THE PARABOLA section. Developing the Equation


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1 80 (0) Chapter Nonlinear Sstems and the Conic Sections. THE PARABOLA In this section Developing the Equation Identifing the Verte from Standard Form Smmetr and Intercepts Graphing a Parabola Maimum or Minimum Value of The parabola is one of four different curves that can be obtained b intersecting a cone and a plane as in Fig... These curves, called conic sections, are the parabola, circle, ellipse, and hperbola. You studied parabolas in Section.6, but in this section ou will stud parabolas in more detail. You will see how the parabola and the other conic sections are used in applications such as satellite dishes, telescopes, spotlights, and navigation. Parabola Parabola Hperbola Focus Verte Circle Ellipse FIGURE. Developing the Equation FIGURE. Directri In Section.6 we called the graph of a b c a parabola. This equation is the standard equation of a parabola. In this section ou will see that the following geometric definition describes the same curve as the equation. Parabola Focus Given a line (the directri) and a point not on the line (the focus), the set of all points in the plane that are equidistant from the point and the line is called a parabola. FIGURE. p > 0 (0, p) (, ) (0, 0) = p (, p) FIGURE.6 The verte of the parabola is the midpoint of the line segment joining the focus and the directri, perpendicular to the directri. See Fig... The focus of a parabola is important in applications. When parallel ras of light travel into a parabolic reflector, the are reflected toward the focus as in Fig... This propert is used in telescopes to see the light from distant stars. If the light source is at the focus, as in a searchlight, the light is reflected off the parabola and projected outward in a narrow beam. This reflecting propert is also used in camera lenses, satellite dishes, and eavesdropping devices. To develop an equation for a parabola, given the focus and directri, choose the point (0, p), where p 0 as the focus and the line p as the directri, as shown in Fig..6. The verte of this parabola is (0, 0). For an arbitrar point (, ) on the parabola the distance to the directri is the distance from (, ) to(, p).
2 . The Parabola () 8 = p (, p) (0, 0) (, ) (0, p) p < 0 FIGURE.7 The distance to the focus is the distance between (, ) and (0, p). We use the fact that these distances are equal to write the equation of the parabola: ( 0) ( p) ( ) ( (p)) To simplif the equation, first remove the parentheses inside the radicals: p p p p p p p p p Square each side. p a Let a p. Subtract and p from each side. So a curve that satisfies the geometric definition of parabola has an equation of the form a b c. If the focus is at (0, p) with p 0 and the directri is p, then the parabola opens downward as shown in Fig..7. Deriving the equation from the, which is of the form. Note that a and p have the same sign because a. distances (as was just done for p 0) again ields p a for a p If a parabola has verte (0, 0) and opens up or down, then its equation is of the form a. In general, if (h, k) is the verte, (h, k p) is the focus, and k p is the directri, then we can develop the equation a( h) k for the parabola just as we developed the equation a. Graphs of a( h) k are shown in Fig..8. p a > 0 (h, k + p) (h, k) a p = a( h) + k Directri: = k p FIGURE.8 a < 0 Directri: = k p (h, k) = a( h) + k (h, k + p) Parabolas in the Form a( h) k The graph of the equation a( h) k (a 0) is a parabola with verte (h, k), focus (h, k p), and directri k p, where a. If a 0, the parabola opens upward; if a 0, the parabola opens downward. p Note that the locations of the focus and directri determine the value of a and the shape and opening of the parabola. CAUTION For a parabola that opens upward, p 0, and the focus (h, k p) is above the verte (h, k). For a parabola that opens downward, p 0, and the focus (h, k p) is below the verte (h, k). In either case the distance from the verte to the focus and the verte to the directri is p.
