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Algebra 2

Algebra 2. Chapter 9 Conic Sections: Circles and Parabolas. 9-3 Parabolas. WARMUP: Determine the distance between the given point and line. Draw a sketch if necessary. ( 3, 4 ); x-axis ( -1, 2 ); y-axis ( -2, 3 ); x = 1 ( 5, -4 ); y = -2 ( 1, -3 ); x = -4. 9-3 Parabolas.

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Algebra 2

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  1. Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas

  2. 9-3 Parabolas • WARMUP: • Determine the distance between the given point and line. Draw a sketch if necessary. • ( 3, 4 ); x-axis • ( -1, 2 ); y-axis • ( -2, 3 ); x = 1 • ( 5, -4 ); y = -2 • ( 1, -3 ); x = -4

  3. 9-3 Parabolas • OBJECTIVE: To learn the relationships among the focus, directrix, vertex, and axis of a parabola and the equation of a parabola.

  4. 9-3 Parabolas • Cool Parabola sites: • http://www.ies.co.jp/math/java/conics/focus/focus.html • http://www.mathwarehouse.com/quadratic/parabola/interactive-parabola.php • http://www.ies.co.jp/math/java/conics/draw_parabola/draw_parabola.html

  5. 9-3 Parabolas • A new definition for a parabola: A parabola is the set of all points equidistant from a fixed line, called the directrix, and a fixed point not on the line, called the focus.

  6. 9-3 Parabolas • Look closer:

  7. 9-3 Parabolas • IMPORTANT! • The distance between the focus and the vertex (call it c) is the same as the distance between the vertex and the directrix! • The parabola ALWAYS opens away from the directrix, and around the focus!!!

  8. 9-3 Parabolas • Typical problem at this stage: • The vertex of a parabola is ( -5, 1 ) and the directrix is the line y = -2. Find the focus of the parabola.

  9. 9-3 Parabolas • The equation of a parabola is: where ( h, k ) is the vertex of the parabola, and a determines how the curve opens, and a basic shape.

  10. 9-3 Parabolas • What about a parabola that looks like this?

  11. 9-3 Parabolas • Let’s just get right to it: • A parabola that opens left or right will have an equation in the form: • What is different? • Is this a function?

  12. 9-3 Parabolas • Some basics: • If a>0, the parabola will open to the right. • If a<0 the parabola will open to the left. • ( h, k ) is still the vertex, as always. • The axis of symmetry will be y=k. • The directrix will be x=?.

  13. 9-3 Parabolas • Look at example 2 in the book on page 413.

  14. 9-3 Parabolas • IMPORTANT!!! If the distance between the vertex and the focus of the parabola is |c|, then it can be shown that in the equation of the parabola.

  15. 9-3 Parabolas • The parabola whose equation is opens upward if a>0, downward if a<0 has vertex V( h, k ) focus F( h, k + c ) directrix y = k – c and axis of symmetry x = h.

  16. 9-3 Parabolas • The parabola whose equation is opens to the right if a>0, to the left if a<0 has vertex V( h, k ) focus F( h + c, k ) directrix x = h – c and axis of symmetry y = k.

  17. 9-3 Parabolas • STEPS TO SOLVE!!! • ALWAYS - Draw a picture with the info you are given! THIS WILL HELP!! • From the picture, determine which way your parabola will open. Roughly sketch it. • Determine the value of c. • Determine a. • Write your equation and all the pieces.

  18. 9-3 Parabolas • Let’s look at some problems:

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