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Conics parabola
- 1. CONICS: PARABOLA SIDI GABERSON D.EMPAS, LPT
- 2. OBJECTIVE • Define parabola •Graph the parabola given an equation in vertex form
- 3. What difficulties did youencounter in doing the previous activities?
- 4. PRE-ASSESSMENT Direction: Choose theletter of the correct answer.
- 5. QUESTION 1 A conicthat consists all points in the plane equidistant from a fixed point called focus F and a fixed line l called directrix, not containing F. A. CIRCLE B. PARABOLA C. HYPERBOLA D. ELLIPSE
- 6. QUESTION 1 QUESTION 2The opening of the parabola given by the equation 𝑦 = − 𝑥 − 3 2 − 5. A. upward B. downward C. To the right D. To the left
- 7. QUESTION 1 QUESTION 2 QUESTION3 The graph of a parabola that has a line of symmetry 𝑦 = −4. A. B. C. D.
- 8. QUESTION 1 QUESTION 2 QUESTION3 QUESTION 4 The equivalent vertex form of the equation 𝑥 = 𝑦2 + 4𝑦 − 10 is ____________. A. 𝑥 = 𝑦 + 2 2 − 14 B. 𝑥 = 𝑦 + 2 2 + 14 C.𝑥 = 𝑦 − 2 2 − 14 D.𝑥 = 𝑦 − 2 2 + 14
- 9. QUESTION 1 QUESTION 2 QUESTION3 QUESTION 4 QUESTION 5 The coordinates of the vertex of the parabola that is represented by the Equation 𝑥 = −𝑦2 − 10. A. 0,10 B. 0, −10 C. −10,0 D. 10,0
- 10. QUESTION 1 QUESTION 2 QUESTION3 QUESTION 4 QUESTION 5 QUESTION 6 Find the value of k if the point (0, −3) is on the graph of 𝑥 = 3𝑦2 + 𝑘𝑦. A. 3 B. −3 C. 9 D. −9
- 11. QUESTION 1 QUESTION 2 QUESTION3 QUESTION 4 QUESTION 5 QUESTION 6 QUESTION 7 The coordinates of the vertex of the parabola that is represented by the equation (𝑥 + 1)2 = 𝑦 + 4. A. (1, −4) B. (−1, −4) C. (−1,4) D. (−1,4)
- 12. QUESTION 1 QUESTION 2 QUESTION3 QUESTION 4 QUESTION 5 QUESTION 6 QUESTION 7 QUESTION 8 The length of the latus rectum of the parabola represented by the equation 𝑦 = 4𝑥2 . A. 2 𝑢𝑛𝑖𝑡𝑠 B. 4 𝑢𝑛𝑖𝑡𝑠 C. 1 2 𝑢𝑛𝑖𝑡𝑠 D.1 𝑢𝑛𝑖𝑡
- 13. QUESTION 1 QUESTION 2 QUESTION3 QUESTION 4 QUESTION 5 QUESTION 6 QUESTION 7 QUESTION 8 QUESTION 9 The graph of the parabola represented by the equation 𝑥 = − 𝑦 − 5 2 + 3. A. B. C. D.
- 14. QUESTION 1 QUESTION 2 QUESTION3 QUESTION 4 QUESTION 5 QUESTION 6 QUESTION 7 QUESTION 8 QUESTION 9 QUESTION 10 The equation of the directrix of the parabola given by the equation 𝑥 = −16𝑦2 . A. 𝑦 = 4 B. 𝑦 = −4 C. 𝑥 = 4 D.𝑥 = −4
- 15. QUADRATIC FUNCTION A function ofthe form𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 is a quadratic function where 𝑎, 𝑏, 𝑐 are real numbers and 𝑎 ≠ 0.
- 16. VERTEX FORM OF A QUADRATIC FUNCTION Aquadratic function of the form 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 can be transformed in the vertex form using completing the square.