3 8 () Chapter Nonlinear Sstems and the Conic Sections E X A M P L E ( 0, = ( = FIGURE.9 Finding the verte, focus, and directri, given an equation Find the verte, focus, and directri for the parabola. Compare to the general formula a( h) k. We see that h 0, k 0, and a. So the verte is (0, 0). Because a, we can use a to p get, p or p. Use (h, k p) to get the focus 0,. Use the equation k p to get as the equation of the directri. See Fig..9. E X A M P L E ( 7, ( (, ) = Finding an equation, given a focus and directri Find the equation of the parabola with focus (, ) and directri. Because the verte is halfwa between the focus and directri, the verte is, 7. See Fig..0. The distance from the verte to the focus is. Because the focus is above the verte, p is positive. So p, and a. The equation is p ( ()) 7. Simplif to get the equation. FIGURE.0 The equation of a parabola can be written in two different forms. To change the form a( h) k to the form a b c, we square the binomial and combine like terms, as in Eample. To change from a b c to a( h) k, we complete the square. E X A M P L E Converting a b c to a( h) k Write in the form a( h) k and identif the verte, focus, and directri of the parabola. Use completing the square to rewrite the equation: ( ) ( ) Complete the square. ( ) ( ) Move () outside the parentheses. The verte is (, ). Because a, we have p, p
4 . The Parabola () 8 The graphs of and ( ) appear to be identical. This supports the conclusion that the equations are equivalent. calculator closeup 0 0 and p 8. Because the parabola opens upward, the focus is 8 unit above the verte at, 8, or, 8, and the directri is the horizontal line unit below the verte, or. 8 CAUTION Be careful when ou complete a square within parentheses as in Eample. For another eample, consider the equivalent equations ( ), ( ), and ( ). Identifing the Verte from Standard Form If the equation of a parabola is in the form a( h) k, then its verte is (h, k). If we complete the square on the standard equation of the parabola a b c to get it into the form a( h) k, we will find the verte in terms of a, b, and c: a b c a b a c a b b b a a a c b a b a a b b b a a c Move a a b a a b b b a a c a a a b a a ac b a Factor a out of the first two terms. and b b a a outside the parentheses. Build up the denominator. Simplif. The equation is now in the form a( h) k, and the verte is b, ac b a a. We summarize these results as follows. Parabolas in the Form a b c The graph of a b c (for a 0) is a parabola opening upward if a 0 and downward if a 0. The coordinate of the verte is b. a b You can use ac to get the coordinate of the verte, but it is usuall a easier to get it from a b c for b. We use the second approach in a Eample.
5 8 () Chapter Nonlinear Sstems and the Conic Sections E X A M P L E A calculator graph can be used to check the verte and opening of a parabola. Ais of smmetr calculator closeup = b a 0 Finding the features of a parabola from standard form Find the verte, focus, and directri of the parabola 9, and determine whether the parabola opens upward or downward. The coordinate of the verte is b 9 9 a ( ) 6. To find the coordinate of the verte, let in 9 : The verte is, 7. Because a, the parabola opens downward. To find the focus, use to get p p. The focus is of a unit below the verte at, 7 or,. The directri is the horizontal line of a unit above the verte, 7 or. 6 Smmetr and Intercepts The graph of shown in Fig..9 is said to be smmetric about the ais because the two halves of the parabola would coincide if the paper were folded on the ais. In general, the vertical line through the verte is called the ais of smmetr for the parabola. Because the coordinate of the verte for a b c is b, the ais of smmetr is b. See Fig... a a The parabola a b c has eactl one intercept. If 0 in the equation, then a(0) b(0) c c. So the intercept is (0, c). To find the intercepts, we let 0 in the equation a b c and solve the quadratic equation a b c 0. The number of intercepts ma be 0,, or, depending on the number of solutions to the quadratic equation. See Fig... FIGURE. No intercepts One intercept FIGURE. Two intercepts Graphing a Parabola When graphing a parabola, we can use the features that we have discussed to improve accurac and understanding. These features are summarized in the following bo as a strateg for graphing parabolas.
