Chapter 9 Conic Sections: Circles and Parabolas

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Chapter 9 Conic Sections: Circles and Parabolas 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 OBJECTIVE: To learn the relationships among the focus, directrix, vertex, and axis of a parabola and the equation of a parabola.

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

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.

9-3 Parabolas Look closer:

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!!!

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.

9-3 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.

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

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?

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=?.

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

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.

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.

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.

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.

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

9-3 Parabolas

9-3 Parabolas

9-3 Parabolas