Graph and Write Equations of Parabolas

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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 left or right. Wh? So ou can model sound projection, as in E. 56. Ke Vocabular focus directri parabola, p. 236 verte, p. 236 You know that the graph of 5 a 2 is a parabola that opens up or down with verte (0, 0) and ais of smmetr 5 0. On an parabola, each point is equidistant from a point called the focus and a line called the directri. The focus lies on the ais of smmetr. ais of smmetr The verte lies halfwa between the focus and the directri. The directri is perpendicular to the ais of smmetr. The equation of a parabola that opens up or down and has verte (0, 0) can also be written in the form 2 5 4p. Parabolas can open left or right as well, in which case the equation has the form 2 5 4p when the verte is (0, 0). Note below that for an parabola, the focus and directri each lie p units from the verte. verte: (0, 0) (0, p) directri: 5 2p verte: (0, 0) directri: 5 2p (0, p) 2 5 4p, p > 0 2 5 4p, p < 0 IDENTIFY FUNCTIONS Notice that parabolas that open left or right do not represent functions. directri: 5 2p verte: (0, 0) (p, 0) (p, 0) directri: 5 2p verte: (0, 0) 2 5 4p, p > 0 2 5 4p, p < 0 620 Chapter 9 Quadratic Relations and Conic Sections

KEY CONCEPT For Your Notebook Standard Equation of a Parabola with Verte at the Origin The standard form of the equation of a parabola with verte at (0, 0) is as follows: Equation Focus Directri Ais of Smmetr 2 5 4p (0, p) 52p Vertical ( 5 0) 2 5 4p (p, 0) 52p Horizontal ( 5 0) E XAMPLE Graph an equation of a parabola Graph 52 } 8 2. Identif the focus, directri, and ais of smmetr. Solution STEP Rewrite the equation in standard form. 52 } 8 2 Write original equation. STEP 2 28 5 2 Multipl each side b 28. Identif the focus, directri, and ais of smmetr. The equation has the form 2 5 4p where p 522. The focus is (p, 0), or (22, 0). The directri is 52p, or 5 2. Because is squared, the ais of smmetr is the -ais. (22, 0) 4 52 SOLVE FOR Y To fill in the table, note that because 28 5 2, 56Ï } 28. The value of will be a real number onl when 0. STEP 3 Draw the parabola b making a table of values and plotting points. Because p < 0, the parabola opens to the left. So, use onl negative -values. 2 22 23 24 25 62.83 64 64.90 65.66 66.32 at classzone.com E XAMPLE 2 Write an equation of a parabola Write an equation of the parabola shown. Solution The graph shows that the verte is (0, 0) and the directri is 52p 52} 3. Substitute } 3 for p in the 2 2 verte directri standard form of the equation of a parabola. 2 5 4p Standard form, vertical ais of smmetr 2 5 4 3 } 2 2 Substitute 3 } 2 for p. 2 5 6 Simplif. 9.2 Graph and Write Equations of Parabolas 62

GUIDED PRACTICE for Eamples and 2 Graph the equation. Identif the focus, directri, and ais of smmetr of the parabola.. 2 526 2. 2 5 2 3. 52 } 4 2 4. 5 } 3 2 Write the standard form of the equation of the parabola with verte at (0, 0) and the given directri or focus. 5. Directri: 5 2 6. Directri: 5 4 7. Focus: (22, 0) 8. Focus: (0, 3) PARABOLIC REFLECTORS Parabolic reflectors have cross sections that are parabolas. Incoming sound, light, or other energ that arrives at a parabolic reflector parallel to the ais of smmetr is directed to the focus (Diagram ). Similarl, energ that is emitted from the focus of a parabolic reflector and then strikes the reflector is directed parallel to the ais of smmetr (Diagram 2). Focus Focus Diagram Diagram 2 E XAMPLE 3 TAKS REASONING: Multi-Step Problem SOLAR ENERGY The EuroDish, developed to provide electricit in remote areas, uses a parabolic reflector to concentrate sunlight onto a high-efficienc engine located at the reflector s focus. The sunlight heats helium to 6508C to power the engine. 4.5 m 8.5 m Write an equation for the EuroDish s cross section with its verte at (0, 0). How deep is the dish? Solution STEP Write an equation for the cross section. The engine is at the focus, which is p 5 4.5 meters from the verte. Because the focus is above the verte, p is positive, so p 5 4.5. An equation for the cross section of the EuroDish with its verte at the origin is as follows: 2 5 4p Standard form, vertical ais of smmetr 2 5 4(4.5) Substitute 4.5 for p. STEP 2 2 5 8 Simplif. Find the depth of the EuroDish. The depth is the -value at the dish s outside edge. The dish etends 8.5 } 2 5 4.25 meters to either side of the verte (0, 0), so substitute 4.25 for in the equation from Step. 2 5 8 Equation for the cross section (4.25) 2 5 8 Substitute 4.25 for..0 ø Solve for. c The dish is about meter deep. 622 Chapter 9 Quadratic Relations and Conic Sections