6 . The Parabola () 8 Graphing the Parabola a b c To graph the parabola a b c, use the following facts:. The parabola opens upward if a 0 and opens downward if a 0.. The coordinate of the verte is b. a. The graph is smmetric about the vertical line b. a. The intercept is (0, c).. The intercepts are found b solving a b c 0. CAUTION The focus and directri are important features of a parabola, but the are not part of the curve itself. So we do not usuall find the focus and directri when graphing a parabola from an equation. E X A M P L E helpful hint When drawing a parabola such as the one in Fig.. b hand, use our hand like a compass and draw it in two steps. If ou are righthanded, sketch the left half of the parabola first. Then turn our paper upside down to sketch the right half. = + 6 FIGURE. Graphing a parabola Determine whether the parabola 6 opens upward or downward, and find the verte, ais of smmetr, intercepts, and intercept. Find several additional points on the parabola, and sketch the graph. Because a, this parabola opens downward. The coordinate of the verte is b ( ) a ( ). If, then 6 6. So the verte is,. The ais of smmetr is the vertical line. To find the intercepts, let 0 in the original equation: ( )( ) 0 0 or 0 or The intercepts are (, 0) and (, 0). The intercept is (0, 6). Using all of this information and the additional points (, ), (, ), and (, 6), we get the graph shown in Fig... Maimum or Minimum Value of On a parabola that opens upward, the minimum value that attains is the coordinate of the verte. On a parabola that opens downward, the maimum value of is again the coordinate of the verte. We use the fact that the coordinate of the verte is b to obtain the maimum or minimum value of. a
7 86 (6) Chapter Nonlinear Sstems and the Conic Sections E X A M P L E 6 calculator closeup The graph of = ( ) supports the conclusion that the maimum value of occurs when =. Maimizing a product Find two numbers that have the maimum product, subject to the condition that their sum is. Because their sum is, we can let represent one number and represent the other. If represents their product, we can write the equation ( ) or. This equation is the equation of a parabola that opens downward. Its highest point, which is the maimum value of, occurs when b a ( ). If, then. The two numbers are and. Among the numbers with a sum of, the give the maimum product. So no two numbers with a sum of can have a product larger than. CAUTION Be sure to answer the question that is asked in a maimumminimum problem. If a b c, then the maimum (or minimum) value of is the coordinate of the verte. The value of that causes to reach its maimum (or minimum) value is the coordinate of the verte. E X A M P L E 7 Maimizing revenue A manufacturer of inline skates uses the formula R 60 to predict the weekl revenue in dollars that will be produced when the skates are priced at dollars per pair. Find the price that will produce the maimum revenue, and find the maimum possible revenue. The graph of R 60 is a parabola that opens downward as shown in Fig.. on the net page. To find the verte of the parabola, use a and b 60 in b : a () Now use 90 in R 60 to find R: R 60(90) 90 6,00 So if the skates are priced at $90 per pair, then the manufacturer will receive the maimum weekl revenue $6,00.
8 . The Parabola (7) 87 Revenue (in thousands of dollars) R Price (in dollars) FIGURE. WARMUPS True or false? Eplain our answer.. There is a parabola with focus (, ), directri, and verte (0, 0). False. The focus for the parabola is (0, ). True. The graph of ( ) is a parabola with verte (, ). True. The graph of 6 is a parabola. False. The graph of 9 is a parabola opening upward. False 6. For the verte and intercept are the same point. True 7. A parabola with verte (, ) and focus (, ) has no intercepts. True 8. The parabola 9 has no intercept. False 9. If (, ) satisfies a( ) k, then so does (, ). True 0. The parabola ( ) has onl one intercept. True. EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. What is the definition of a parabola given in this section? A parabola is the set of all points in a plane that are equidistant from a given line and a fied point not on the line.. What is the location of the verte? The verte is the midpoint of the line segment joining the focus and directri, perpendicular to the directri.. What are the two forms of the equation of a parabola? A parabola can be written in the forms a b c or a( h) k.. What is the distance from the focus to the verte in an parabola of the form a b c? The distance from the focus to the directri is p, where a (p).. How do we convert an equation of the form a b c into the form a( h) k? We use completing the square to convert a b c into a( h) k. 6. How do we convert an equation of the form a( h) k into the form a b c? To convert a( h) k into the form a b c, square the binomial, multipl b a, then add like terms. Find the verte, focus, and directri for each parabola. See Eample. 7. Verte (0, 0), focus 0, 8, directri Verte (0, 0), focus 0,, directri.
