Mastering Parabolas: Definitions, Formulas, and Examples

1. Worksheet Key 9.2: Parabolas

2. Worksheet Key 9.2: Parabolas

3. Section 11.3 CONIC SECTIONS: PARABOLAS 9.2: Parabolas

4. Shape of a parabola from a cone 9.2: Parabolas

5. Real-Life Examples 9.2: Parabolas

6. Definitions • Parabola:Graph of a quadratic equation to which a set of points in a plane that are the same distance from a given point • Vertex:Midpoint of the graph; the turning point • Focus:Distance from the vertex; located inside the parabola • Directrix:A fixed line used to define its shape; located outside of the parabola • Axis of Symmetry:A line that divides a plane figure or a graph into congruent reflected halves. • Latus Rectum:A line segment through the foci of the shape in which it is perpendicular through the major axis and endpoints of the ellipse • Eccentricity:Ratio to describe the shape of the conic, e = 1 9.2: Parabolas

7. Formulas to know: Horizontal Axis Standard Form: If the ‘x’ is not being squared: 9.2: Parabolas

8. Formulas to know: Vertical Axis Standard Form: If the ‘y’ is not being squared: 9.2: Parabolas

9. Formulas to know: All Standard Form Equations 9.2: Parabolas

10. Review of Parent Function Parabolas y = x2 or x2 = y x = y2 or y2 = x 9.2: Parabolas

11. The Parabola Brief Clip Where do we see the focus point used in real-life? 9.2: Parabolas

12. Horizontal Parabola Vertex: (h, k) Focus point: (p, 0) Directrix: x = –p Axis of Symmetry: y = k Length of Latus Rectum: |4p| Latus Rectum: (h + p, k+ 2p) Horizontal parabola due to its ‘Axis of Symmetry’ (h, k) F (p, 0) y = k x = –p 9.2: Parabolas

13. Vertical Parabola Vertex: (h, k) Focus point: (0, p) Directrix: y = –p Axis of Symmetry: x = k Length of Latus Rectum: |4p| Latus Rectum: (h+ 2p, k + p) (p, 0) F (h, k) y = –p Vertical parabola due to its ‘Axis of Symmetry’ x = k 9.2: Parabolas

14. Review of Parent Function Parabolas y = x2 or x2 = y x = y2 or y2 = x 9.2: Parabolas

15. Steps in Writing Equations of Parabolas • Identify whether the equation opens Up/Down or Left/Right • Divide the coefficient (if necessary) to keep the variable by itself • On the side without the squared into the equation (which usually is a fraction), drop off all the variables • Multiply the coefficient (not involved with squared) with ¼ to solve for p • Put it in standard form and graph 9.2: Parabolas

16. Example 1 Determine the vertex, focus point, directrix, axis of symmetry, latus rectum, and graph the parabola for y = 4x2. First, figure out what variable is squared? Put into the suitable equation. What is the vertex? (0, 0) 9.2: Parabolas

17. Example 1 Determine the vertex, focus point, directrix, axis of symmetry, latus rectum, and graph the parabola for y = 4x2. To find p: Isolate the equation to where the variable squared has a coefficient of 1. 9.2: Parabolas

18. Example 1 Determine the vertex, focus point, directrix, axis of symmetry, latus rectum, and graph the parabola for y = 4x2. To find p: Take the coefficient in front of the isolated un-squared variable and multiply it by ¼ 9.2: Parabolas

19. Example 1 Determine the vertex, focus point, directrix, axis of symmetry, latus rectum, and graph the parabola for y = 4x2. To determine the Latus Rectum: 9.2: Parabolas

20. Example 1 Determine the vertex, focus point, directrix, axis of symmetry, latus rectum, and graph the parabola for y = 4x2. (0, 0) 1/16 Vertical Vertex: P: Focus Point: Directrix: Axis of Symmetry: Latus Rectum: (0, 1/16) F y = –1/16 x = 0 (+1/8, 1/16) 9.2: Parabolas

21. Your Turn Determine the vertex, focus point, directrix, axis of symmetry, latus rectum, and graph the parabola for x = (1/20)y2. (0, 0) 5 Vertex: P: Focus Point: Directrix: Axis of Symmetry: Latus Rectum: (5, 0) F x = –5 y= 0 (5, +10) 9.2: Parabolas

22. Example 2 Write in standard form equation of a parabola with the vertex is at the origin and the focus is at (2, 0). F P = 2 9.2: Parabolas

23. Your Turn Write a standard form equation of a parabola where the directrix is y = 6 and focus point (0, –6). 9.2: Parabolas

24. Example 3 Determine the vertex, focus point, directrix, axis of symmetry, latus rectum, and graph the parabola for (y + 1)2 = 8(x + 1). (–1, –1) 2 Vertex: P: Focus Point: Directrix: Axis of Symmetry: Latus Rectum: (1, –1) x = –3 F y = -1 (1, 3), (1, –5) 9.2: Parabolas

25. Example 4 Determine the vertex, focus point, directrix, axis of symmetry, latus rectum, and graph the parabola for (x – 7)2 = –8(y – 2). (7, 2) –2 Vertex: P: Focus Point: Directrix: Axis of Symmetry: Latus Rectum: (7, –2) y = 4 F x = 7 (7, 5), (7, 9) 9.2: Parabolas

26. Your Turn Determine the vertex, focus point, directrix, axis of symmetry, latus rectum, and graph the parabola for (y – 1)2 = –4(x – 1). (1, 1) –1 Vertex: P: Focus Point: Directrix: Axis of Symmetry: Latus Rectum: (0, 1) F x = 2 y = 1 (0, 3), (0, –1) 9.2: Parabolas

27. Example 5 Write a standard form equation of a parabola where the vertex is (–7, –3) and focus point (2, –3). F 9.2: Parabolas

28. Example 6 Write a standard form equation of a parabola where the vertex is (–2, 1) and the directrix is at x = 1. 9.2: Parabolas

29. Your Turn Write a standard form equation of a parabola where the axis of symmetry is at y = –1, directrix is at x = 2 and the focus point (4, –1). 9.2: Parabolas

30. Assignment Worksheet 9.2: Parabolas