Deriving the Standard-Form Equation of a Parabola

Transcription

1 Name Class Date. Parabolas Essential Question: How is the distance formula connected with deriving equations for both vertical and horizontal parabolas? Eplore Deriving the Standard-Form Equation of a Parabola A parabola is defined as a set of points equidistant from a line (called the directri) and a point (called the focus). The focus will alwas lie on the ais of smmetr, and the directri will alwas be perpendicular to the ais of smmetr. This definition can be used to derive the equation for a horizontal parabola opening to the right with its verte at the origin using the distance formula. (The derivations of parabolas opening in other directions will be covered later.) (-p, ) (-p, 0) Directri d Resource Locker (, ) d (p, 0) Focus A The coordinates for the focus are given b. B Write down the epression for the distance from a point (, ) to the coordinates of the focus: ( - ) d = + ( - ) C The distance from a point to a line is measured b drawing a perpendicular line segment from the point to the line. Find the point where a horizontal line from (, ) intersects the directri (defined b the line = -p for a parabola with its verte on the origin). D Write down the epression for the distance from a point, (, ) to the point from Step C: ( - ) d = + ( - ) Houghton Mifflin Harcourt Publishing Compan E F Setting the two distances the same and simplifing gives. ( - p) + = ( + p) To continue solving the problem, square both sides of the equation and epand the squared binomials. + p + p + = + p + p Collect terms. + p + p + = 0 G Finall, simplif and arrange the equation into the standard form for a horizontal parabola (with verte at (0, 0)): = Module 175 Lesson

2 Reflect 1. Wh was the directri placed on the line = -p?. Discussion How can the result be generalized to arrive at the standard form for a horizontal parabola with a verte at (h, k) : ( - k) = p ( - h)? Eplain 1 Writing the Equation of a Parabola with Verte at (0, 0) The equation for a horizontal parabola with verte at (0, 0) is written in the standard form as = p. It has a vertical directri along the line = -p, a horizontal ais of smmetr along the line = 0, and a focus at the point (p, 0). The parabola opens toward the focus, whether it is on the right or left of the origin (p > 0 or p < 0). Vertical parabolas are similar, but with horizontal directrices and vertical aes of smmetr: Parabolas with Vertices at the Origin Vertical Horizontal Equation in standard form = p = p p > 0 Opens upward Opens rightward p < 0 Opens downward Opens leftward Focus (0, p) (p, 0) Directri = -p = -p Ais of Smmetr = 0 = 0 Houghton Mifflin Harcourt Publishing Compan Module 176 Lesson

3 Eample 1 Find the equation of the parabola from the description of the focus and directri. Then make a sketch showing the parabola, the focus, and the directri. A Focus (, 0), directri = A vertical directri means a horizontal parabola. Confirm that the verte is at (0, 0) : a. The -coordinate of the verte is the same as the focus: 0. b. The -coordinate is halfwa between the focus () and the directri (+): 0. c. The verte is at (0, 0) Use the equation for a horizontal parabola, = p, and replace p with the coordinate of the focus: = () Simplif: = -3 Plot the focus and directri and sketch the parabola. B Focus (0, -), directri = A [vertical/horizontal] directri means a [vertical/horizontal] parabola. Confirm that the verte is at (0, 0) : a. The -coordinate of the verte is the same as the focus: 0. b. The -coordinate is halfwa between the focus, and the directri, : c. The verte is at (0, 0). Use the equation for a vertical parabola,, and replace p with the -coordinate of the focus: = Houghton Mifflin Harcourt Publishing Compan Simplif: = Plot the focus, the directri, and the parabola. Your Turn Find the equation of the parabola from the description of the focus and directri. Then make a sketch showing the parabola, the focus, and the directri. 3. Focus (, 0), directri = -. Focus ( 0, - 1_ ), directri = 1_ Module 177 Lesson

4 Eplain Writing the Equation of a Parabola with Verte at (h, k) The standard equation for a parabola with a verte (h, k) can be found b translating from (0, 0) to (h, k): substitute ( - h) for and ( - k) for. This also translates the focus and directri each b the same amount. Parabolas with Verte (h, k) Vertical Horizontal Equation in standard form ( - h) = p ( - k) ( - k) = p ( - h) p > 0 Opens upward Opens rightward p < 0 Opens downward Opens leftward Focus (h, k + p) (h + p, k) Directri = k - p = h - p Ais of Smmetr = h = k p is found halfwa from the directri to the focus: For vertical parabolas: p = For horizontal parabolas: p = ( value of focus) - ( value of directri) ( value of focus) - ( value of directri) The verte can be found from the focus b relating the coordinates of the focus to h, k, and p. Eample Find the equation of the parabola from the description of the focus and directri. Then make a sketch showing the parabola, the focus, and the directri. A Focus (3, ), directri = 0 A horizontal directri means a vertical parabola. ( value of focus) - ( value of directri) p = = _ - 0 = 1 h = the -coordinate of the focus = 3 Solve for k: The -value of the focus is k + p, so k + p = k + 1 = k = 1 Write the equation: ( - 3) = ( - 1) Plot the focus, the directri, and the parabola Houghton Mifflin Harcourt Publishing Compan Module 17 Lesson

