A 6. B(4, 8) and M(-2, 5), find A

Midpoint and Distance Formula Class Work M is the midpoint of A and B. Use the given information to find the missing point. 1. A(, 2) and B(3, -8), find M 2. A(5, 7) and B( -2, -), find M (3. 5, 3) (1.

### Standard Form of Conics

When we teach linear equations in Algebra1, we teach the simplest linear function (the mother function) as y = x. We then usually lead to the understanding of the effects of the slope and the y-intercept

### 2. Determine the domain of the function. Verify your result with a graph. f(x) = 25 x 2

29 April PreCalculus Final Review 1. Find the slope and y-intercept (if possible) of the equation of the line. Sketch the line: y = 3x + 13 2. Determine the domain of the function. Verify your result with

### PARAMETRIC EQUATIONS AND POLAR COORDINATES

10 PARAMETRIC EQUATIONS AND POLAR COORDINATES PARAMETRIC EQUATIONS & POLAR COORDINATES 10.5 Conic Sections In this section, we will learn: How to derive standard equations for conic sections. CONIC SECTIONS

### Introduction to Computer Graphics (Lecture No 07) Ellipse and Other Curves

Introduction to Computer Graphics (Lecture No 07) Ellipse and Other Curves 7.1 Ellipse An ellipse is a curve that is the locus of all points in the plane the sum of whose distances r1 and r from two fixed

### January 21, 2018 Math 9. Geometry. The method of coordinates (continued). Ellipse. Hyperbola. Parabola.

January 21, 2018 Math 9 Ellipse Geometry The method of coordinates (continued) Ellipse Hyperbola Parabola Definition An ellipse is a locus of points, such that the sum of the distances from point on the

### The 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

### Math 190 (Calculus II) Final Review

Math 90 (Calculus II) Final Review. Sketch the region enclosed by the given curves and find the area of the region. a. y = 7 x, y = x + 4 b. y = cos ( πx ), y = x. Use the specified method to find the

### REVIEW OF KEY CONCEPTS

REVIEW OF KEY CONCEPTS 8.1 8. Equations of Loci Refer to the Key Concepts on page 598. 1. Sketch the locus of points in the plane that are cm from a circle of radius 5 cm.. a) How are the lines y = x 3

### 8.6 Translate and Classify Conic Sections

8.6 Translate and Classify Conic Sections Where are the symmetric lines of conic sections? What is the general 2 nd degree equation for any conic? What information can the discriminant tell you about a

### 9-4 Ellipses. Write an equation of each ellipse. 1. ANSWER: ANSWER:

Write an equation of each ellipse. 5. CCSS SENSE-MAKING An architectural firm sent a proposal to a city for building a coliseum, shown at the right. 1. a. Determine the values of a and b. b. Assuming that

### Chapter 1 Analytic geometry in the plane

3110 General Mathematics 1 31 10 General Mathematics For the students from Pharmaceutical Faculty 1/004 Instructor: Dr Wattana Toutip (ดร.ว ฒนา เถาว ท พย ) Chapter 1 Analytic geometry in the plane Overview:

### Chapter 10: Conic Sections; Polar Coordinates; Parametric Equations

Chapter 10: Conic Sections; Polar Coordinates; Parametric Equations Section 10.1 Geometry of Parabola, Ellipse, Hyperbola a. Geometric Definition b. Parabola c. Ellipse d. Hyperbola e. Translations f.

### Fall Exam 4: 8&11-11/14/13 - Write all responses on separate paper. Show your work for credit.

Math Fall - Exam : 8& - // - Write all responses on separate paper. Show your work for credit. Name (Print):. Convert the rectangular equation to polar coordinates and solve for r. (a) x + (y ) = 6 Solution:

### Conic 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

### Pre-Calculus Final Exam Review Name: May June Use the following schedule to complete the final exam review.

Pre-Calculus Final Exam Review Name: May June 2015 Use the following schedule to complete the final exam review. Homework will be checked in every day. Late work will NOT be accepted. Homework answers

### Introduction to conic sections. Author: Eduard Ortega

Introduction to conic sections Author: Eduard Ortega 1 Introduction A conic is a two-dimensional figure created by the intersection of a plane and a right circular cone. All conics can be written in terms

### y 1 x 1 ) 2 + (y 2 ) 2 A circle is a set of points P in a plane that are equidistant from a fixed point, called the center.

Ch 12. Conic Sections Circles, Parabolas, Ellipses & Hyperbolas The formulas for the conic sections are derived by using the distance formula, which was derived from the Pythagorean Theorem. If you know

### Concept Category 4 (textbook ch. 8) Parametric Equations

Concept Category 4 (textbook ch. 8) Parametric Equations Parametric Equations Write parametric equations. Graph parametric equations. Determine an equivalent rectangular equation for parametric equations.

