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Parabolas

Parabolas. Finding the Rule and Graphing. Definitions. The parabola has a focus and a directrix l , which is the line NOT passing through the focus. There are four cases according to the concavity. Can have the vertex at the origin, or at (h,k). d(V,F) = d(V,l) = c.

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Parabolas

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  1. Parabolas Finding the Rule and Graphing

  2. Definitions The parabola has a focus and a directrix l, which is the line NOT passing through the focus There are four cases according to the concavity Can have the vertex at the origin, or at (h,k) d(V,F) = d(V,l) = c The axis of symmetry passes through the vertex and the focus. It is perpendicular to the directrix.

  3. 4 Cases - Centered at Origin x2 = 4cy F(0,c) l: y= - c

  4. 4 Cases - Centered at Origin x2 = -4cy l: y= c F(0,-c)

  5. 4 Cases - Centered at Origin y2 = 4cx F(c,0) l: x = - c

  6. 4 Cases - Centered at Origin y2 = -4cx F(-c,0) l: x = c

  7. Observations When x2, it opens up or down The axis of symmetry is the y-axis When y2, it opens left or right The axis of symmetry is the x-axis

  8. Example 1 – Represent the following: y2 = 2x Which way are we opening? To the right (y2 and positive) F(0.5,0) What is c? c = 2/4 = 0.5 Check: 4c = 4(0.5) = 2! l: x = -0.5 What is l? l: x = -c so x=-0.5

  9. 4 Cases - Centered at (h,k) (x-h)2 = 4c(y-k) F(h,c+k) V(h,k) l: y= k - c

  10. 4 Cases - Centered at (h,k) (x-h)2 = -4c(y-k) l: y= k+ c V(h,k) F(h,k-c)

  11. 4 Cases - Centered at (h,k) (y-k)2 = 4c(x-h) V(h,k) F(h+c, k) l: x = h - c

  12. 4 Cases - Centered at (h,k) (y-k)2 = -4c(x-h) V(h,k) F(h-c,k) l: x = h+c

  13. Example 2 – Represent: (x-2)2 = 8(y-4) What is the vertex? (2,4) F(2,6) Which way are we opening? UP! (x2 and positive) What are c and F? V(2,4) c = 8/4 = 2 F(h, c+k)=(2,6) l: y= 2 What is l? l: y =k-c so y=2

  14. HOMEWORK – The only way to make these easier is to practice… Workbook p. 346 #1,2,3,4,5 (intersections like you always do them!) p. 348-349 #6,7,8,9 (for #9, use the table of values to make your parabola more accurate!) NEXT CLASS: Hand in p.343 #20 on a loose leaf! We will move on to general form and inequalities

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