9 88 (8) Chapter Nonlinear Sstems and the Conic Sections 9. Verte (0, 0), focus (0, ), directri 0. Verte (0, 0), focus (0, ), directri. ( ) Verte (, ), focus (,.), directri.. ( ) Verte (, ), focus (, ), directri 6. ( ) 6 Verte (, 6), focus (,.7), directri 6.. ( ) Verte (, ), focus,, directri Find the equation of the parabola with the given focus and directri. See Eample.. Focus (0, ), directri 8 6. Focus (0, ), directri 7. Focus 0,, directri 8. Focus 0, 8, directri 8 9. Focus (, ), directri 6 0. Focus (, ), directri. Focus (, ), directri 8 8. Focus (, ), directri 8. Focus (,.), directri Focus, 7 8, directri 0 8 Write each equation in the form a( h) k. Identif the verte, focus, and directri of each parabola. See Eample.. 6 ( ) 8, verte (, 8), focus (, 7.7), directri ( ), verte (, ), focus (, 0.7), directri. 7. ( ), verte (, ), focus (,.87), directri ( ) 0, verte (, 0), focus, 9, directri ( ), verte (, ), focus, 7 8, directri ( ) 0, verte (, 0), focus, 9, directri 0. 0 ( ) 80, verte (, 80), focus, , directri , verte,, focus, 9 9 8, directri 0 8 Find the verte, focus, and directri of each parabola (without completing the square), and determine whether the parabola opens upward or downward. See Eample.. Verte (, ), focus,, directri, upward. 6 7 Verte (, 6), focus,, directri 6, upward. Verte (, ), focus,, directri, downward 6. 9 Verte (, ), focus,, directri, downward 7. 6 Verte (, ), focus,, directri, upward 8. Verte (, ), focus, 7 8, directri 8, upward 9. Verte, 7 downward 0., focus,, directri 9, Verte,, focus,, directri, downward. Verte (0, ), focus 0,, directri, upward. 6 Verte (0, 6), focus 0, 6 8, directri 7 8, downward
10 . The Parabola (9) 89 Find the verte, ais of smmetr, intercepts, and intercept for each parabola. Find several additional points on the parabola, and then sketch its graph. See Eample ais of Verte (, ), ais of Verte, smmetr smmetr, intercepts (0, ), (, 0), (, 0), intercepts (0, 8), (, 0), (, 0) Verte (, 0), ais of Verte (, 0), ais of smmetr, smmetr, intercepts (0, ), (, 0) intercepts (0, 9), (, 0).. 9 Verte, 0, ais of Verte, 0, ais of Verte (, 9), ais of Verte (, 6), ais of smmetr, smmetr, intercepts (0, 8), (, 0), intercepts (0, ), (, 0) (, 0), (, 0) smmetr, intercepts (0, ),, 0 smmetr, intercepts, 0, (0, 9) 7. ( ) 8. ( ) Verte (, ), ais of Verte (, ), ais of smmetr, smmetr, intercept (0, ) intercepts (0, ), 6, 0, 6, 0. Verte is,, ais of smmetr, intercepts (0, 0), (, 0). 9 Verte, 7, ais of smmetr, intercepts (0, 0), (, 0)
11 90 (0) Chapter Nonlinear Sstems and the Conic Sections. 6. Verte (0, ), ais of Verte (0, ), ais of smmetr 0, smmetr 0, intercept (0, ) intercepts (0, ), 6, 0, 6, Verte (, ), ais of Verte (, ), ais of smmetr, smmetr, interintercepts (0, ), cepts (0, ), (, 0), (, 0), (, 0) (, 0) 6. Maimum product. What is the maimum product that can be obtained b two numbers that a have a sum of 6? 9 6. Maimum area. A gardener plans to enclose a rectangular area with 60 feet (ft) of fencing. What dimensions for the length and width would maimize the area? 0 ft b 0 ft 6. Maimum area. Another gardener has a footwide gate for her rectangular garden in place. If she has 00 ft of fencing in addition to the gate, then what dimensions for the length and width would maimize the area? 6 ft b 6 ft 6. Minimum hpotenuse. If the total length of the legs of a right triangle is 6 ft, then what lengths for the legs would minimize the square of the hpotenuse? ft each 66. Maimum revenue. A concert promoter uses the formula R 700p 0p to calculate the total revenue for a concert if the ticket price is p dollars each. What ticket price will maimize the revenue? See the accompaning figure. $ 9. ( ) 60. ( ) Verte (, 0), ais of Verte (, ), ais of smmetr, smmetr, intercepts (0, ), (, 0) intercepts (0, ),, 0,, 0 Revenue from concert tickets (in thousands of dollars) R Price of tickets (in dollars) FIGURE FOR EXERCISE Maimum height. If a punter kicks a football straight up at a velocit of 8 feet per second (ft/sec) from a height of 6 ft, then the football s distance above the earth after t seconds is given b the formula s 6t 8t 6. What is the maimum height reached b the ball? How long does the ball take to reach its maimum height? Could the punter Solve each problem. See Eamples 6 and Maimum product. Find two numbers that have the maimum possible product among numbers that have a sum of 8. and Height of football (ft) s t Time (sec) FIGURE FOR EXERCISE 67