### Unit 2 Quadratics. Mrs. Valentine Math 3

Unit 2 Quadratics Mrs. Valentine Math 3 2.1 Factoring and the Quadratic Formula Factoring ax 2 + bx + c when a = ±1 Reverse FOIL method Find factors of c that add up to b. Using the factors, write the

### Algebra 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)

### Intermediate Math Circles Wednesday, April 5, 2017 Problem Set 8

Intermediate Math Circles Wednesday, April 5, 2017 Problem Set 8 1. Determine the coordinates of the vertices and foci for each of the following ellipses. (a) + 9y 2 = 36 We want equation to be of the

### Algebra II Final Exam Semester II Practice Test

Name: Class: Date: Algebra II Final Exam Semester II Practice Test 1. (10 points) A bacteria population starts at,03 and decreases at about 15% per day. Write a function representing the number of bacteria

### CIRCLES: #1. What is an equation of the circle at the origin and radius 12?

1 Pre-AP Algebra II Chapter 10 Test Review Standards/Goals: E.3.a.: I can identify conic sections (parabola, circle, ellipse, hyperbola) from their equations in standard form. E.3.b.: I can graph circles

### Math Conic Sections

Math 114 - Conic Sections Peter A. Perry University of Kentucky April 13, 2017 Bill of Fare Why Conic Sections? Parabolas Ellipses Hyperbolas Shifted Conics Goals of This Lecture By the end of this lecture,

### Skills 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

### Math 160 Final Exam Info and Review Exercises

Math 160 Final Exam Info and Review Exercises Fall 2018, Prof. Beydler Test Info Will cover almost all sections in this class. This will be a 2-part test. Part 1 will be no calculator. Part 2 will be scientific

### 1 x. II. CHAPTER 2: (A) Graphing Rational Functions: Show Asymptotes using dotted lines, Intercepts, Holes(Coordinates, if any.)

FINAL REVIEW-014: Before using this review guide be sure to study your test and quizzes from this year. The final will contain big ideas from the first half of the year (chapters 1-) but it will be focused

### -,- 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

### Pure Math 30: Explained! 81

4 www.puremath30.com 81 Part I: General Form General Form of a Conic: Ax + Cy + Dx + Ey + F = 0 A & C are useful in finding out which conic is produced: A = C Circle AC > 0 Ellipse A or C = 0 Parabola

### Systems of Nonlinear Equations and Inequalities: Two Variables

Systems of Nonlinear Equations and Inequalities: Two Variables By: OpenStaxCollege Halley s Comet ([link]) orbits the sun about once every 75 years. Its path can be considered to be a very elongated ellipse.

### CRASH COURSE IN PRECALCULUS

CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 2012, Brooks/Cole

### Graph and Write Equations of Ellipses. You graphed and wrote equations of parabolas and circles. You will graph and write equations of ellipses.

TEKS 9.4 a.5, A.5.B, A.5.C Before Now Graph and Write Equations of Ellipses You graphed and wrote equations of parabolas and circles. You will graph and write equations of ellipses. Wh? So ou can model

### SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET

SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET 017-018 Name: 1. This packet is to be handed in on Monday August 8, 017.. All work must be shown on separate paper attached to the packet. 3.

### 9.6 PROPERTIES OF THE CONIC SECTIONS

9.6 Properties of the Conic Sections Contemporary Calculus 1 9.6 PROPERTIES OF THE CONIC SECTIONS This section presents some of the interesting and important properties of the conic sections that can be

### MATH10000 Mathematical Workshop Project 2 Part 1 Conic Sections

MATH10000 Mathematical Workshop Project 2 Part 1 Conic Sections The aim of this project is to introduce you to an area of geometry known as the theory of conic sections, which is one of the most famous

### Find: sinθ. Name: Date:

Name: Date: 1. Find the exact value of the given trigonometric function of the angle θ shown in the figure. (Use the Pythagorean Theorem to find the third side of the triangle.) Find: sinθ c a θ a a =

### Math 1720 Final Exam REVIEW Show All work!

Math 1720 Final Exam REVIEW Show All work! The Final Exam will contain problems/questions that fit into these Course Outcomes (stated on the course syllabus): Upon completion of this course, students will:

### Logs and Exponential functions e, ln, solving exponential functions, solving log and exponential equations, properties of logs

Page 1 AM1 Final Exam Review Packet TOPICS Complex Numbers, Vectors, and Parametric Equations Change back and forth from and to polar and rectangular forms. Raise a term in polar form to a power (DeMoivre).

