4.2 Parabolas. Explore Deriving the Standard-Form Equation. Houghton Mifflin Harcourt Publishing Company. (x - p) 2 + y 2 = (x + p) 2
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1 COMMON CORE. d Locker d LESSON Parabolas Common Core Math Standards The student is epected to: COMMON CORE A-CED.A. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate aes with labels and scales. Also A-CED.A.3, G-GPE.A. Mathematical Practices COMMON CORE. MP.7 Using Structure Language Objective Eplain to a partner what the focus and directri of a parabola are. Fill in and label a graphic organizer describing different tpes of parabolas. 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.) A The coordinates for the focus are given b (p, 0). B (-p, ) (-p, 0) Directri Write down the epression for the distance from a point (, ) to the coordinates of the focus: ( - p ) + ( - 0 ) d = d Resource Locker (, ) d (p, 0) Focus 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. Houghton Mifflin Harcourt Publishing Compan C E F 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, ) D Setting the two distances the same and simplifing gives. ( - p) + = ( + p) 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. + - p + p + = + p + p Collect terms p + 0 p + = 0 G Finall, simplif and arrange the equation into the standard form for a horizontal parabola (with verte at (0, 0)): = p PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and how the shape of a parabola can be used to build a microphone. Then preview the Lesson Performance Task. Module 75 Lesson Name Class Date. Parabolas Essential Question: How is the distance formula connected with deriving equations for both vertical and horizontal parabolas? A-CED.A. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate aes with labels and scales. Also A-CED.A.3, G-GPE.A. 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, ) 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)): = Resource Directri (-p, ) (, ) Focus (-p, 0) (p, 0) Module 75 Lesson p -p p 0 HARDCOVER PAGES 5 3 Turn to these pages to find this lesson in the hardcover student edition. 75 Lesson.
2 Reflect. 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 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 PRACTICES Focus on Patterns MP. 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 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 PRACTICES Focus on Math Connections MP. Eplain that for an equation in the form = p, the graph opens upward if is positive and p downward if is negative. For an equation in the p form = p, the graph opens to the right if p is positive and to the left if is negative. p Module 76 Lesson PROFESSIONAL DEVELOPMENT Integrate Mathematical Practices This lesson provides an opportunit to address Mathematical Practice MP.7, which calls for students to look for and make use of structure. 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 76
3 QUESTIONING STRATEGIES How can ou find the directri of a parabola with an equation in the form = p or = p? The directri is p units from the verte. Remember that the parabola opens awa from the directri. 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 (, 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 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). = p 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, - _ ), directri = _ p = -coordinate of the focus = = () = p = -coordinate of the focus = ( _ ) = ( _ ) = Module 77 Lesson A_MNLESE359_UM0L 77 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. 6//5 :36 PM 77 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. Focus (3, ), directri = 0 A horizontal directri means a vertical parabola. ( value of focus) - ( value of directri) p = = _ - 0 = h = the -coordinate of the focus = 3 Solve for k: The -value of the focus is k + p, so k + p = k + = k = Write the equation: ( - 3) = ( - ) Plot the focus, the directri, and the parabola Module 7 Lesson Houghton Mifflin Harcourt Publishing Compan EXPLAIN Writing the Equation of a Parabola with Verte at (h, k) INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 The focus of a parabola can be found at (h + p, k) or (h, k + p). Alternativel, students can graph the verte, find the focus b determining the opening direction of the parabola, then count p units in the appropriate direction. 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. 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 7
5 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 (-, -), directri = 5 A vertical directri means a horizontal parabola ( value of focus) - ( value of directri) p = = = -3 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: ( + ) = - ( - ) 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. 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, -), directri = Focus (-, 0), directri = Eplain 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 _ p = 5 - (-3) = k = - h + p = h + () = 5 h = ( + ) = 6 ( - ) _ p = 0 - = - h = - k + p = k + (-) = 0 k = ( + ) = ( - ) 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 79 Lesson A_MNLESE359_UM0L.indd 79 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. 9/03/ :6 PM 79 Lesson.
6 Eample 3 Convert the equation to the standard form of a parabola and graph the parabola, the focus, and the directri = 0 Isolate the terms and complete the square on. Isolate the terms. - = - 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 =. h =, k =. Verte = (h, k) = (, ). Focus = (h, k + p) = (, + ) = (, 3). Directri: = k - p = -, or = = 0 Isolate the terms. + = 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) = (- 3, - ) Directri: = h - p or = Houghton Mifflin Harcourt Publishing Compan Module 0 Lesson Parabolas 0
7 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 = = = = ( - ) = - 60 ( - ) = ( - 5) Verte = (5, ), Focus = (, ), Directri: = = = ( + ) = ( + ) = 6 ( + ) 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 Lesson A_MNLESE359_UM0L.indd /05/ 5:33 PM 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 0 mm and the mirror has a 70 mm diameter (width), what is the depth of the bowl of the mirror?? 70 mm 0 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 = mm The bowl is approimatel.9 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 0 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 - = ( - 0), in ards. Does the team win? - = (35-0) =.75 Since.75 is greater than 3.3, the ball goes over the goalpost and the team wins the game. Houghton Mifflin Harcourt Publishing Compan Module Lesson Parabolas
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