12 . The Parabola () 9 hit the roof of the New Orleans Superdome, which is 7 ft above the plaing field? 6 ft, sec, no 68. Maimum height. If a baseball is hit straight upward at 0 ft/sec from a height of ft, then its distance above the earth after t seconds is given b the formula s 6t 0t. What is the maimum height attained b the ball? How long does the ball take to reach its maimum height? 6.6 ft,.687 sec v 0 ft/sec 7. Arecibo Observator. The largest radio telescope in the world uses a,000ft parabolic dish, suspended in a valle in Arecibo, Puerto Rico. The antenna hangs above the verte of the dish on cables stretching from two towers. The accompaning figure shows a cross section of the parabolic dish and the towers. Assuming the verte is at (0, 0), find the equation for the parabola. Find the distance from the verte to the antenna located at the focus ,. ft Antenna at focus ft 00 ft 00 ft FIGURE FOR EXERCISE Minimum cost. It costs Acme Manufacturing C dollars per hour to operate its golf ball division. An analst has determined that C is related to the number of golf balls produced per hour,, b the equation C What number of balls per hour shouldacme produce to minimize the cost per hour of manufacturing these golf balls? Maimum profit. A chain store manager has been told b the main office that dail profit, P, is related to the number of clerks working that da,, according to the equation P 00. What number of clerks will maimize the profit, and what is the maimum possible profit? 6 clerks, $ World s largest telescope. The largest reflecting telescope in the world is the 6meter (m) reflector on Mount Pastukhov in Russia. The accompaning figure shows a cross section of a parabolic mirror 6 m in diameter with the verte at the origin and the focus at (0, ). Find the equation of the parabola. 6 0 (0, ) 000 ft FIGURE FOR EXERCISE 7 Graph both equations of each sstem on the same coordinate aes. Use elimination of variables to find all points of intersection (, ), (, ) (, ), (, ) (, ) (, 6) 6 m FIGURE FOR EXERCISE 7
13 9 () Chapter Nonlinear Sstems and the Conic Sections ,, (, 0),, (, 0) 8. (0, 0) (, ) (, 0) 8. (, 0) (0, ) (, ) (, ), (, 6) (, 6), (, ) Write an equation in the form a b c for each given parabola. 8. (0, ) (, 0) (, 0) 8. (, ) (, ) (0, 0) GETTING MORE INVOLVED 8. Eploration. Consider the parabola with focus ( p, 0) and directri p for p 0. Let (, ) be an arbitrar point on the parabola. Write an equation epressing the fact that the distance from (, ) to the focus is equal to the distance from (, ) to the directri. Rewrite the equation in the form a, where a. p 86. Eploration. In general, the graph of a( h) k for a 0 is a parabola opening left or right with verte at (k, h). a) For which values of a does the parabola open to the right, and for which values of a does it open to the left? b) What is the equation of its ais of smmetr? c) Sketch the graphs ( ) and ( ). a) Right for a 0 and left for a 0 b) h GRAPHING CALCULATOR EXERCISES 87. Graph using the viewing window with and 0. Net graph using the viewing window and 7. Eplain what ou see. The graphs have identical shapes. 88. Graph and 6 9 in the viewing window and 0. Does the line appear to be tangent to the parabola? Solve the sstem and 6 9 to find all points of intersection for the parabola and the line. Intersection (, 9)
THE PARABOLA 13.2. section
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