### 1. Determine the length of the major & minor axis. List the coordinates of vertices and co-vertices of the following ellipses. Vertices: Co-Vertices:

1. Sec 6.3 Conic Sections Ellipses Name: An ELLIPSE could be accurately described as circle that has been stretched or compressed by a constant ratio towards a diameter of a circle. A circle is actually

### 1. Graph each of the given equations, state the domain and range, and specify all intercepts and symmetry. a) y 3x

MATH 94 Final Exam Review. Graph each of the given equations, state the domain and range, and specify all intercepts and symmetry. a) y x b) y x 4 c) y x 4. Determine whether or not each of the following

### April 30, Name: Amy s Solutions. Discussion Section: N/A. Discussion TA: N/A

Math 1151, April 30, 010 Exam 3 (in-class) Name: Amy s Solutions Discussion Section: N/A Discussion TA: N/A This exam has 8 multiple-choice problems, each worth 5 points. When you have decided on a correct

### A. Correct! These are the corresponding rectangular coordinates.

Precalculus - Problem Drill 20: Polar Coordinates No. 1 of 10 1. Find the rectangular coordinates given the point (0, π) in polar (A) (0, 0) (B) (2, 0) (C) (0, 2) (D) (2, 2) (E) (0, -2) A. Correct! These

### IUPUI Department of Mathematical Sciences Departmental Final Examination PRACTICE FINAL EXAM VERSION #1 MATH Trigonometry

IUPUI Department of Mathematical Sciences Departmental Final Examination PRACTICE FINAL EXAM VERSION #1 MATH 15400 Trigonometry Exam directions similar to those on the departmental final. 1. DO NOT OPEN

### QUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola)

QUESTION BANK ON CONIC SECTION (Parabola, Ellipse & Hyperbola) Question bank on Parabola, Ellipse & Hyperbola Select the correct alternative : (Only one is correct) Q. Two mutually perpendicular tangents

### Lecture 15 - Orbit Problems

Lecture 15 - Orbit Problems A Puzzle... The ellipse shown below has one focus at the origin and its major axis lies along the x-axis. The ellipse has a semimajor axis of length a and a semi-minor axis

### SKILL BUILDER TEN. Graphs of Linear Equations with Two Variables. If x = 2 then y = = = 7 and (2, 7) is a solution.

SKILL BUILDER TEN Graphs of Linear Equations with Two Variables A first degree equation is called a linear equation, since its graph is a straight line. In a linear equation, each term is a constant or

### Hi AP AB Calculus Class of :

Hi AP AB Calculus Class of 2017 2018: In order to complete the syllabus that the College Board requires and to have sufficient time to review and practice for the exam, I am asking you to do a (mandatory)

### Math 370 Semester Review Name

Math 370 Semester Review Name 1) State the following theorems: (a) Remainder Theorem (b) Factor Theorem (c) Rational Root Theorem (d) Fundamental Theorem of Algebra (a) If a polynomial f(x) is divided

### Things You Should Know Coming Into Calc I

Things You Should Know Coming Into Calc I Algebraic Rules, Properties, Formulas, Ideas and Processes: 1) Rules and Properties of Exponents. Let x and y be positive real numbers, let a and b represent real

### Conic Sections in Polar Coordinates

Conic Sections in Polar Coordinates MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction We have develop the familiar formulas for the parabola, ellipse, and hyperbola

### Math 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?

### b = 2, c = 3, we get x = 0.3 for the positive root. Ans. (D) x 2-2x - 8 < 0, or (x - 4)(x + 2) < 0, Therefore -2 < x < 4 Ans. (C)

SAT II - Math Level 2 Test #02 Solution 1. The positive zero of y = x 2 + 2x is, to the nearest tenth, equal to (A) 0.8 (B) 0.7 + 1.1i (C) 0.7 (D) 0.3 (E) 2.2 ± Using Quadratic formula, x =, with a = 1,

### ALGEBRA 2 X. Final Exam. Review Packet

ALGEBRA X Final Exam Review Packet Multiple Choice Match: 1) x + y = r a) equation of a line ) x = 5y 4y+ b) equation of a hyperbola ) 4) x y + = 1 64 9 c) equation of a parabola x y = 1 4 49 d) equation

### 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

### Calculus I Sample Exam #01

Calculus I Sample Exam #01 1. Sketch the graph of the function and define the domain and range. 1 a) f( x) 3 b) g( x) x 1 x c) hx ( ) x x 1 5x6 d) jx ( ) x x x 3 6 . Evaluate the following. a) 5 sin 6

### Notes 10-3: Ellipses

Notes 10-3: Ellipses I. Ellipse- Definition and Vocab An ellipse is the set of points P(x, y) in a plane such that the sum of the distances from any point P on the ellipse to two fixed points F 1 and F

9.1 Circles and Parabolas Copyright Cengage Learning. All rights reserved. What You Should Learn Recognize a conic as the intersection of a plane and a double-napped cone. Write equations of circles in

### Standardized Test Practice

Standardized Test Practice. A store uses a matrix to show their inventory of jeans by waist size (in inches) and style of leg. What is a 3? A a. straight boot cut flared tapered 3 3 3 3 3 3 7 3 3 9 b.

