Transcript Slide 1
Homework, Page 641 Find the vertex, focus, directrix, and focal width of the parabola. 1. x2 6 y Vertex : 0,0 Focus : 0,1.5 4 p 6 p 6 1.5 4 Directrix : y 1.5 Focal width : 4 p 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 1 Homework, Page 641 Find the vertex, focus, directrix, and focal width of the parabola. 2 5. 3x 4 y Vertex : 0,0 4 1 4 1 Focus : 0, 4 p p 3 3 3 4 3 1 Directrix : y 3 4 4 Focal width : 3 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 2 Homework, Page 641 Match the graph with its equation. y 2 5x 9. y (a) x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 3 Homework, Page 641 Find an equation in standard form for the parabola that satisfies the given conditions. 13. Vertex 0,0 , directrix y 4 p 4 x2 4 py x2 16 y Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 4 Homework, Page 641 Find an equation in standard form for the parabola that satisfies the given conditions. 17. Vertex 0,0 , opens to the right, focal width 8 4p 8 2 y 2 4 px y 2 8x y 8 x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 5 Homework, Page 641 Find an equation in standard form for the parabola that satisfies the given conditions. 21. Focus 2, 4 ; Vertex 4, 4 y k 4 p x h 2 p 2 4 2 y 4 4 2 x 4 y 42 8 x 4 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 6 Homework, Page 641 Find an equation in standard form for the parabola that satisfies the given conditions. Vertex 4,3 ; Directrix x 6 25. y k 4 p x h p 4 6 2 y 3 2 2 4 2 x 4 y 32 8 x 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 7 Homework, Page 641 Find an equation in standard form for the parabola that satisfies the given conditions. 29. Vertex 1, 4 , opens to the left, focal width 10 y k 4 p x h 4 p 10 p 2.5 2 y 4 2 4 2.5 x 1 y 4 10 x 1 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 8 Homework, Page 641 Sketch the graph of the parabola by hand 2 x 4 12 y 1 33. (-4, -1) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 9 Homework, Page 641 Graph the parabola using a function grapher. 2 37. y 4 x (0, 0) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 10 Homework, Page 641 Graph the parabola using a function grapher. 2 12 y 1 x 3 41. 12 y 12 x 3 2 12 y x 3 12 2 x 3 y 2 12 12 (3, -1) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 11 Homework, Page 641 Graph the parabola using a function grapher. 2 y 3 12 x 2 45. y 3 2 12 x 2 y 3 12 x 2 y 3 12 x 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 12 Homework, Page 641 Prove that the graph of the equation is a parabola, and find its vertex, focus, and directrix. 2 x 2x y 3 0 49. x2 2 x y 3 x2 2 x 1 y 3 1 x 1 y 2 General form for a parabola 1 3 4 p 1 p Vertex 1, 2 , Focus 1,2 1 , Directrix y 1 4 4 4 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 13 Homework, Page 641 Write an equation for the parabola. 53. (-6, -4) 2 4 p x h y k a x h 2 y 2 a x 0 2 4 2 a 6 2 (0, 2) y k 36 6a a 6 y 2 6 x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Slide 8- 14 Homework, Page 641 57. Explain why the derivation of x2 = 4py is valid regardless of whether p > 0 or p < 0. The derivation on page 635 only requires that p be a real number, its sign is of no significance in the derivation, although in the equation in standard form it tells us the direction in which the parabola opens. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 15 Homework, Page 641 61. The Sports Channel uses a parabolic microphone to capture the sounds from golf tournaments. If one of its microphones has a parabolic surface generated by the parabola x2 = 10y, locate the focus (the electronic receiver) of the parabola. x2 10 y x2 4 py. 4 p 10 p 2.5 The focus is located at 0, 2.5 . Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 16 Homework, Page 641 65. axis. Every point on a parabola is the same distance from its focus and its False. Every point on a parabola is equidistant from its focus and its directrix, a line perpendicular to the axis. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 17 Homework, Page 641 69. a. b. c. d. e. The focus of y2 = 12x is (3, 3) (3, 0) (0, 3) (0, 0) y 2 12 x y 2 4 px 4 p 12 p 3 3, 3 Axis is x-axis, vertex is 0,0 , parabola opens to right, focus is three units right of the origin at 3,0 . Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 18 What you’ll learn about Geometry of an Ellipse Translations of Ellipses Orbits and Eccentricity Reflective Property of an Ellipse … and why Ellipses are the paths of planets and comets around the Sun, or of moons around planets. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 19 Ellipse An ellipse is the set of all points in a plane whose distance from two fixed points in the plane have a constant sum. The fixed points are the foci (plural of focus) of the ellipse. The line through the foci is the focal axis. The point on the focal axis midway between the foci is the center. The points where the ellipse intersects its axis are the vertices of the ellipse. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 20 Key Points on the Focal Axis of an Ellipse Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 21 Ellipse - Additional Terms The major axis is the chord connecting the vertices of the ellipse. The semimajor axis is the distance from the center of the ellipse and to one of the vertices. The minor axis is the chord perpendicular to the major axis and passing through the center of the ellipse. The semiminor axis is the distance from the center of the ellipse to one end of the minor axis, sometimes called a minor vertex. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 22 Ellipse with Center (0,0) Focal axis Foci x2 y 2 2 1 2 a b x-axis ( c,0) y 2 x2 2 1 2 a b y -axis (0, c) Vertices Semimajor axis ( a,0) a (0, a) a Standard equation Semiminor axis b Pythagorean relation a 2 b2 c2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley b a 2 b2 c2 Slide 8- 23 Pythagorean Relation Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 24 Example Finding the Vertices and Foci of an Ellipse Find the vertices and the foci of the ellipse 9 x 2 4 y 2 36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 25 Example Finding an Equation of an Ellipse Find an equation of the ellipse with foci ( 2,0) and (2,0) whose minor axis has length 2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 26 Ellipse with Center (h,k) Standard equation x h 2 2 Focal axis Foci a yk ( h c, k ) Vertices Semimajor axis ( h a, k ) a y k Semiminor axis b Pythagorean relation a 2 b2 c2 b y k 2 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 a 2 2 x h b 2 2 1 xh ( h, k c) (h, k a) a b a 2 b2 c2 Slide 8- 27 Ellipse with Center (h,k) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 28 Example Locating Key Points of an Ellipse Find the center, vertices, and foci of the ellipse x 1 4 2 y 1 9 2 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 29 Example Finding the Equation of an Ellipse Find the equation of the ellipse with major axis endpoints 3, 7 and 3,3 and minor axis length 6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 30 Example Graphing an Ellipse Using Parametric Equations Parameterize and graph the ellipse whose equation is: Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley y 1 16 2 x 2 9 2 1 Slide 8- 31 Example Proving an Ellipse Prove that the graph of the equation is an ellipse, and find its center, vertices, and foci: 9 x 2 16 y 2 54 x 32 y 47 0 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 32 Elliptical Orbits Around the Sun Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 33 Eccentricity of an Ellipse c a b The eccentricity of an ellipse is e , a a where a is the semimajor axis, b is the semiminor 2 2 axis, and c is the distance from the center of the ellipse to either focus. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 34 Homework Homework Assignment #17 Review Section 8.2 Page 653, Exercises: 1 – 69 (EOO) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 35 8.3 Hyperbolas Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Quick Review 1. Find the distance between the points (a, b) and (c, 4). 2 2 y x 2. Solve for y in terms of x. 1 16 2 Solve for x algebraically. 3. 3 x 12 3 x 8 10 4. 6 x 12 6 x 1 1 2 2 5. Solve the system of equations: ca 2 c a 16a / c 2 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 37 Quick Review Solutions 1. Find the distance between the points (a, b) and (c, 4). a c b 4 2 2 2 2 y x 2. Solve for y in terms of x. 1 y 8 x 16 16 2 Solve for x algebraically. 2 3. 3 x 12 3 x 8 10 no solution 4. 6 x 12 6 x 1 1 x 2 2 222 6 5. Solve the system of equations: ca 2 c a 16a / c no solution 2 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 38 What you’ll learn about Geometry of a Hyperbola Translations of Hyperbolas Eccentricity and Orbits Reflective Property of a Hyperbola Long-Range Navigation … and why The hyperbola is the least known conic section, yet it is used astronomy, optics, and navigation. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 39 Hyperbola A hyperbola is the set of all points in a plane whose distances from two fixed points in the plane have a constant difference. The fixed points are the foci of the hyperbola. The line through the foci is the focal axis. The point on the focal axis midway between the foci is the center. The line through the center and perpendicular to the focal axis is the conjugate axis. The points where the hyperbola intersects its focal axis are the vertices of the hyperbola. The line collinear with the focal axis and connecting the vertices is the transverse axis. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 40 Hyperbola Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 41 Hyperbola Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 42 Hyperbola with Center (0,0) Standard equation Focal axis Foci Vertices Semitransverse axis Semiconjugate axis Pythagorean relation Asymptotes x2 y 2 2 1 2 a b x-axis (c,0) ( a,0) a b c2 a 2 b2 b y x a Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley y 2 x2 2 1 2 a b y -axis (0, c) (0, a) a b c2 a 2 b2 a y x b Slide 8- 43 Hyperbola Centered at (0,0) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 44 Example Finding the Vertices and Foci of a Hyperbola Find the vertices and the foci of the hyperbola 9x2 4 y 2 36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 45 Example Sketching a Hyperbola by Hand Sketch the graph of the hyperbola 9x2 16 y 2 144. y x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 46 Example Finding an Equation of a Hyperbola Find an equation of the hyperbola with foci (0,4) and (0, 4) whose conjugate axis has length 2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 47 Hyperbola with Center (h,k) Standard equation x h 2 2 Focal axis Foci a yk ( h c, k ) Vertices Semimajor axis Semiminor axis ( h a, k ) a b Pythagorean relation Asymptotes y k b 2 2 c2 a 2 b2 b y ( x h) k a Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 y k a 2 2 x h b 2 2 1 xh (h, k c) ( h, k a ) a b c2 a 2 b2 a y ( x h) k b Slide 8- 48 Hyperbola with Center (h,k) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 49 Example Locating Key Points of a Hyperbola Find the center, vertices, and foci of the hyperbola Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley x 1 4 2 y2 1. 9 Slide 8- 50 Example Locating a Point Using Hyperbolas Radio signals are sent simultaneously from three transmitters located at O, Q, and R. R is 80 miles due north of O and Q is 100 miles due east of O. A ship receives the transmission from O 323.27 μsec after the signal from R and 258.61 μsec after the signal from Q. What is the ship’s bearing and distance from O? Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 51 Example Locating a Point Using Hyperbolas Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 52 Eccentricity of a Hyperbola c a b The eccentricity of a hyperbola is e , a a where a is the semitransverse axis, b is the semiconjugate axis, and c is the distance from the center to either focus. 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Slide 8- 53