Section 9.1 Video Guide Distance and Midpoint Formulas

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.