. d d Locker LESSON 5.1 Parabolas Teas Math Standards The student is epected to: A..B Write the equation of a parabola using given attributes, including verte, focus, directri, ais of smmetr, and direction of opening. Mathematical Processes A.1.F Analze mathematical relationships to connect and communicate mathematical ideas. Language Objective Eplain to a partner what the focus and directri of a parabola are. 1.E.,.F.3, 5.B.1, 5.B. Fill in and label a graphic organizer describing different tpes of parabolas. ENGAGE Essential Question: How is the distance formula connected with deriving equations for both vertical and horizontal parabolas? Possible answer: When ou use the distance formula to describe all the points that are equidistant from a given point and a horizontal line ou get the equation of a vertical parabola. Similarl, when ou use the distance formula to describe all the points that are equidistant from a given point and a vertical line, ou get the equation of a horizontal parabola. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and how the shape of a parabola can be used to design a microphone. Then preview the Lesson Performance Task. Houghton Mifflin Harcourt Publishing Compan Name Class Date 5.1 Parabolas Essential Question: How is the distance formula connected with deriving equations for both vertical and horizontal parabolas? Eplore A..B Write the equation of a parabola using given attributes, including focus, directri 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.) The coordinates for the focus are given b (p, 0). 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). (-p, ) Setting the two distances the same and simplifing gives: ( - p) + = ( + p) Directri d (-p, ) (-p, 0) Resource Locker (, ) d (p, 0) Write down the epression for the distance from a point (, ) to the coordinates of the focus: ( - p ) + ( - 0 ) d = Write down the epression for the distance from a point, (, ) to the point from Step C: ( - -p ) + ( - ) d = To continue solving the problem, square both sides of the equation and epand the squared binomials. 1 + - p + 1 p + = 1 + p + 1 p Collect terms. 0 + - p + 0 p + = 0 Finall, simplif and arrange the equation into the standard form for a horizontal parabola (with verte at (0, 0)): Module 5 1 Lesson 1 Name Class Date 5.1 Parabolas Essential Question: How is the distance formula connected with deriving equations for both vertical and horizontal parabolas? A..B Write the equation of a parabola using given attributes, including focus, directri Eplore Deriving the Standard-Form Equation Houghton Mifflin Harcourt Publishing Compan 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.) The coordinates for the focus are given b (p, 0) 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). (-p, ) 1 = Resource Directri (-p, ) (, ) Focus (-p, 0) (p, 0) Write down the epression for the distance from a point (, ) to the coordinates of the focus: d = ( - ) + ( - ) Write down the epression for the distance from a point, (, ) to the point from Step C: d = ( - ) + ( - ) 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. Finall, simplif and arrange the equation into the standard form for a horizontal parabola + p + p + = 0 (with verte at (0, 0)): = 0 - - 1 0 1 1 p -p p 0 p HARDCOVER PAGES 171 10 Turn to these pages to find this lesson in the hardcover student edition. Focus Module 5 1 Lesson 1 1 Lesson 5.1
Reflect 1. Wh was the directri placed on the line = -p? The directri had to be as far from the verte (at the origin) as the focus, but on the opposite side. So if the focus is at (p, 0), the directri has to intersect the -ais at (-p, 0). The line = -p is perpendicular to the ais of smmetr (the line connecting the focus and the origin) and contains the point (-p, 0).. 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)? A parabola with a verte at (h, k) can be described b a horizontal shift of h to the right and a vertical shift of k upward, which can be achieved for an graph b substituting ( - k) for and ( - h) for. 