- 17. VERTEX FORM OF A QUADRATIC FUNCTION 𝑦= 𝑎𝑥2 + 𝑏𝑥 + 𝑐 𝑦 = 𝑎 𝑥2 + 𝑏 𝑎 𝑥 + 𝑏 2𝑎 2 + 𝑐 − 𝑎 𝑏 2𝑎 2 𝑦 = 𝑎 𝑥 + 𝑏 2𝑎 2 + 𝑐 − 𝑏2 4𝑎 𝑦 = 𝑎 𝑥 + 𝑏 2𝑎 2 + 4𝑎𝑐 − 𝑏2 4𝑎 𝑦 = 𝑎 𝑥 − − 𝑏 2𝑎 2 + 4𝑎𝑐 − 𝑏2 4𝑎 So, we have ℎ = − 𝑏 2𝑎 ; 𝑘 = 4𝑎𝑐−𝑏2 4𝑎
- 18. VERTEX OF A QUADRATICFUNCTION The point (ℎ ,𝑘) the parabola which is the graph of 𝑦 = − 𝑎 𝑥 − ℎ 2 + 𝑘 is the vertex, where ℎ = − 𝑏 2𝑎 , and 𝑘 = 4𝑎𝑐−𝑏2 4𝑎
- 19. ACTIVITY 1: This activitywill enable you to review the vertex of a parabola. Determine the highest and lowest value of each function by identifying the y-coordinate of the vertex. Match Column A with Column B 1. 𝑦 = 𝑥2 + 3𝑥 − 6 2. 𝑦 = −𝑥2 − 5𝑥 + 7 3. 𝑦 = −3(𝑥 − 5)2 + 2 4. 𝑦 = (𝑥 + 1)2−3 5. 𝑦 = −(𝑥 − 7)2 − 11 𝑦 = −8.25, 𝑙𝑜𝑤𝑒𝑠𝑡 𝑝𝑜𝑖𝑛𝑡 𝑦 = 13.25, ℎ𝑖𝑔ℎ𝑒𝑠𝑡 𝑝𝑜𝑖𝑛𝑡 𝑦 = 2, ℎ𝑖𝑔ℎ𝑒𝑠𝑡 𝑝𝑜𝑖𝑛𝑡 𝑦 = −3, 𝑙𝑜𝑤𝑒𝑠𝑡 𝑝𝑜𝑖𝑛𝑡 𝑦 = −11, 𝑙𝑜𝑤𝑒𝑠𝑡 𝑝𝑜𝑖𝑛𝑡
- 20. DEFINITION OF A PARABOLA(Desmos) A parabola is the set of all points in the plane equidistant from a fixed point, F, and a fixed line l not containing F.
- 21. Parts of theParabola (Desmos)
- 22. REMEMBER: Standard Form ofthe Equation of a Parabola with Vertex at the Origin FOCUS EQUATION PARABOLAS OPENS DIRECTRIX AXIS OF SYMMETRY (𝑎, 0) 𝑦 = 4𝑎𝑥2 Upward 𝑦 = 𝑎 𝑥 = 0 (−𝑎, 0) 𝑦 = −4𝑎𝑥2 Downward 𝑦 = −𝑎 𝑥 = 0 (0, 𝑎) 𝑥 = 4𝑎𝑦2 To the right 𝑥 = 𝑎 𝑦 = 0 (0, −𝑎) 𝑥 = −4𝑎𝑦2 To the left 𝑥 = −𝑎 𝑦 = 0
- 23. Example 1 Determine thecoordinates of the vertex, axis of symmetry, focal distance (𝑎), length of latus rectum and endpoints of the latus rectum of the parabola 𝑥2 = − 4𝑦 . Sketch the graph.
- 24. Solution Vertex 𝑉(0,0) Focus 𝐹(0,−2) Focal distance (a) 2 Latus rectum (𝑙𝑙) 4 endpoints of the latus rectum −2, −1 , (2, −1) Axis of symmetry 𝑦 = 0 𝐹 𝑙𝑙 𝑉 𝑦 = 0 𝐸1 𝐸2
- 25. Example 2 Determine thevertex, axis of symmetry, focus, focal distance (𝑎), length and endpoints of the latus rectum of the parabola 𝑦2 = 12𝑥. Graph the parabola.
- 26. Solution Vertex 𝑉(0,0) Focus 𝐹(3,0) Focaldistance (a) 6 Latus rectum (𝑙𝑙) 12 endpoints of the latus rectum −2, −1 , (2, −1) Axis of symmetry 𝑦 = 0 𝐹 𝑉 𝐸1 𝐸2 𝑦 = 0 𝑙𝑙
- 27. ACTIVITY 2: Give MyParts Direction: Choose from inside the box the corresponding parts of the given graph of a parabola. The equation of the parabola is 3𝑦2 + 4𝑥 − 24𝑦 + 44 = 0.