5 B Focus (-1, -1), directri = 5 A vertical directri means a parabola. - ( value of focus) - ( value of directri) p = = = k = the -coordinate of the focus = Solve for h: The -value of the focus is h + p, so h + p = h + (-3) = h = Write the equation: ( + 1) = ( - ) Your Turn Find the equation of the parabola from the description of the focus and directri. Then make a sketch showing the parabola, the focus, and the directri. 5. Focus (5, -1), directri = Focus (-, 0), directri = Houghton Mifflin Harcourt Publishing Compan Eplain 3 Rewriting the Equation of a Parabola to Graph the Parabola A second-degree equation in two variables is an equation constructed b adding terms in two variables with powers no higher than. The general form looks like this: a + b + c + d + e = 0 Epanding the standard form of a parabola and grouping like terms results in a second-degree equation with either a = 0 or b = 0, depending on whether the parabola is vertical or horizontal. To graph an equation in this form requires the opposite conversion, accomplished b completing the square of the squared variable. Module 179 Lesson

6 Eample 3 Convert the equation to the standard form of a parabola and graph the parabola, the focus, and the directri. A = 0 Isolate the terms and complete the square on. Isolate the terms. - = - 1 Add _- ( ) to both sides. - + = - Factor the perfect square trinomial on the left side. ( - ) = - Factor out from the right side. ( - ) = ( - ) This is the standard form for a vertical parabola. Now find p, h, and k from the standard form ( - h ) = p( - k) in order to graph the parabola, focus, and directri. p =, so p = 1. h =, k =. Verte = (h, k) = (, ). Focus = (h, k + p) = (, + 1) = (, 3). Directri: = k - p = - 1, or = B = 0 Isolate the terms. + = 1 Add ( ) _ to both sides. + + = Factor the perfect square trinomial. ( + ) Factor out on the right. ( + ) = = ( + ) Identif the features of the graph using the standard form of a horizontal parabola, ( k ) = p( h): p =, so p =. h =, k = Verte = (h, k) = (, ) Focus = (h + p, k) = (, ) Directri: = h - p or = Houghton Mifflin Harcourt Publishing Compan Module 10 Lesson

7 Your Turn Convert the equation to the standard form of a parabola and graph the parabola, the focus, and the directri = = Eplain Solving a Real-World Problem Parabolic shapes occur in a variet of applications in science and engineering that take advantage of the concentrating propert of reflections from the parabolic surface at the focus. Houghton Mifflin Harcourt Publishing Compan A Parabolic microphones are so-named because the use a parabolic dish to bounce sound waves toward a microphone placed at the focus of the parabola in order to increase sensitivit. The dish shown has a cross section dictated b the equation = 3 where and are in inches. How far from the center of the dish should the microphone be placed? The cross section matches the standard form of a horizontal parabola with h = 0, k = 0, p =. Therefore the verte, which is the center of the dish, is at (0, 0) and the focus is at (, 0), inches awa. Module 11 Lesson

8 B A reflective telescope uses a parabolic mirror to focus light ras before creating an image with the eepiece. If the focal length (the distance from the bottom of the mirror s bowl to the focus) is 10 mm and the mirror has a 70 mm diameter (width), what is the depth of the bowl of the mirror?? 70 mm 10 mm parabolic mirror eepiece plane mirror prime focus The distance from the bottom of the mirror s bowl to the focus is p. The verte location is not specified (or needed), so use (0, 0) for simplicit. The equation for the mirror is a horizontal parabola (with the distance along the telescope and the position out from the center). ( - ) = p ( - ) = Since the diameter of the bowl of the mirror is 70 mm, the points at the rim of the mirror have -values of 35 mm and -35 mm. The -value of either point will be the same as the -value of the point directl above the bottom of the bowl, which equals the depth of the bowl. Since the points on the rim lie on the parabola, use the equation of the parabola to solve for the -value of either edge of the mirror. = mm The bowl is approimatel.19 mm deep. Your Turn 9. A football team needs one more field goal to win the game. The goalpost that the ball must clear is 10 feet (~3.3 d) off the ground. The path of the football after it is kicked for a 35-ard field goal is given b the equation - 11 = ( - 0), in ards. Does the team win? Houghton Mifflin Harcourt Publishing Compan Module 1 Lesson

