5-4. Focus and Directrix of a Parabola. Key Concept Parabola VOCABULARY TEKS FOCUS ESSENTIAL UNDERSTANDING

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1 5- Focus and Directri of a Parabola TEKS FOCUS VOCABULARY TEKS ()(B) Write the equation of a parabola using given attributes, including verte, focus, directri, ais of smmetr, and direction of opening. TEKS ()(B) Use a problem solving model that incorporates analzing given information, formulating a plan or strateg, determining a solution, justifing the solution, and evaluating the problem solving process and the reasonableness of the solution. Additional TEKS ()(A), ()(E) Directri the fied line used to Focus (plural: foci) of a parabola define a parabola the fied point used to define a parabola Focal length the distance between the verte and the focus of a parabola Formulate create with careful Reasonableness the qualit effort and purpose. You can formulate a plan or strateg to solve a problem. of being within the realm of common sense or sound reasoning. The reasonableness of a solution is whether or not the solution makes sense. Strateg a plan or method for solving a problem ESSENTIAL UNDERSTANDING ach point of a parabola is equidistant from a point called the focus and a line called E the directri. Ke Concept Parabola Definition A parabola is the set of all points in a plane that are the same distance from a fied line and a fied point not on the line. The fied point is called the focus of a parabola. The fied line is called the directri. The distance between the verte and the focus is the focal length of the parabola. You can find the equation of a vertical parabola with verte at the origin. If ou denote the focus b (0, c), the directri is the line with equation = -c. The equation will be of the form = a2. Then a = c. 70 Lesson 5- Focus and Directri of a Parabola Graph ais of smmetr Focus Focus ais of smmetr directri directri Vertical Parabola Horizontal Parabola focus (0, c) O directri c (, ) (, c)

2 Ke Concept Transformations of a Parabola Vertical Parabola Verte (0, 0) Verte (h, k) Equation = c 2 = c ( - h)2 + k Focus (0, c) (h, k + c) Directri = -c = k - c Horizontal Parabola Verte (0, 0) Verte (h, k) Equation = c 2 = c ( - k)2 + h Focus (c, 0) (h + c, k) Directri = -c = h - c Problem Parabolas With Equation = a 2 A What is an equation of the parabola with verte at the origin and focus (0, 2)? How can ou tell if this is a vertical or a horizontal parabola? The focus and the verte are on the ais of smmetr. The both lie on the -ais, so the parabola is vertical. The focus is directl above the verte. This is a vertical parabola with verte at the origin. The focus is (0, c), so c = 2. = c 2 = (2) 2 = (0, 2) O 2 What does the sign of a tell ou about the graph? Since a is negative, the parabola opens downward. B What are the focus and directri of the parabola with equation = 2 2? This is a vertical parabola with verte at the origin and a = - 2. a = c = - 2 c = -2 c = -3 Since the verte is at the origin, knowing c, ou can conclude that the focus is the point (0, -3) and the directri is the line with equation = (0, 3) PearsonTEXAS.com 7

3 Problem 2 Parabolas With Equation = a 2 A What is an equation of a parabola with verte at the origin and directri =.25?.25 O 2 c.25 How can ou tell if this is a vertical or a horizontal parabola? The directri is parallel to the -ais, so this is a horizontal parabola. What does the sign of a tell ou about the graph? Since a is positive, the parabola opens to the right. The directri lies directl to the right of the verte. The parabola is horizontal. The directri has equation = -c, so c = Thus, = c 2 Use the equation for a horizontal parabola. = (-.25) 2 Substitute -.25 for c. = Simplif. The equation is = Check for Reasonableness The graph is reasonable since it opens in the negative direction and a 6 0. B What are the verte, focus, and directri of the parabola with equation = ? c 3 3 (, 0) 3 2 This is a horizontal parabola. The verte is at the origin and a = Thus, a = c = 0.75 Substitute 0.75 for a. c = 0.75 Solve for c. c = 3. Knowing c, ou can conclude that the focus is the point 3, 02. The directri is the line with equation = Lesson 5- Focus and Directri of a Parabola

4 Problem 3 TEKS Process Standard ()(A) What is the shape of the solar reflector? A cross section is part of a parabola and is 8 ft across. Using Parabolas to Solve Problems STEM Solar Reflector The parabolic solar reflector pictured has a depth of 2 feet at the center. How far from the verte is the focus? (What is the focal length?) Graph the parabola in a coordinate sstem with verte (0, 0). The vertical parabola has the form = c 2. Substitute either the point (-, 2) or the point (, 2). (, 2) (, 2) O 2 8 ft 2 = c ()2 Substitute. 2 = 6 c Simplif. 8c = 6 Solve for c. c = 2 Therefore the focus is at (0, 2), 2 ft from the verte. The focal length is 2 ft. PearsonTEXAS.com 73

