Parabola PowerPoint

Debra Schablik

Western Governor’s University

Lesson 10.2Parabolas

Goal: Graph and write equations of parabolas

Creation of a Parabola

A conic section is a curve formed by the intersection of a plane a double-napped cone

(Zoebel, 1997-2006)

Definition of a Parabola

(Larson, Boswell, Kanold & Stiff, 2005)

Where are parabolas?(Internet Access is Required)

•They’re everywhere.

•Put arrow on icon and click.

•Click power point icon on task bar to continue with slide show after video is finished.

(Part 1-They’re Out There!!!, 2008)

Parabolas

Parabolas with vertex at (0,0) and open up or down are in the form:

pyx 42

4py

If positive, the parabola opens up

If negative, the parabola opens down

Parabolas with vertex at (0,0) and open right or left are in the form:

pxy 42

4px

If positive, the parabola opens to the right

If negative, the parabola opens to the left

The Axis of Symmetry

For parabolas that open up or down, the axis of symmetry is the line x = the x-coordinate of the vertex.

For parabolas that open right or left, the axis of symmetry is the line y = the y- coordinate of the vertex.

The Focus

The focus is an ordered pair (x,y), and is INSIDE the parabola and on the axis of symmetry.

The Directrix

The directrix is a line that is perpendicular to the axis of symmetry and is always OUTSIDE the parabola.

4p

4p is the number in front of the variable that has a coefficient of 1.

is the distance from the vertex to the focus and/or the distance from the vertex to the directrix.

p

The Vertex

The vertex lies halfway between the focus ( x, y) and the directrix (line).

Focal Chord

The focal chord, 2p, is measured from the focus and gives the true width of the parabola.

#32 Identify the focus and directrix of the parabola. 

x y2 8x y2 8

opens up, with vertex at origin, to get the focus, plot the point 2 units inside the parabola and on the axis of symmetry, thus the focus is .

4 8 2p p

( , )0 2

)2,0(

The directrix is perpendicular to the axis of symmetry and is also 2 units away from the vertex, so the equation of the directrix is

y 2

2y

y x2 16

#34 Identify the focus and directrix of the parabola. 

y x2 16 y x2 16

4 16 4p p

( , ) 4 0

opens left, with the vertex at origin. To find the focus, plot the point 4 units inside the parabola and on the axis of symmetry, thus the focus is .

( , ) 4 0

xy 162

)0,4(

x 4

The directrix is perpendicular to the axis of symmetry and is also 4 units away from the vertex, so the equation of the directrix is

x 4

4x

From the graph, the vertex is at the origin, (0,0), and the directrix is 2 units away from the vertex.

 The parabola opens up, so the equation is in form. Since p = 2 , the equation is

Example #2 Writing the equation of a parabola

x py2 4

p 2

yx )2(42

yx 82 (Larson, Boswell, Kanold & Stiff, 2005)

pyx 42

#10 Write the standard form of the equation of the parabola with the given focus or directrix with the vertex at (0,0). Focus

)3,0(

Since the focus has to be inside the parabola and lie on the axis of symmetry, this parabola opens up, and is the form

pyx 42 The distance p is the distance from the vertex to the focus, or in this case 3.

So the equation is

x y x y2 24 3 12 ( )

References

Zoebel, Edward A. (1997-2006) Retrieved April 13, 2008. Welcome to Zona Land

http://id.mind.net/~zona/mmts/miscellaneousMath/conicSections/para

bolaPic1.jpg

Larson, Ron, Laurie Boswel, Timothy Kanold and Lee Stif. (2005). Algebra 2. Evanston Illinois: McDougall Little.

They’re Out There! (n.d.) retrieved April 20 , 2008 from http://www.youtube.com/watch?v=pQHxjJxQCzI