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 Equation in standard form = p = p Horizontal EXPLORE Deriving the Standard Form Equation of a Parabola INTEGRATE MATHEMATICAL PROCESSES Focus on Patterns Eplain that if the equation of a parabola contains an term the parabola opens either up or down, while an equation that contains a term opens either right or left. EXPLAIN 1 Writing the Equation of a Parabola with Verte at (0, 0) 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 INTEGRATE MATHEMATICAL PROCESSES Focus on Math Connections 1 Eplain that for an equation in the form = p, the graph opens upward if 1 is positive and p downward if 1 is negative. For an equation in the p 1 form = p, the graph opens to the right if 1 p is positive and to the left if 1 is negative. p Module 5 Lesson 1 PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunit to address Mathematical Process TEKS A.1.F, which calls for students to analze mathematical relationships to connect and communicate mathematical ideas. Students learn the relationships between quadratic equations and their graphs. Students learn that equations in the forms ( - k) = p ( - h) and ( - h) = p ( - k) have vertices (h, k), focus at either (h + p, k) or (h, k + p), and have the directri = k - p or = h - p. Parabolas
QUESTIONING STRATEGIES How can ou find the directri of a parabola 1 with an equation in the form = p or 1 = p? The directri is p units from the verte. Remember that the parabola opens awa from the directri. 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. 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 epression 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. - 0 - 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, : 0-0 - c. The verte is at (0, 0). = p Use the epression 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, - ), directri = p = -coordinate of the focus = - 0 - = () = - 0 - p = -coordinate of the focus = ( ) = ( ) = Module 5 3 Lesson 1 A_MTXESE353930_UM05L1 3 COLLABORATIVE LEARNING Peer-to-Peer Activit Have students work in pairs. Instruct each student to create a design using graphs of parabolas. Students echange designs and write the equations for the parabolas in the partner s design, including the domain and range of each curve. 1/7/15 1:1 PM 3 Lesson 5.1
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) EXPLAIN Writing the Equation of a Parabola with Verte at (h, k) INTEGRATE MATHEMATICAL PROCESSES Focus on Critical Thinking The focus of a parabola is (h + p, k) for a horizontal parabola and (h, k + p) for a vertical parabola. Alternativel, students can graph the verte and find the focus b determining the opening direction of the parabola, then count p units in the appropriate direction. 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. 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. - 0 - Houghton Mifflin Harcourt Publishing Compan QUESTIONING STRATEGIES Given values of h, k, and p, describe the similarities and differences between the graph of a parabola with an equation in the form ( - k) = p ( - h) and an equation in the form ( - h) = p ( - k). Similarities: Both graphs have a verte at (h, k) and the distance to the focus is the same. Differences: The graph of the equation in the form ( - k) = p ( - h) opens to either the left or the right, while the graph of the equation in the form ( - h) = p ( - k) opens either upward or downward. Module 5 Lesson 1 CONNECT VOCABULARY Help students to understand the meanings of focus, directri, and ais of smmetr b labeling these on the graph of a parabola. DIFFERENTIATE INSTRUCTION Modeling Students can write equations which model parabolic shapes that eist in the real world. These include bridges, arcs, and the paths traced in projectile motion. Critical Thinking Have students eplain how to tell if the graph of a quadratic equation in standard form is a circle or parabola. Parabolas
EXPLAIN 3 Rewriting the Equation of a Parabola to Graph the Parabola INTEGRATE TECHNOLOGY Students can solve equations of parabolas for and graph the corresponding function(s) on their graphing calculators. If the equation is for a parabola that opens left or right, the parabola needs to be graphed using two functions. B Focus (-1, -1), directri = 5 A vertical directri means a horizontal parabola. -1-5 ( value of focus) - ( value of directri) p = = = -3 k = the -coordinate of the focus = -1 Solve for h: The -value of the focus is h + p, so h + p = h + (-3) = h = Write the equation: ( + 1) = -1 ( - ) Your Turn -1-1 - 0 - 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. AVOID COMMON ERRORS Some students ma include both positive and negative values of p ( - h) when taking the square root of both sides of an equation in the form ( - k) = p ( - h). Remind them that when equations of this form are solved for, the resulting equation should be in the form = ± p ( - h) + k. Houghton Mifflin Harcourt