Math 3 Unit 7: Parabolas & Circles
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1 Math 3 Unit 7: Parabolas & Circles Unit Title Standards 7.1 Introduction to Circles G.GPE.1, F.IF.8.a, F.IF Converting Circles to Descriptive Form G.GPE.1, A.CED.4, F.IF Definition of a Parabola from the Focus and Directri G.GPE Sidewas Parabolas from the Focus and Directri G.GPE.2, A.CED Transformations of Parabolas G.GPE.2, A.CED Converting Parabolas from General to Descriptive Form G.GPE.2, A.CED.4, F.IF.8.a 7.7 Converting Parabolas and Circles to Descriptive Form G.GPE.1, G.GPE.2, A.CED.4, F.IF.8.a Unit 7 Performance Task Unit 7 Review Parabola Calculation Challenge Additional Clovis Unified Resources Clovis Unified is dedicated to helping ou be successful in Math 3. On the website above ou will find videos from Clovis Unified teachers on lessons, homework, and reviews. Digital copies of the worksheets, as well as hperlinks to the videos listed on the back are also available at this site.
2 Math 3 Unit 7: Online Resources 7.1 Introduction to Circles Khan Academ: Features of a Circle from its Standard Equation Khan Academ: Graphing a Circle from its Standard Equation Purple Math: Circles: Introduction & Drawing Khan Academ: Completing the Square Purple Math: Solving Quadratic Equations: Solving b Completing the Square Converting Circles to Descriptive Form 7.3 Definition of a Parabola from the Focus and Directri 7.4 Sidewas Parabolas from the Focus and Directri 7.5 Transformations of Parabolas 7.6 Converting Parabolas from General to Descriptive Form 7.7 Converting Parabolas and Circles to Descriptive Form Khan Academ: Features of a Circle from its Epanded Equation Purple Math: Completing the Square: Circle Equations Patrick JMT: Finding the Center-Radius Form of a Circle b Completing the Square Khan Academ: Intro to Focus & Directri Lawrence Math Academ: Equation of Parabola from Focus and Directri Purple Math: Parabolas: Introduction (note: p = c) Mario s Math Tutoring: Graphing using Focal Chord Patrick JMT: Conic Sections, Parabola: Sketch Graph b Finding Focus, Directri, Points (note: p = c) and Coolmath.com: Sidewas Parabolas (Pages 1 5) Purple Math: Parabolas: Introduction (note: p = c) Patrick JMT: Conic Sections: Parabolas, Part 2 (Directri and Focus) -start at 3:10 minutes Purple Math: Parabolas: Finding the Equation from Information Khan Academ: Finding the Verte of a Parabola in Standard Form Patrick JMT: Conic Sections: Parabolas, Part 1 Khan Academ: Verte & Ais of Smmetr of a Parabola Purple Math: Conic Sections Parabolas: Finding Information from the Equation Patrick JMT: Conic Sections: Parabolas, Part 1
3 Math 3 Unit 7 Worksheet 1 Name: Introduction to Circles Date: Per: [1-7] Identif the center and radius for the circle. Sketch the circle and be sure to identif the scale being used = = = ( + 4) 2 + ( 7) 2 = 4 5. ( 8) 2 + ( + 3) 2 = ( 1) 2 < ( + 5) [8-11] Write the equation of the circle with the information given: 8. The circle is shifted from the origin 3 units left and 5 units up, with a radius of 3 9. The circle is shifted from the origin 12 units right and 8 units down, with a radius of The circle is shifted from the origin 4 units down, with a radius of The circle is shifted from the origin 5 units right, with a radius of 5 3. Math 3 Unit 7 Worksheet 1
4 12. Is the point (2, 4) inside, outside, or on the circle, ( + 1) 2 + ( 1) 2 = 16? Justif our response. 13. Is the point (5, 5) inside, outside, or on the circle, ( + 1) 2 + ( + 2) 2 = 49? Justif our response. [14-19] Complete the square and write the trinomial as a perfect square. No decimals allowed = = = 5 + ( ) 2 = ( ) 2 = ( ) 2 = = = = 1 + ( ) 2 = ( ) 2 = ( ) 2 = = = ( ) 2 = ( ) 2 = [22-24] Rewrite the following as ( aa) 2 = bb b completing the square = = = 0 Math 3 Unit 7 Worksheet 1