GUIDED PRACTICE for Eample 3 9. MICROWAVES A parabolic microwave antenna is 6 feet in diameter. Find an equation for the cross section of the antenna with its verte at the origin and its focus 0 feet to the right of its verte. Then find the antenna s depth. 9.2 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 5, 27, and 57 5 TAKS PRACTICE AND REASONING Es. 25, 38, 5, 59, 6, and 62. VOCABULARY Cop and complete: A parabola is the set of all points in a plane equidistant from a point called the? and a line called the?. 2. WRITING Compare the graphs of 2 5 4p and 2 5 4p. EXAMPLE on p. 62 for Es. 3 25 GRAPHING Graph the equation. Identif the focus, directri, and ais of smmetr of the parabola. 3. 2 5 6 4. 2 526 5. 2 5 20 6. 2 5 28 7. 2 520 8. 2 5 30 9. 2 522 0. 2 5236. 2 5 2 2. 22 5 2 3. 5 4 2 4. 2 2 5 48 5. 5 2 525 6. 2 2 5 8 7. 224 5 3 2 8. 4 5 6 2 9. } 8 2 2 5 0 20. 4 2 2 5 0 2. 5 2 2 5 0 22. 25 } 3 2 5 0 ERROR ANALYSIS Describe and correct the error in graphing the parabola. 23. 26 2 5 0 24. 0.5 2 5 0 4 (0,.5) 4 = 0.5 2 (0.5, 0) 2 =.5 25. TAKS REASONING What is the directri of the parabola 5 3 2 5 0? A 525 B 52.25 C 52.25 D 5.25 EXAMPLE 2 on p. 62 for Es. 26 50 WRITING EQUATIONS Write the standard form of the equation of the parabola with the given focus and verte at (0, 0). 26. (2, 0) 27. (25, 0) 28. (3, 0) 29. (0, 24) 30. (0, 8) 3. (0, 20) 32. (0, 26) 33. (29, 0) 34. 0, 7 } 4 2 35. 0, 23 } 8 2 36. 5 } 2, 02 37. 2 9 } 6, 02 9.2 Graph and Write Equations of Parabolas 623

38. TAKS REASONING What is an equation of the parabola with focus at (28, 0) and verte at (0, 0)? A 2 5232 B 2 520.5 C 2 528 D 2 5232 WRITING EQUATIONS Write the standard form of the equation of the parabola with the given directri and verte at (0, 0). 39. 5 3 40. 527 4. 525 42. 5 2 43. 524 44. 522 45. 5 6 46. 5 47. 52 3 } 2 48. 5 5 } 2 49. 52 } 6 50. 52 } 8 5. TAKS REASONING Predict how the indicated change in a will affect the focus, directri, and shape of the given equation s graph. Then graph both the original and revised equations in the same coordinate plane. a. 2 5 a; a changes from to 4 b. 2 5 a; a changes from 6 to 2 } 2 52. WRITING Suppose that 2 5 4p and 5 a 2 represent the same parabola. Eplain how a and p are related. 53. VISUAL THINKING As p increases, how does the width of the graph of 2 5 4p change? Eplain. 54. CHALLENGE Consider the parabola with focus (0, p) and directri 52p. Let (, ) be an point on the parabola. Use the fact that (, ) is equidistant from the focus and directri to show that 2 5 4p. PROBLEM SOLVING EXAMPLE 3 on p. 622 for Es. 55 59 55. SOLAR ENERGY Solar energ can be concentrated using long troughs that have a parabolic cross section. The collected energ s uses include heating buildings, producing electricit, and producing fresh water from seawater. Write an equation for the cross section of the trough shown. How deep is it? focus 6 ft 7 ft 56. BIOLOGY Scientists studing dolphin echolocation can simulate the projection of a dolphin s clicking sounds using computer models. The models originate the sounds at the focus of a parabolic reflector. The parabola in the graph models the cross section (with units in inches) of the reflector used to simulate sound projection for a bottlenose dolphin. What is the focal length (the distance from the verte to the focus)? focus (3, 4) 624 5 WORKED-OUT SOLUTIONS on p. WS 5 TAKS PRACTICE AND REASONING

57. MULTI-STEP PROBLEM The parabolic antenna used b a television station to transmit is 46 inches in diameter. Its focus is 48 inches from the verte. a. Sketch the antenna twice: once opening upward and once opening left. b. Use our sketches from part (a) to write two equations for the antenna s cross section: one of the form 2 5 4p and one of the form 2 5 4p. c. How deep is the antenna s dish? Does it matter which equation from part (b) ou use to find our answer? Eplain. 58. RADIO TELESCOPES The Ver Large Arra in New Meico consists of 27 radio telescopes. For each parabolic telescope dish, the diameter is 25 meters and the distance between the verte and focus is 0.36 times the diameter. Write an equation for the cross section of a dish opening upward with its verte at the origin. How deep is each dish? at classzone.com 59. TAKS REASONING Searchlights use parabolic reflectors to project their beams. The cross section of a 9.5-inch-deep searchlight reflector has equation 2 5 0.5. a. How wide is the beam of light projected from the searchlight s reflector? b. Write an equation for the cross section of a reflector that has the same depth as the original reflector, but which projects a wider beam. Eplain how ou found our answer. How wide is the new reflector s beam? c. Repeat part (b) for a beam narrower than the original. 60. CHALLENGE The latus rectum of a parabola is the line segment that is parallel to the directri, passes through the focus, and has endpoints that lie on the parabola. Find the length in terms of p of the latus rectum of a parabola with equation 2 5 4p. latus rectum 2 5 4p verte (0, 0) 52p MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW TAKS Preparation p. 324; TAKS Workbook 6. TAKS PRACTICE The figure shows a triangular cit park. What is the perimeter of the park? TAKS Obj. 8 A 70 d C 200 d B 90 d D 273 d 85 d 75 d REVIEW TAKS Preparation p. 46; TAKS Workbook 62. TAKS PRACTICE Sarah has 9 points less than she needs to make a grade of A in her mathematics course. Her point total for the course is 423 points. How man points are possible in the course? (Assume she needs 90% of the total possible points for an A.) TAKS Obj. 9 F 460 G 470 H 480 J 490 EXTRA PRACTICE for Lesson 9.2, p. 08 ONLINE QUIZ at classzone.com 625

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