Publishing Compan 5. Focus (5, -1), directri = -3 6. Focus (-, 0), directri = Eplain 3-0 - _ p = 5 - (-3) = k = -1 h + p = h + () = 5 h = 1 ( + 1) = 16 ( - 1) - 0 - 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. p = 0 - = - h = - k + p = k + (-) = 0 k = ( + ) = ( - ) Module 5 5 Lesson 1 A_MTXESE353930_UM05L1 5 LANGUAGE SUPPORT Connect Vocabular Have students work in pairs to fill in a graphic organizer. Write the word parabola in a circle in the middle of a sheet of paper. Fold the paper in fourths. Write opens upward in one corner of the paper, opens downward in another corner, opens to the right and opens to the left in the remaining corners. Have the students work together to sketch a parabola and write an equation for each kind of graph. 0/0/1 :05 AM 5 Lesson 5.1
Eample 3 Convert the equation to the standard form of a parabola and graph the parabola, the focus, and the directri. - - + 1 = 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. ( - ) = ( - ) QUESTIONING STRATEGIES How would ou solve an equation in the form ( - h) = p ( - k) for in order to graph the equation on our graphing calculator? Divide both sides of the equation b p and then add k to both sides of the equation. 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 = 1 + + + 1 = 0 Isolate the terms. + = 1 Add ( ) _ to both sides. + + 16 = Factor the perfect square trinomial. ( ) + = Factor out - on the right. ( + ) - 1 = ( + ) - 0 - Identif the features of the graph using the standard form of a horizontal parabola, ( k ) = p( h): p = -, so p =. h = -1, k = - - _ Verte = (h, k) = ( -1, - ) Focus = (h + p, k) = ( - 3, -) Directri: = h - p, or = - _ - 0 - Houghton Mifflin Harcourt Publishing Compan Module 5 6 Lesson 1 Parabolas 6
EXPLAIN Solving a Real-World Problem CONNECT VOCABULARY Remind students that placing constraints on the values of is equivalent to restricting the domain. Similarl, placing constraints on the values of is equivalent to restricting the range. Your Turn Convert the equation to the standard form of a parabola and graph the parabola, the focus, and the directri. 7. - 1 - + 6 = 0. + - 16 - = 0-0 - - = 1-6 - + = 1-6 + ( - ) = 1-60 ( - ) = 1 ( - 5) Verte = (5, ), Focus = (, ), Directri: = - 0 - + = 16 + + + 16 = 16 + + 16 ( + ) = 16 + 6 ( + ) = 16 ( + ) Verte = (-, -), Focus = (-, 0), 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 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 5 7 Lesson 1 A_MTXESE353930_UM05L1.indd 7 1/11/15 :09 AM 7 Lesson 5.1
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). ( - 0 ) = p ( - 0 ) = 560 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. 35 = 560.19 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 = -0.015 ( - 0), in ards. Does the team win? - 11 = -0.015 (35-0) =.175 Since.175 is greater than 3.3, the ball goes over the goalpost and the team wins the game. Houghton Mifflin Harcourt Publishing Compan Module 5 Lesson 1 Parabolas
ELABORATE INTEGRATE MATHEMATICAL PROCESSES Focus on Math Connections Eplain that there are alternate forms for the equations of parabolas. Parabolas with vertices at the origin ma be written in the forms = a (for vertical parabolas) and = a (for horizontal parabolas). Parabolas with vertices at points other than the origin ma be written in the forms ( - h) = p ( - k ) (vertical parabolas) and ( - k) = p ( - h) (horizontal parabolas). 1 In these forms, a = p. SUMMARIZE THE LESSON Eplain how a parabola can be graphed given its equation. Use the equation to graph the verte of the parabola. Then find the value of p and determine the direction in which the parabola opens. Graph the focus and directri accordingl. Substitute an -value to find a point on the parabola. Graph the reflection of the point over the ais of smmetr. Then complete the graph. Houghton Mifflin Harcourt Publishing Compan 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. The parabola gets wider as the focus moves awa from the verte. To convert from the standard form of a parabola to the standard form of a quadratic, isolate : = p = p = a a = p 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? Write the epressions for the distance from a point on the parabola to each of the focus and the directri. Then equate the two distances per the definition of a parabola. 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. 1. Focus at (3, 0), directri: = -3. Focus at (0, -5), directri: = 5-0 - - 0 - = p = (3) = 1-0 - 3. Focus at (-1, 0), directri: = 1. Focus at (0, ), directri: = - = p = (-1) = - - 0 - Online Homework Hints and Help Etra Practice = p = (-5) = -0 = p = () = Module 5 9 Lesson 1 A_MTXESE353930_UM05L1.indd 9 1/11/15 :09 AM 9 Lesson 5.1