5 Math 3 Unit 7 Worksheet 2 Name: Converting Circles to Descriptive Form Date: Per: [1-6] Convert the following circle equations to descriptive form, ( h) 2 + ( kk) 2 = rr 2, b completing the square. Identif the center and the radius for each circle. Sketch the circle and be sure to label the scale being used = = = = > 0 [7-12] Convert the following circle equations to descriptive form, ( h) 2 + ( kk) 2 = rr 2, b completing the square. Identif the center and the radius for each circle = = 0 Math 3 Unit 7 Worksheet 2
6 = = = = Show that the circle with equation = 6( 3) is congruent to the circle with center at (0, 0) and radius 4. {i.e. Show the radius from the first circle is congruent to the radius from the second, and indicate the translation required to map the first circle to the second circle.} 14. Show that the circle with equation 208 = ( 8) + ( + 2) is a dilation image of the circle with center at (4, 1) and radius 6. {i.e. Show the center of the first circle is same as the center of the second, and find the ratio of dilation.} Math 3 Unit 7 Worksheet 2
7 Math 3 Unit 7 Worksheet 3 Name: Definition of a Parabola from the Focus and Directri Date: Per: 1. Graph the parabola 2 = 16. Find and graph the focus, directri, and focal chord endpoints. Verte: Directri: 2. Graph the parabola = Find and graph the focus, directri, and focal chord endpoints. Verte: Directri: 3. Graph the parabola 2 = 8. Find and graph the focus, directri, and focal chord endpoints. Verte: Directri: Math 3 Unit 7 Worksheet 3
8 4. Graph the parabola = Find and graph the focus, directri, and focal chord endpoints. Verte: Directri: 5. Graph the parabola 2 = 10. Find and graph the focus, directri, and focal chord endpoints. Verte: Directri: 6. Graph the parabola = Find and graph the focus, directri, and focal chord endpoints. Verte: Directri: Math 3 Unit 7 Worksheet 3
9 For each of the following, use the formal definition of a parabola to derive the equation in focal width form. 7. the parabola with focus (0,4) and directri = the parabola with focus (0,-12) and directri = the parabola with focus (0,10) and directri = -10 Math 3 Unit 7 Worksheet 3
10 Math 3 Unit 7 Worksheet 3
11 Math 3 Unit 7 Worksheet 4 Name: Sidewas Parabolas from the Focus and Directri Date: Per: 1. Graph the parabola = Find and graph the focus, directri, and focal chord endpoints. Verte: Directri: 2. Graph the parabola 2 = 1. Find and graph the focus, directri, and focal chord endpoints. 2 Verte: Directri: 3. Graph the parabola = Find and graph the focus, directri, and focal chord endpoints. Verte: Directri: Math 3 Unit 7 Worksheet 4
12 4. Graph the parabola 2 = 3. Find and graph the focus, directri, and focal chord endpoints. Verte: Directri: [5-6] For each of the following, make a sketch and use the formal definition of a parabola to derive the equation in verte (descriptive) form. 5. the parabola with focus (2,0) and directri = the parabola with focus (-8,0) and directri = 8 [7-8] Complete the square for each circle to get the equation into standard/descriptive form. Identif center and radius = = 0 Math 3 Unit 7 Worksheet 4
13 Math 3 Unit 7 Worksheet 5 Name: Transformations of Parabolas Date: Per: 1. Graph the parabola ( + 3) 2 = 12( 1). Find and graph the verte, focus, directri, and focal chord endpoints. Verte: Directri: 2. Graph the parabola = 1 2 ( 2)2 4. Find and graph the verte, focus, directri, and focal chord endpoints. Verte: Directri: 3. Graph the parabola ( 3) 2 = 8. Find and graph the verte, focus, directri, and focal chord endpoints. Verte: Directri: Math 3 Unit 7 Worksheet 5
14 4. Graph the parabola = Find and graph the verte, focus, directri, and focal chord endpoints. Verte: Directri: 5. Graph the parabola( + 2) 2 = 3( 1). Find and graph the verte, focus, directri, and focal chord endpoints. Verte: Directri: 6. Graph the parabola = 1 10 ( + 1)2 4. Find and graph the verte, focus, directri, and focal chord endpoints. Verte: Directri: Math 3 Unit 7 Worksheet 5
15 [7-9] For each of the following, make a sketch and use the formal definition of a parabola to derive the equation in descriptive form. 7. the parabola with focus (2,1) and directri = the parabola with focus (-8,3) and directri = the parabola with focus (1,2) and directri = 3 [10-11] Complete the square for each circle. Identif center and radius = = 0 Math 3 Unit 7 Worksheet 5
