Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.
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Transcript of Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.
![Page 1: Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.](https://reader036.fdocuments.in/reader036/viewer/2022081504/5697c0211a28abf838cd2a6b/html5/thumbnails/1.jpg)
Algebra 2
Chapter 9 Conic Sections: Circles and Parabolas
![Page 2: Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.](https://reader036.fdocuments.in/reader036/viewer/2022081504/5697c0211a28abf838cd2a6b/html5/thumbnails/2.jpg)
9-3 Parabolas
WARMUP: Determine the distance between the given
point and line. Draw a sketch if necessary.
1. ( 3, 4 ); x-axis
2. ( -1, 2 ); y-axis
3. ( -2, 3 ); x = 1
4. ( 5, -4 ); y = -2
5. ( 1, -3 ); x = -4
![Page 3: Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.](https://reader036.fdocuments.in/reader036/viewer/2022081504/5697c0211a28abf838cd2a6b/html5/thumbnails/3.jpg)
9-3 Parabolas
OBJECTIVE: To learn the relationships among the focus, directrix, vertex, and axis of a parabola and the equation of a parabola.
![Page 4: Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.](https://reader036.fdocuments.in/reader036/viewer/2022081504/5697c0211a28abf838cd2a6b/html5/thumbnails/4.jpg)
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
![Page 5: Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.](https://reader036.fdocuments.in/reader036/viewer/2022081504/5697c0211a28abf838cd2a6b/html5/thumbnails/5.jpg)
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.
![Page 6: Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.](https://reader036.fdocuments.in/reader036/viewer/2022081504/5697c0211a28abf838cd2a6b/html5/thumbnails/6.jpg)
9-3 Parabolas
Look closer:
![Page 7: Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.](https://reader036.fdocuments.in/reader036/viewer/2022081504/5697c0211a28abf838cd2a6b/html5/thumbnails/7.jpg)
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!!!
![Page 8: Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.](https://reader036.fdocuments.in/reader036/viewer/2022081504/5697c0211a28abf838cd2a6b/html5/thumbnails/8.jpg)
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.
![Page 9: Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.](https://reader036.fdocuments.in/reader036/viewer/2022081504/5697c0211a28abf838cd2a6b/html5/thumbnails/9.jpg)
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.
2( ) ( )y k a x h
![Page 10: Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.](https://reader036.fdocuments.in/reader036/viewer/2022081504/5697c0211a28abf838cd2a6b/html5/thumbnails/10.jpg)
9-3 Parabolas
What about a parabola that looks like this?
![Page 11: Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.](https://reader036.fdocuments.in/reader036/viewer/2022081504/5697c0211a28abf838cd2a6b/html5/thumbnails/11.jpg)
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?
2( ) ( )x h a y k
![Page 12: Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.](https://reader036.fdocuments.in/reader036/viewer/2022081504/5697c0211a28abf838cd2a6b/html5/thumbnails/12.jpg)
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=?.
2( ) ( )x h a y k
![Page 13: Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.](https://reader036.fdocuments.in/reader036/viewer/2022081504/5697c0211a28abf838cd2a6b/html5/thumbnails/13.jpg)
9-3 Parabolas
Look at example 2 in the book on page 413.
![Page 14: Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.](https://reader036.fdocuments.in/reader036/viewer/2022081504/5697c0211a28abf838cd2a6b/html5/thumbnails/14.jpg)
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.
1
4a
c
![Page 15: Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.](https://reader036.fdocuments.in/reader036/viewer/2022081504/5697c0211a28abf838cd2a6b/html5/thumbnails/15.jpg)
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.
2 1( ) ( ) where
4y k a x h a
c
![Page 16: Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.](https://reader036.fdocuments.in/reader036/viewer/2022081504/5697c0211a28abf838cd2a6b/html5/thumbnails/16.jpg)
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.
2 1( ) ( ) where
4x h a y k a
c
![Page 17: Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.](https://reader036.fdocuments.in/reader036/viewer/2022081504/5697c0211a28abf838cd2a6b/html5/thumbnails/17.jpg)
9-3 Parabolas
STEPS TO SOLVE!!!
1. ALWAYS - Draw a picture with the info you are given! THIS WILL HELP!!
2. From the picture, determine which way your parabola will open. Roughly sketch it.
3. Determine the value of c.
4. Determine a.
5. Write your equation and all the pieces.
![Page 18: Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.](https://reader036.fdocuments.in/reader036/viewer/2022081504/5697c0211a28abf838cd2a6b/html5/thumbnails/18.jpg)
9-3 Parabolas
Let’s look at some problems:
![Page 19: Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.](https://reader036.fdocuments.in/reader036/viewer/2022081504/5697c0211a28abf838cd2a6b/html5/thumbnails/19.jpg)
9-3 Parabolas
![Page 20: Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas.](https://reader036.fdocuments.in/reader036/viewer/2022081504/5697c0211a28abf838cd2a6b/html5/thumbnails/20.jpg)
9-3 Parabolas
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9-3 Parabolas