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Understanding Parabolas: Vertex, Focus, and Directrix Analysis

This review focuses on the geometric definition of parabolas and how to find key features such as the vertex, focus, axis of symmetry, and directrix. Using the standard forms of parabolas, we demonstrate through multiple examples how to rewrite equations, identify the direction of opening, and calculate distances from the vertex to the focus and directrix. The information is reinforced by various practice problems aimed at enhancing comprehension of this quadratic shape in the Cartesian plane.

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Understanding Parabolas: Vertex, Focus, and Directrix Analysis

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  1. 8.2Parabolas May 10,2013

  2. Review Graph: x = 3 Graph: y = -5

  3. Parabola Geometric Definition: The collection of all the points (x,y) in a plane that are the same distance from the focus as they are from the directrix. Axis of symmetry Focus p Vertex (h,k) p directrix p: distance from the vertex to the focus or the directrix.

  4. STANDARD FORM (x2) when 4p is positive when 4p is negative

  5. STANDARD FORM (y2) When 4p is positive when 4p is negative

  6. Ex 1: Find the vertex, focus, axis of symmetry and directrix of y Rewrite in standard form: -4 -4 Since 4p is positive graph opens up Focus p 4 p 2. Vertex: (0, 4) Focus: First find P (distance from focus to vertex) From , 4p = 1 4 4 Focus (0, 4) 2 Axis of symmetry (vertical line) Directrix (horizontal line) x

  7. Ex 2: Find the vertex, focus, axis of symmetry and directrix of y Rewrite in standard form: -2y -2y Since 4p is negative graph opens down 1 2. Vertex: (0, 0) Focus: First find P (distance from focus to vertex) From , 4p = -2 4 4 Focus (0, - ) p x p Focus -1 Axis of symmetry (vertical line) Directrix (horizontal line)

  8. Ex 3: Find the vertex, focus, axis of symmetry and directrix of Rewrite in standard form: +2y - 29 +2y -29 Since 4p is positive graph opens up Complete the square Focus 2. Vertex: (5, 2) Focus: First find p (distance from focus to vertex) From 4p = 2 4 4 Focus (5, 2 ) Axis of symmetry (vertical line) Directrix (horizontal line)

  9. Ex 4: Find the vertex, focus, axis of symmetry and directrix of Rewrite in standard form: Complete the square

  10. Since 4p is positive graph opens right 2. Vertex: (-1, -5) Focus: First find p (distance from focus to vertex) From Mult to both sides Focus (-) Focus Axis of symmetry (horizontal line) y = -5 Directrix (vertical line)

  11. Homework: Worksheet 8.2 odd only “I stayed up all night to see where the sun went. Then it dawned on me.”

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