16 Math 3 Unit 7 Worksheet 5
17 Math 3 Unit 7 Worksheet 6 Name: Converting Parabolas from General to Descriptive Form Date: Per: For each equation, identif the direction the parabola opens (left, right, up or down); complete the square to write the equation in descriptive form; and indicate the parabola s verte. Show all work! 1. = Direction: Verte: Equation: 2. = Direction: Verte: Equation: 3. = Direction: Verte: Equation: 4. = Direction: Verte: Equation: 5. = Direction: Verte: Equation: Math 3 Unit 7 Worksheet 6
18 6. = Direction: Verte: Equation: 7. = Direction: Verte: Equation: 8. = Direction: Verte: Equation: 9. = Direction: Verte: Equation: Verte answers: Not In Order (2, 25) (49, 3) (72, 12) ( 4, 25) (4, 7) ( 1, 1 ) 8 2 (107, 6) ( 10, 48) ( 29, 3) Selected answers for equations: 2. = 5( + 3) opens left 6. = 1 ( )2 48 opens up 8. = 3( 6) opens left Math 3 Unit 7 Worksheet 6
19 Math 3 Unit 7 Worksheet 7 Name: Converting Parabolas and Circles to Descriptive Form Date: Per: Show all appropriate work. {It might be possible to sketch and/or answer the follow-up information before converting to descriptive form. You ma do this, but ou must still do the algebraic manipulation needed to convert each to descriptive form.} Descriptive form reminder: Parabola: = aa( h) 2 + kk oooo = aa( kk) 2 + h & Circle: ( h) 2 + ( kk) 2 = rr 2 [1-4]: A) Convert to descriptive form, B) sketch, and identif the following for each: C) Verte D) Line/Ais of smmetr E) Focus F) Directri G) Focal Chord Endpoints. 1) ( 3) 2 = 8( 5) Verte: Line of Smmetr: Directri: 2) ( + 2) 2 = 4( + 1) Verte: Line of Smmetr: Directri: 3) = Verte: Line of Smmetr: Directri: 4) = Verte: Line of Smmetr: Directri: Math 3 Unit 7 Worksheet 7
20 [5-10]: A) Convert to descriptive form, B) sketch, and identif the following for each: C) Verte D) Ais/Line of smmetr E) Number of -intercepts F) Number of -intercepts. 5) ( 5) 2 + 3( 2) = 0 Verte: Line of Smmetr: Number of -intercepts: Number of -intercepts: 6) = Verte: Line of Smmetr: Number of -intercepts: Number of -intercepts: 7) = Verte: Line of Smmetr: Number of -intercepts: Number of -intercepts: 8) = Verte: Line of Smmetr: Number of -intercepts: Number of -intercepts: Math 3 Unit 7 Worksheet 7
21 9) ( 4) 2 = 12 Verte: Line of Smmetr: Number of -intercepts: Number of -intercepts: 10) ( + 4) 2 + 6( + 2) = 0 Verte: Line of Smmetr: Number of -intercepts: Number of -intercepts: [11-12]: A) Convert to descriptive form, B) sketch, and identif the following for each: C) Center D) Radius E) Number of -intercepts F) Number of -intercepts. 11) = 0 Center: Radius: Number of -intercepts: Number of -intercepts: 12) = 0 Center: Radius: Number of -intercepts: Number of -intercepts: Math 3 Unit 7 Worksheet 7
22 Math 3 Unit 7 Worksheet 7
23 Math 3 Unit 7 Review Worksheet 1 Name: Parabolas & Circles Date: Per: 1. Is the point (3, 10) on the parabola, + 6 = ( + 1) 2? Justif our response. 2. Is the point (1, 4) inside, outside, or on the circle, ( + 2) 2 + ( 1) 2 = 16? Justif our response. 3. Is the point ( 7, 5) inside, outside, or on the circle, ( + 1) 2 + ( 2) 2 = 49? Justif our response. 4. What is the verte and the length of the focal chord for the parabola, = 1 12 ( 3)2 1? Verte: Focal chord length: 5. What is the center and the length of the radius for the circle, = 0? Center: Radius: 6. What is the verte and the length of the focal chord for the parabola, = 0? Verte: Focal chord length: 7. Write the equation for the circle with center (4, 1) and diameter Using the distance formula and the definition of parabola, write the equation for the parabola in focal width form, ( kk) 2 = 4cc( h), that has a focus at the point ( 7, 2) and a directri of = 1. Math 3 Unit 7 Review Worksheet 1