9 Elaborate 10. Eamine the graphs in this lesson and determine a relationship between the separation of the focus and the verte, and the shape of the parabola. Demonstrate this b finding the relationship between p for a vertical parabola with verte of (0, 0) and a, the coefficient of the quadratic parent function = a. 11. Essential Question Check-In How can ou use the distance formula to derive an equation relating and from the definition of a parabola based on focus and directri? Evaluate: Homework and Practice Find the equation of the parabola with verte at (0, 0) from the description of the focus and directri and plot the parabola, the focus, and the directri. Online Homework Hints and Help Etra Practice 1. Focus at (3, 0), directri: = -3. Focus at (0, -5), directri: = 5 Houghton Mifflin Harcourt Publishing Compan Focus at (-1, 0), directri: = 1. Focus at (0, ), directri: = Module 13 Lesson

10 Find the equation of the parabola with the given information. 5. Verte: (-3, 6) ; Directri: = Verte: (6, 0) ; Focus: (6, 11) Find the equation of the parabola with verte at (h, k) from the description of the focus and directri and plot the parabola, the focus, and the directri. 7. Focus at (5, 3), directri: = 7. Focus at (-3, 3), directri: = Convert the equation to the standard form of a parabola and graph the parabola, the focus, and the directri = = Houghton Mifflin Harcourt Publishing Compan 11. Communications The equation for the cross section of a parabolic satellite television dish is = 50, measured in inches. How far is the focus from the verte of the cross section? 1 Module 1 Lesson

11 1. Engineering The equation for the cross section of a spotlight is + 5 = 1, measured in inches. The bulb is located at the focus. How far is the bulb from the verte of the cross section? When a ball is thrown into the air, the path that the ball travels is modeled b the parabola - 7 = ( - 0), measured in feet. What is the maimum height the ball reaches? How far does the ball travel before it hits the ground? 1. A cable for a suspension bridge is modeled b - 55 = 0.005, where is the horizontal distance, in feet, from the support tower and is the height, in feet, above the bridge. How far is the lowest point of the cable above the bridge? Houghton Mifflin Harcourt Publishing Compan Image Credits: J. Aa./ Shutterstock 15. Match each equation to its graph. + 1 = 1_ 16 ( - ) A B. - 1 = 1_ ( 16 + ) C. + 1 = - 1_ Module 15 Lesson

12 Derive the equation of the parabolas with the given information. 16. An upward-opening parabola with a focus at (0, p) and a directri = -p. 17. A leftward-opening parabola with a focus at (-p, 0) and directri = p. H.O.T. Focus on Higher Order Thinking 1. Multi-Step A tennis plaer hits a tennis ball just as it hits one end line of the court. The path of the ball is modeled b the equation - = ( - 39) where = 0 is at the end line. The tennis net is 3 feet high, and the total length of the court is 7 feet. a. How far is the net located from the plaer? b. Eplain wh the ball will go over the net. c. Will the ball land "in," that is, inside the court or on the opposite endline? Houghton Mifflin Harcourt Publishing Compan Module 16 Lesson

13 19. Critical Thinking The latus rectum of a parabola is the line segment perpendicular to the ais of smmetr through the focus, with endpoints on the parabola. Find the length of the latus rectum of a parabola. Justif our answer. Hint: Set the coordinate sstem such that the verte is at the origin and the parabola opens rightward with the focus at (p, 0). 0. Eplain the Error Lois is finding the focus and directri of the parabola - = - 1 ( + ). Her work is shown. Eplain what Lois did wrong, and then find the correct answer. h = -, k = p = - 1_, so p = - 1_, or p = Focus = (h, k + p) = (-, 7.75) Directri: = k - p, or =.15 Houghton Mifflin Harcourt Publishing Compan Module 17 Lesson

14 Lesson Performance Task Parabolic microphones are used for field audio during sports events. The microphones are manufactured such that the equation of their cross section is = 1 3, in inches. The feedhorn part of the microphone is located at the focus. a. How far is the feedhorn from the edge of the parabolic surface of the microphone? b. What is the diameter of the microphone? Eplain our reasoning. c. If the diameter is increased b 5 inches, what is the new equation of the cross section of the microphone? Houghton Mifflin Harcourt Publishing Compan Image Credits: Scott Boehm/AP Images Module 1 Lesson