- 28. Remember: Parabolas whose vertexis at (𝒉,𝒌) PARABOLAS OPENS EQUATION FOCUS DIRECTRIX To the right (𝑦 − 𝑘)2= 4𝑎(𝑥 − ℎ) (ℎ + 𝑎, 𝑘) 𝑥 = ℎ − 𝑎 To the left (𝑦 − 𝑘)2 = −4𝑎(𝑥 − ℎ) (ℎ − 𝑎, 𝑘) 𝑥 = ℎ + 𝑎 Upward (𝑥 − ℎ)2 = 4𝑎(𝑦 − 𝑘) (ℎ, 𝑘 + 𝑎) 𝑦 = 𝑘 − 𝑎 Downward (𝑦 − ℎ)2= 4𝑎(𝑦 − 𝑘) (ℎ, 𝑘 − 𝑎) 𝑦 = 𝑘 + 𝑎
- 29. ACTIVITY 3: Findmy Pair Match the equation of the parabola to the correct graph. Equations Equations 1. 𝑦2 = 𝑥 − 4 2. 𝑦 = −𝑥2 + 6𝑥 3. 𝑦 = 𝑥2 − 10𝑥 + 29 4. 𝑥 = −𝑦2 + 2𝑦 + 1 5. 𝑦 = 𝑥2 A B C D E
- 30. EXAMPLE 4 (Desmos) Determinethe vertex, focus, focal distance (𝑎), length of latus rectum, endpoints of the latus rectum and axis of symmetry of the parabola (𝑦 − 1)2 = 8(𝑥 − 4). Graph the parabola. Vertex 𝑉(4,1) Focus 𝐹(6,1) Focal distance (a) 2 Latus rectum (𝑙𝑙) 8 endpoints of the latus rectum 6,5 , (6, −3) Axis of symmetry 𝑥 = 2
- 31.
- 32. QUESTION 1 A conicthat consists all points in the plane equidistant from a fixed point called focus F and a fixed line l called directrix, not containing F. A. CIRCLE B. PARABOLA C. HYPERBOLA D. ELLIPSE
- 33. QUESTION 1 QUESTION 2The opening of the parabola given by the equation 𝑦 = − 𝑥 − 3 2 − 5. A. upward B. downward C. To the right D. To the left
- 34. QUESTION 1 QUESTION 2 QUESTION3 The graph of a parabola that has a line of symmetry 𝑦 = −4. A. B. C. D.
- 35. QUESTION 1 QUESTION 2 QUESTION3 QUESTION 4 The equivalent vertex form of the equation 𝑥 = 𝑦2 + 4𝑦 − 10 is ____________. A. 𝑥 = 𝑦 + 2 2 − 14 B. 𝑥 = 𝑦 + 2 2 + 14 C.𝑥 = 𝑦 − 2 2 − 14 D.𝑥 = 𝑦 − 2 2 + 14
- 36. QUESTION 1 QUESTION 2 QUESTION3 QUESTION 4 QUESTION 5 The coordinates of the vertex of the parabola that is represented by the Equation 𝑥 = −𝑦2 − 10. A. 0,10 B. 0, −10 C. −10,0 D. 10,0
- 37. QUESTION 1 QUESTION 2 QUESTION3 QUESTION 4 QUESTION 5 QUESTION 6 Find the value of k if the point (0, −3) is on the graph of 𝑥 = 3𝑦2 + 𝑘𝑦. A. 3 B. −3 C. 9 D. −9
- 38. QUESTION 1 QUESTION 2 QUESTION3 QUESTION 4 QUESTION 5 QUESTION 6 QUESTION 7 The coordinates of the vertex of the parabola that is represented by the equation (𝑥 + 1)2 = 𝑦 + 4. A. (1, −4) B. (−1, −4) C. (−1,4) D. (−1,4)
- 39. QUESTION 1 QUESTION 2 QUESTION3 QUESTION 4 QUESTION 5 QUESTION 6 QUESTION 7 QUESTION 8 The length of the latus rectum of the parabola represented by the equation 𝑦 = 4𝑥2 . A. 2 𝑢𝑛𝑖𝑡𝑠 B. 4 𝑢𝑛𝑖𝑡𝑠 C. 1 2 𝑢𝑛𝑖𝑡𝑠 D.1 𝑢𝑛𝑖𝑡
- 40. QUESTION 1 QUESTION 2 QUESTION3 QUESTION 4 QUESTION 5 QUESTION 6 QUESTION 7 QUESTION 8 QUESTION 9 The graph of the parabola represented by the equation 𝑥 = − 𝑦 − 5 2 + 3. A. B. C. D.
- 41. QUESTION 1 QUESTION 2 QUESTION3 QUESTION 4 QUESTION 5 QUESTION 6 QUESTION 7 QUESTION 8 QUESTION 9 QUESTION 10 The equation of the directrix of the parabola given by the equation 𝑥 = −16𝑦2 . A. 𝑦 = 4 B. 𝑦 = −4 C. 𝑥 = 4 D.𝑥 = −4
- 42.
- 43. Problem: Write an equationof the parabola with vertical axis of symmetry, vertex at the point (5,1), and passing through the point (1,3).
- 44. DIRECTION IN ANSWERING THEPROBLEM, USE A SEPARATE CLEAN SHEET OF PAPER AND SHOW YOUR SOLUTION. ORGANIZED, NEAT AND CLEAN PRESENTATION OF THE SOLUTION IS A PLUS IN YOUR POINTS.
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- 46. THANK YOU ☺☺ ☺