### 10.1 Review of Parametric Equations

10.1 Review of Parametric Equations Recall that often, instead of representing a curve using just x and y (called a Cartesian equation), it is more convenient to define x and y using parametric equations

### Convert the equation to the standard form for an ellipse by completing the square on x and y. 3) 16x y 2-32x - 150y = 0 3)

Math 370 Exam 5 Review Name Graph the ellipse and locate the foci. 1) x 6 + y = 1 1) foci: ( 15, 0) and (- 15, 0) Objective: (9.1) Graph Ellipses Not Centered at the Origin Graph the ellipse. ) (x + )

### Conic section. Ans: c. Ans: a. Ans: c. Episode:43 Faculty: Prof. A. NAGARAJ. 1. A circle

Episode:43 Faculty: Prof. A. NAGARAJ Conic section 1. A circle gx fy c 0 is said to be imaginary circle if a) g + f = c b) g + f > c c) g + f < c d) g = f. If (1,-3) is the centre of the circle x y ax

### Section 8.2: Integration by Parts When you finish your homework, you should be able to

Section 8.2: Integration by Parts When you finish your homework, you should be able to π Use the integration by parts technique to find indefinite integral and evaluate definite integrals π Use the tabular

### Conic Sections: THE ELLIPSE

Conic Sections: THE ELLIPSE An ellipse is the set of all points,such that the sum of the distance between, and two distinct points is a constant. These two distinct points are called the foci (plural of

### FGCU 6th Annual Math Competition 2008 Precalculus - Individual Exam

FGCU 6th Annual Math Competition 008 Precalculus - Individual Eam Find the domain of the rational function. ) f() = + + 9 A) all real numbers B) { -, } C){ -,, -} D) { 0, -9} Solve the equation b epressing

### 8.8 Conics With Equations in the Form

8.8 Conics With Equations in the Form ax + b + gx + f + c = 0 The CF-18 Hornet is a supersonic jet flown in Canada. It has a maximum speed of Mach 1.8. The speed of sound is Mach 1. When a plane like the

### Chapter 8. Orbits. 8.1 Conics

Chapter 8 Orbits 8.1 Conics Conic sections first studied in the abstract by the Greeks are the curves formed by the intersection of a plane with a cone. Ignoring degenerate cases (such as a point, or pairs

### Have a Safe and Happy Break

Math 121 Final EF: December 10, 2013 Name Directions: 1 /15 2 /15 3 /15 4 /15 5 /10 6 /10 7 /20 8 /15 9 /15 10 /10 11 /15 12 /20 13 /15 14 /10 Total /200 1. No book, notes, or ouiji boards. You may use

### CP Pre-Calculus Summer Packet

Page CP Pre-Calculus Summer Packet Name: Ø Do all work on a separate sheet of paper. Number your problems and show your work when appropriate. Ø This packet will count as your first homework assignment

### Chapter 11 Parametric Equations, Polar Curves, and Conic Sections

Chapter 11 Parametric Equations, Polar Curves, and Conic Sections ü 11.1 Parametric Equations Students should read Sections 11.1-11. of Rogawski's Calculus [1] for a detailed discussion of the material

### Some Highlights along a Path to Elliptic Curves

11/8/016 Some Highlights along a Path to Elliptic Curves Part : Conic Sections and Rational Points Steven J Wilson, Fall 016 Outline of the Series 1 The World of Algebraic Curves Conic Sections and Rational

### MA 162 FINAL EXAM PRACTICE PROBLEMS Spring Find the angle between the vectors v = 2i + 2j + k and w = 2i + 2j k. C.

MA 6 FINAL EXAM PRACTICE PROBLEMS Spring. Find the angle between the vectors v = i + j + k and w = i + j k. cos 8 cos 5 cos D. cos 7 E. cos. Find a such that u = i j + ak and v = i + j + k are perpendicular.