24 9. Using the distance formula and the definition of parabola, write the equation for the parabola in verte/descriptive form, = aa( h) 2 + kk, that has a directri of = 2 and a focus at the point ( 1, 8). 10. Sketch the graph for the parabola, 2 = 2( 1). Find, graph, and identif the focus, directri, and focal chord endpoints. Verte: Directri: 11. Sketch the graph for the parabola, + 4 = 1 6 ( 1)2. Find, graph, and identif the focus, directri, and focal chord endpoints. Verte: Directri: 12. Sketch the graph for the circle, = 0. Find, graph, and identif the center and radius. How man times does the circle intersect with the -ais? the -ais? Center: Radius: Number of -intercepts: Number of -intercepts: 13. Sketch the graph for the parabola, = 0. Find, graph, and identif the focus and the directri. How man times does the parabola intersect with the -ais? the -ais? Directri: Number of -intercepts: Number of -intercepts: Math 3 Unit 7 Review Worksheet 1
25 Math 3 Unit 7 Review Worksheet 2 Name: Parabolas & Circles Date: Per: Show all valid & appropriate work. 1. Find the center and the radius for the circle = 9. Is the point (6, 5) on the circle? Justif/eplain. Center: Radius: Is (6, 5) on the circle? Y or N 2. Write the equation for the circle with center (5, 2) and with diameter Which one of the three is the correct response: The point ( 3, 2) is inside / outside / on the circle with center (5, 2) and diameter Justif/eplain. Equation: 3. Write the equation for the circle with endpoints of a diameter (3, 12) and ( 5, 2). Hint: Find the center first! How man times does this circle intersect with the -ais? How man times does this circle intersect with the -ais? Center: Number of -intercepts: Number of -intercepts: Equation: 4. What is the focus and the equation of the directri for 4 = 1 ( )2? How man times does this parabola intersect with the -ais? the -ais? Directri: Number of -intercepts: Number of -intercepts: 5. What is the focus and the equation of the directri for ( 2) 2 = 2( + 3)? How man times does this parabola intersect with the -ais? the -ais? Directri: Number of -intercepts: Number of -intercepts: Math 3 Unit 7 Review Worksheet 2
26 6. What is the verte and the equation for the line of smmetr for the parabola = 0? Verte: Line of Smmetr: [7-8]: Using the distance formula and the definition for parabola, write the equation for each parabola in either Focal Width form or a variation of Verte/Descriptive form. Reminders Focal Width form: ( h) 2 = 4cc( kk) oooo ( kk) 2 = 4cc( h) Verte/Descriptive form: = aa( h) 2 + kk oooo = aa( kk) 2 + h 7. Focus is at ( 5, 0) and the equation for the directri is = 5. Sketch the parabola. 8. Focus is at ( 4, 5) and the equation for the directri is = 3. Sketch the parabola. [9-10]: Convert to either Focal Width form or a variation of Verte/Descriptive form. Once ou have done this, sketch the parabola, find and graph the focus, directri, and focal chord endpoints = = 4 14 Math 3 Unit 7 Review Worksheet 2
27 Parabola Calculation Challenge: The Golden Gate bridge is a suspension bridge in San Francisco, California. The towers are 1280 meters apart and rise 160 meters above the road. The cable just touches the sides of the road midwa between the towers. What is the height of the cable 200 meters from a tower? 1. Sketch the bridge, two towers, and the cable between them on grid paper. 2. Draw a coordinate ais onto our grid so that the origin is at the point where the cable touches the road. 3. Label the points at the top of each tower with the correct coordinates based on the information given in the problem. 4. Use these points to write the equation of the parabola in verte form. Things to think about: a. Should it be an = or = equation? b. Should a be positive or negative? 5. On our graph, mark a point, P, on the roadwa 200 meters from the tower. Find the coordinates of that point, based on the information given in the problem. 6. On our graph, mark the point on the cable directl above point P. Things to think about: a. How does point P relate to the question ou are tring to answer? b. Which part of the coordinate of P do ou alread know? c. Discuss the answers to these questions with our partner or group. 7. Use all of the information ou have gathered, including the equation ou wrote for the parabola made b the cable, to find the height of the cable 200 meters from a tower.