5 Problem TEKS Process Standard ()(B) Writing an Equation Given the Focus and Directri What is an equation of a parabola with focus (2, 3) and directri =? Wh do ou substitute for 2 in the epression on the right? The directri is = -. An point that lies on that line is of the form (-, ). Wh don t ou epand in? The equation of a horizontal parabola contains ( - k) 2, so ou can use ( + 3) 2 for that part of the equation. Analze the Given Information You know the coordinates of the focus and the equation of the directri. An point on the parabola is equidistant from the focus and the directri. Determine a Solution Formulate a Plan Because ou know the distances are equal, use the Distance Formula. 2( - ) 2 + ( - ) 2 = 2( - 2 ) 2 + ( - 2 ) 2 Distance Formula 2 ( - 2) 2 + [ - ( -3)] 2 = 2[ - (-)] 2 + ( - ) 2 Substitute. 2( - 2) 2 + ( + 3) 2 = 2( + ) 2 Simplif. ( - 2) 2 + ( + 3) 2 = ( + ) 2 Square each side ( + 3) 2 = Epand in. ( + 3) 2 - = Subtract 2 and. ( + 3) 2 = Add to each side. ( + 3) 2 = 2( + ) Distributive Propert = 2 ( + 3)2 - Standard Quadratic Form Evaluate the Reasonableness of the Solution The equation can be written as = # 3 ( + 3)2 -, so c is 3, k = -3, and h is -. So, the focus of this parabola is (h + c, k) or (2, -3) and the directri is = h - c or = -. The equation is reasonable. 7 Lesson 5- Focus and Directri of a Parabola

6 Problem 5 Writing an Equation of a Parabola Multiple Choice Which is an equation of the parabola with verte (3, 7) and focus (5, 7)? How do ou determine which equation to use? Use the focus and the verte to determine the orientation of the parabola. = ( - 7)2 + 3 = 8 ( - 7)2 + 3 = 8 ( - 7)2 + 3 = 8 ( + 7)2-3 The focus is to the right of the verte, so the parabola is horizontal. Also, (h, k) is (3, 7) and (h + c, k) is (5, 7), so c = 2. Substitute all this information into the equation for a horizontal parabola, = c ( - k)2 + h, to get = 8 ( - 7) The correct answer is C. ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. Write an equation of a parabola with verte at the origin and the given focus. For additional support when completing our homework, go to PearsonTEXAS.com.. focus at (6, 0) 2. focus at (0, - ) 3. focus at (0, 7). focus at (-, 0) 5. focus at (2, 0) 6. focus at (0, -5) Identif the verte, the focus, and the directri of the parabola with the given equation. Then sketch the graph of the parabola. 7. = 2 8. = = 2 0. = 2 2 Write an equation of a parabola with verte at the origin and the given directri.. directri = directri = 5 3. directri = - 3. directri = 9 5. directri = directri = Appl Mathematics ()(A) A cross section of a flashlight reflector is a parabola. The bulb is located at the focus. Suppose the bulb is located in. from the verte of the reflector. Model a cross section of the reflector b writing an equation of a parabola that opens upward and has its verte at the origin. What is an advantage of this parabolic design? Identif the verte, the focus, and the directri of the parabola with the given equation. Then sketch the graph of the parabola. 8. = = = = = = PearsonTEXAS.com 75

7 Write an equation of a parabola with the given verte and focus. 2. verte (, ), focus (6, ) 25. verte (0, 3), focus ( -8, 3) 26. verte ( -5, ), focus ( -5, 0) 27. verte (7, 2), focus (7, -2) 28. The center main cable of the etension bridge shown is parabolic. The -ais represents the bridge roadwa and the directri. The -ais is the ais of smmetr. Write an equation to model the main cable of the bridge. 50 ft Find the equation of the parabola with the given properties. 29. focus ( -3, 5) and directri = focus (, 5) and directri = focus ( -2, -) and directri = 8 Identif the verte, the focus, and the directri of a parabola with each equation. Then sketch a graph of the parabola with the given equation = = - 3. ( - 2)2 = = = = Appl Mathematics ()(A) In some solar collectors, a mirror with a parabolic cross section is used to concentrate sunlight on a pipe, which is located at the focus of the mirror as shown in the diagram. What is an equation of the parabola that models the cross section of the mirror? F 6 ft Appl Mathematics ()(A) The equation d = 0 s relates the depth d (in meters) of the ocean to the speed s (in m/s) at which tsunamis travel. What is the graph of the equation? Use the information in each graph to write the equation for the parabola. 0. F( 2, 0) O Directri Lesson 5- Focus and Directri of a Parabola O 2 F Q, 0R 32

8 3. Appl Mathematics ()(A) Broadcasters use a parabolic microphone on football sidelines to pick up field audio for broadcasting purposes. A certain parabolic microphone has a reflector dish with a diameter of 28 inches and a depth of inches. If the receiver of the microphone is located at the focus of the reflector dish, how far from the verte should the receiver be positioned? Graph each equation = = = = 2 8. = ( - 3) ( - 2) 2 = ( + 3) Write an equation of a parabola with verte at (, ) and the given information. 50. directri = directri = focus at (, 0) 53. Eplain Mathematical Ideas ()(G) Eplain how to find the distance from the focus to the directri of the parabola = Use a Problem-Solving Model ()(B) Use the definition of a parabola to show that the parabola with verte (h, k) and focus (h, k + c) has the equation ( - h) 2 = c( - k). 55. a. What part of a parabola is modeled b the function =? b. State the domain and range for the function in part (a). 56. Justif Mathematical Arguments ()(G) If the radius and depth of a satellite dish are equal, prove that the radius is four times the focal length. TEXAS Test Practice 57. What is the equation of a parabola with verte at the origin and focus at 0, 5 2 2? A. = C. = B. = 0 2 D. = Use the information in the graph to find the equation for the graph. F = 0 H = 0 G. 2-6 = 0 J. 2-6 = Which epression is NOT equivalent to ? A C B. 5 3 D Directri, 3 2 O 2 PearsonTEXAS.com 77