Find the equation of the parabola with the given information. 5. Verte: (-3, 6) ; Directri: = -1.75 6. Verte: (6, 0) ; Focus: (6, 11) ( - h) = p ( - k) ( - 6) = (-9)( - 0) ( - 6) = -36( - 0) ( -k) = p ( - h) ( - 6) = (-1.5) ( + 3) ( - 6) = -5 ( + 3) 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: = 3-0 - - 0 - - 0 - Convert the equation to the standard form of a parabola and graph the parabola, the focus, and the directri. 9. - 0-6 - 51 = 0 10. - 1-1 + 73 = 0-6 = 0 + 51-6 + 9 = 0 + 51 + 9 p = _ 3-7 = - h = 5 k + p = k + (- ) = 3 k = 5 ( - 5) = - ( - 5) ( - 3) = 0 + 60 ( - 3) = 0 ( + 3) Verte = (-3, 3), Focus = (, 3), Directri : = - 0-11. Communications The equation for the cross section of a parabolic satellite television dish is = 1 50, measured in inches. How far is the focus from the verte of the cross section? 1.5 = 50 = The focus is 1.5 in. from the verte of the cross section. - 1 = 1-73 - 1 + 9 = 1-73 + 9 ( - 7) = 1 - ( - 7) = 1 ( - ) Verte = (7, ), Focus = (7, 5), Directri : = -1 _ p = -3-3 = -3 k = 3 h + p = h + (- 3) = -3 h = 0 ( - 3) = -1 Houghton Mifflin Harcourt Publishing Compan EVALUATE ASSIGNMENT GUIDE Concepts and Skills Eplore Deriving the Standard Form Equation of a Parabola Eample 1 Writing the Equation of a Parabola with Verte at (0, 0) Eample Writing the Equation of a Parabola with Verte at (h, k) Eample 3 Rewriting the Equation of a Parabola to Graph the Parabola Eample Solving a Real-World Problem INTEGRATE MATHEMATICAL PROCESSES Focus on Reasoning Practice Eercises 16 17 Eercises 1 Eercises 5, 15 Eercises 9 10 Eercises 11 1 Students can check that their parabolas have the correct widths b verifing that the distance across the parabola at the focus equals p. Module 5 50 Lesson 1 Eercise Depth of Knowledge (D.O.K.) Mathematical Process A_MTXESE353930_UM05L1.indd 50 1/11/15 :09 AM 1 1 Recall of Information 1.D Multiple representations 5 6 1 Recall of Information 1.F Analze relationships 7 1 Recall of Information 1.D Multiple representations 9 10 Skills/Concepts 1.D Multiple representations 11 1 Skills/Concepts 1.A Everda life 15 Skills/Concepts 1.D Multiple representations Parabolas 50
AVOID COMMON ERRORS Some students ma confuse the equations of horizontal and vertical parabolas. It ma help them to make a chart listing the general equations for both horizontal and vertical parabolas. 1. Engineering The equation for the cross section of a spotlight is + 5 = 1 1, measured in inches. Where is the bulb located with respect to the verte of the cross section? + 5 = 1 + 5 = 3 The bulb is 3 in. from the verte of the cross section. 13. When a ball is thrown into the air, the path that the ball travels is modeled b the parabola - 7 = -0.0175 ( - 0), measured in feet. What is the maimum height the ball reaches? How far does the ball travel before it hits the ground? The verte of the parabola is (0, 7), so the maimum height of the ball is 7 feet. 0-7 = -0.0175 ( - 0) ±0 = - 0 = 0, 0 The ball travels 0 feet before it hits the ground. Houghton Mifflin Harcourt Publishing Compan Image Credits: J. Aa./ Shutterstock 1. The equation of the cables 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 cables above the bridge? The verte of the parabola is (0, 55), so the lowest point of the cables is 55 feet above the bridge. 15. Match each equation to its graph. B + 1 = _ 1 16 ( - ) C - 1 = _ 1 16 ( + ) A A. - 0 - B. - 0 - C. + 1 = - 16 ( - ) - 0 - Module 5 51 Lesson 1 Eercise Depth of Knowledge (D.O.K.) Mathematical Process 16 17 3 Strategic Thinking 1.F Analze relationships 1 Skills/Concepts 1.A Everda life 19 3 Strategic Thinking 1.F Analze relationships 0 Skills/Concepts 1.G Eplain and justif arguments 51 Lesson 5.1