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29 Math 3 Unit 7 Worksheet 2 - Selected Answers 1. Center = (5, 2) & r = 3 2. Center = ( 3, 4) & r = 5 3. Center = (4, 1) & r = 2 4. Center = ( 2, 7) & r = Center = (0, 6) & r = Center = (3, 1) & r = Center = ( 7, 3) & r = Center = ( 1, 9 ) & r = Center = ( 4, 0) & r = Center = ( 6, 2) & r = Center = (8, 5 11 ) & r = Center = (, 8) & r = Translation: (, ) ( + 9, 3) 14. Scale Factor = 5 or Math 3 Unit 7 Worksheet 7 - Selected Answers 1) A) = 1 ( 8 3)2 + 5 C-G) V = (3, 5); Ais: = 3; F = (3, 7); Directri: = 3; FC Endpts: ( 1, 7) & (7, 7) 2) A) = 1 ( + 4 2)2 1 C-G) V = ( 1, 2); Ais: = 2; F = (0, 2); Directri: = 2; FC Endpts: (0, 0) & (0, 4) 3) A) = 1 ( 4 10)2 + 5 C-G) V = (10, 5); Ais: = 10; F = (10, 4); Directri: = 6 ; FC Endpts: (8, 4) & (12, 4) 4) A) = 1 ( + 2 4)2 + 5 C-G) V = (5, 4); Ais: = 4; F = (5 1, 4) ; 2 Directri: = 4 1 ; FC Endpts: (5 1, 3) & (5 1, 5) ) A) = 1 ( 3 5)2 + 2 C-F) V = (5, 2); Ais: = 5; # int = 2; # int = 1 6) A) = ( 3) C-F) V = (1, 3); Ais: = 3; # int = 1; # int = 2 7) A) = 3( 1) 2 2 C-F) V = ( 2, 1); Ais: = 1; # int = 1; # int = 0 8) A) = 2( 7) 2 C-F) V = (7, 0); Ais: = 7; # int = 1; # int = 1 9) A) = 1 ( 12 4)2 C-F) V = (0, 4); Ais: = 4; # int = 1; # int = 1 10) A) = 1 ( + 6 4)2 2 C-F) V = ( 4, 2); Ais: = 4; # int = 0; # int = 1 11) A) ( 2) 2 + ( + 5) 2 = 4 C-F) C = (2, 5); r = 2; # int = 0; # int = 1 12) A) ( + 4) 2 + ( 1) 2 = 12 C-F) C = ( 4, 1); r = ; # int = 2; # int = 0 Math 3 Unit 7 Selected Answers
30 Math 3 Unit 7 Review Worksheet 1 Selected Answers 1. Yes, wh? 2. Outside, wh? 3. Inside, wh? 4. V = ( 1, 3); FC = C = ( 4, 3); r = ( 3) 2 = 5 2 ( + 6) or = 2 5 ( 3)2 6 V = (3, 6) & FC = ( 4) 2 + ( + 1) 2 = ( + 2) 2 = 16( + 3) 9. = 1 ( ) F = (0, 1 1 ) ; Dir = 1 ; FC end pts ( 1, 1 1 ) & (1, 1 1 ) F = ( 2 1, 1) ; Dir = 5 1 ; FC end pts ( 2 1, 4) & ( 2 1, 2) C = (5, 1); r = ; ais int = 0; ais int = F = (8, 0); Dir = 2; FC end pts (6, 0) & (10, 0); ais int = 2; ais int = 1 Math 3 Unit 7 Review Worksheet 2 Selected Answers 1) C = (6, 2); r = 7; Yes, (6, 5) is on the circle. Wh? 2) ( 5) 2 + ( + 2) 2 = 75; ( 3, 2) is outside the circle. Wh? 3) ( + 1) 2 + ( 7) 2 = 41; # int = 0; # int = 2 4) F = (1, 1); dir is = 7; # int = 1; # int = 2 5) F = (2, 3.5); dir is = 2.5; # int = 0; # int = 1 6) V = ( 3, 5); eq for L of S is = 5 7) 2 = 20 or = ) ( + 4) 2 = 16( 1) or = 1 ( ) ) ( + 3) 2 = 4( + 5) or = 1 ( + 4 3)2 5; F = ( 4, 3); eq of dir = 6; fc endpts are ( 4, 1) & ( 4, 5) 10) ( 1) 2 = 2( 3) or = 1 ( 2 1)2 + 3; F = (1, 3.5); eq of dir = 2.5; fc endpts are (0, 3.5) & (2, 3.5) Math 3 Unit 7 Selected Answers
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