### Ch 9/10/11/12 Exam Review

Ch 9/0// Exam Review The vector v has initial position P and terminal point Q. Write v in the form ai + bj; that is, find its position vector. ) P = (4, 6); Q = (-6, -) Find the vertex, focus, and directrix

### Math 2412 Final Exam Review

Math 41 Final Exam Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Factor and simplify the algebraic expression. 1) (x + 4) /5 - (x + 4) 1/5

### 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

### December 16, Conic sections in real life.notebook

OCCURRENCE OF THE CONICS Mathematicians have a habit of studying, just for the fun of it, things that seem utterly useless; then centuries later their studies turn out to have enormous scientific value.

### Precalculus 2nd Semester Review 2014

Precalculus 2nd Semester Review 2014 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the coordinates of the center of the ellipse represented by.

### ( x + 1) 2 = ( y + 3)

f(x + h) f(x) h 0 f(x) = 1 1 Simplify the difference quotient, if, for. h 5 x 1 ( 5 x+h ) ( 5 x) 1 h( 5 x) 2 1 ( 5 x+h ) ( 5 x) 1 ( 5 x h) ( 5 x) 1 h( 5 x) 1 ( 5 x h) ( 5 x) 2 Write an equation for the

### 6.3 Ellipses. Objective: To find equations of ellipses and to graph them. Complete the Drawing an Ellipse Activity With Your Group

6.3 Ellipses Objective: To find equations of ellipses and to graph them. Complete the Drawing an Ellipse Activity With Your Group Conic Section A figure formed by the intersection of a plane and a right

### Conic Sections Session 3: Hyperbola

Conic Sections Session 3: Hyperbola Toh Pee Choon NIE Oct 2017 Toh Pee Choon (NIE) Session 3: Hyperbola Oct 2017 1 / 16 Problem 3.1 1 Recall that an ellipse is defined as the locus of points P such that

### THE CONIC SECTIONS: AMAZING UNITY IN DIVERSITY

O ne of the most incredible revelations about quadratic equations in two unknowns is that they can be graphed as a circle, a parabola, an ellipse, or a hyperbola. These graphs, albeit quadratic functions,

### Summary, 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

### Recall, to graph a conic function, you want it in the form parabola: (x x 0 ) 2 =4p(y y 0 ) or (y y 0 ) 2 =4p(x x 0 ), x x. a 2 x x 0.

Warm up Recall, to graph a conic function, you want it in the form parabola: (x x 0 ) =4p(y y 0 ) or (y y 0 ) =4p(x x 0 ), x x ellipse: 0 a + y y 0 b =, x x hyperbola: 0 y y 0 a b =or y y 0 x x 0 a b =

### TARGET QUARTERLY MATHS MATERIAL

Adyar Adambakkam Pallavaram Pammal Chromepet Now also at SELAIYUR TARGET QUARTERLY MATHS MATERIAL Achievement through HARDWORK Improvement through INNOVATION Target Centum Practising Package +2 GENERAL

### Chapter 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

### Chetek-Weyerhaeuser High School

Chetek-Weyerhaeuser High School Advanced Math A Units and s Advanced Math A Unit 1 Functions and Math Models (7 days) 10% of grade s 1. I can make connections between the algebraic equation or description

### MATH-1420 Review Concepts (Haugen)

MATH-40 Review Concepts (Haugen) Unit : Equations, Inequalities, Functions, and Graphs Rational Expressions Determine the domain of a rational expression Simplify rational expressions -factor and then

### KEMATH1 Calculus for Chemistry and Biochemistry Students. Francis Joseph H. Campeña, De La Salle University Manila

KEMATH1 Calculus for Chemistry and Biochemistry Students Francis Joseph H Campeña, De La Salle University Manila February 9, 2015 Contents 1 Conic Sections 2 11 A review of the coordinate system 2 12 Conic

### DAY 139 EQUATION OF A HYPERBOLA

DAY 139 EQUATION OF A HYPERBOLA INTRODUCTION In our prior conic sections lessons, we discussed in detail the two conic sections, the parabola, and the ellipse. The hyperbola is another conic section we

### Precalculus Table of Contents Unit 1 : Algebra Review Lesson 1: (For worksheet #1) Factoring Review Factoring Using the Distributive Laws Factoring

Unit 1 : Algebra Review Factoring Review Factoring Using the Distributive Laws Factoring Trinomials Factoring the Difference of Two Squares Factoring Perfect Square Trinomials Factoring the Sum and Difference

### Rotation of Axes. By: OpenStaxCollege

Rotation of Axes By: OpenStaxCollege As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions,

### Section 7.3 Double Angle Identities

Section 7.3 Double Angle Identities 3 Section 7.3 Double Angle Identities Two special cases of the sum of angles identities arise often enough that we choose to state these identities separately. Identities