Derive the equation of the parabolas with the given information. 16. An upward-opening parabola with a focus at (0, p) and directri = -p. distance from (, ) to focus = distance from (, ) to directri ( - 0) + ( - p) = ( - ) + ( + p) + ( - p) = ( + p) + ( - p) = ( + p) + - p + p = + p + p = p CONNECT VOCABULARY Have students find and label the focus and directri of different parabolas and describe their locations in relation to the ais of smmetr and the verte of each parabola. 17. A leftward-opening parabola with a focus at (-p, 0) and directri = p. distance from (, ) to focus = distance from (, ) to directri ( + p) + ( - 0) = ( - p) + ( - ) ( + p) + = ( - p) + p + p + = - p + p = -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 - = - 151 ( - 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? Because the court is 7 feet long and the net is at the court's midline, the net is 39 feet from the plaer. b. Eplain wh the ball will go over the net. The verte of the parabola is (39, ), so the maimum height the ball reaches is feet. This maimum height occurs at the midline, where the net is located. Since the net is onl 3 feet high, the ball will go over the net. c. Will the ball land in, that is, inside the court or on the opposite endline? Find where the ball's height is 0. 0 = ( 39) 151 = ( 39) 151 151 = ( 39) ±39 = 39 = 0, 7 The ball will travel 7 feet before it hits the ground. Since the ball s path and the court are 7 feet long, the ball will land on the opposite endline, or in. Houghton Mifflin Harcourt Publishing Compan Module 5 5 Lesson 1 Parabolas 5
JOURNAL Have students compare and contrast the methods the have learned for graphing parabolas and writing equations for parabolas. 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). The parabola has the equation: = p. The ais of smmetr of this parabola is the -ais. The line containing the latus rectum is perpendicular to the -ais and goes through the focus so it has an equation of = p. Setting = p in the equation above and solving for we obtain the coordinates of the endpoints of the latus rectum. Their coordinates are (p, p) and (p, -p). The length of this segment is p - (-p) = p as epected for a vertical segment with those endpoints. =p =p p = p = ± p = ±p 0. Eplain the Error Lois is finding the focus and directri of the parabola - = - _ ( + ). Her work is shown. Eplain what Lois did wrong, and then find the correct answer. h = -, k = p = -, so p = -, or p = -0.15 Focus = (h, k + p) = (-, 7.75) Directri: = k - p, or =.15 Houghton Mifflin Harcourt Publishing Compan Lois did not find p correctl. The equation for a vertical parabola is ( - h) = p( - k), so Lois should have begun b finding an equivalent equation with the coefficient in front of - instead of in front of ( + ) : - = - _ ( + ) -( - ) = ( + ). h =, k = p =, so p = _, or p = 0.5 Focus = (h, k + p) = (, 7.5) Directri: = k p, or =.5 Module 5 53 Lesson 1 53 Lesson 5.1
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?.5 a. = 3 = b. The point directl above the focus is at (p, ). Since p =.5, we can plug (.5, ) into the equation of the parabola and find = 17. The radius is 17 inches, so the diameter is 3 inches. c. The new diameter is 39 inches, so the new radius is 19.5 inches. So the point (p, 19.5) directl above the focus is on the parabola. = p p = p (19.5) p = 9.75 = (9.75) = The new equation is = 39 39. Houghton Mifflin Harcourt Publishing Compan Image Credits: Scott Boehm/AP Images INTEGRATE TECHNOLOGY Students can plot parabolas which open to the left or right using a calculator or a graphing program on a computer b first solving the equation = p for, and plotting two functions, = ± p. QUESTIONING STRATEGIES Ask students to derive = p b assuming that the distance from the focus to a point on a parabola is equal to the shortest distance from that point to the directri, and using the distance formula. P(, ) is on the parabola. The focus is (0, p), the distance from P to (0, p) is ( - 0) + ( - p), and the distance from P to the directri is + p. Setting the two equal: + ( - p) = + p + ( - p) = ( + p) = p Module 5 5 Lesson 1 EXTENSION ACTIVITY Sound ras parallel to the ais of a parabolic microphone are reflected off its inner surface and pass through the focal point. To eplore this phenomenon, have students graph the parabola =. The slope of an line tangent to a point on this parabola is m =. Have students pick different points on the parabola (ecept the origin), draw tangent lines through these points, then draw a line parallel to the -ais that ends at the point, forming an acute angle a, which students can measure using a protractor. Then have students draw a second line from the point, forming an angle with the tangent line congruent to a. Encourage the students to draw man lines, to observe that the lines intersect at the focus. Scoring Rubric points: Student correctl solves the problem and eplains his/her reasoning. 1 point: Student shows good understanding of the problem but does not full solve or eplain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Parabolas 5