SWBAT identify conic sections and write equations of parabolas in standard form and graph parabolas (Lesson 1- Chapter 9-1)
Introduction to Conics
A conic is the intersection of a plane and a right circular cone. The four basic types of conics are parabolas, ellipses, circles, and hyperbolas.
In the conics above, the plane does not pass through the vertex of the cone. When the plane does intersect the vertex of the cone, the resulting conic is called a degenerate conic. Degenerate conics include a point, a line, and two intersecting lines.

The equation of every conic can be written in the following form:
Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 .
This is the algebraic definition of a conic. Conics can be classified according to the coefficients of this equation.
The determinant of the equation is B 2 - 4 AC . Assuming a conic is not degenerate, the following conditions hold true: If

B 2 - 4 AC > 0, the conic is a hyperbola

B 2 - 4 AC < 0, the conic is a circle or an ellipse

B 2 - 4 AC = 0, the conic is a parabola.
Another way to classify conics has to do with the product of A and C .
Assuming a conic is not degenerate, the following conditions hold true: If

AC > 0, the conic is an ellipse or a circle

AC < 0, the conic is a hyperbola

AC = 0, and A and C are not both zero, the conic is a parabola

A = C , the conic is a circle.
In the following sections we'll study the other forms in which the equations for certain conics can be written, and what each part of the equation means graphically.
Example 1a) : Is the following conic a parabola, an ellipse, a circle, or a hyperbola: -3 x 2 + y + 2 = 0? b) Is the following conic a parabola, an ellipse, a circle, or a hyperbola:
2 x 2 +3 xy - 4 y 2 + 2 x - 3 y + 1 = 0?

c) Is the following conic a parabola, an ellipse, a circle, or a hyperbola:
2 x 2 - 3 y 2 = 0 d) Is the following conic a parabola, an ellipse, a circle, or a hyperbola: -
3 x 2 + xy - 2 y 2 + 4 = 0? e) Is the following conic a parabola, an ellipse, a circle, or a hyperbola: x = 0?

There are several ways to approach the study of conics.
1) You can define conics in terms of the intersection of planes and cones, as the
Greeks did.
2) You can define conics algebraically, in terms of the general second degree equation
Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0
3) You can define it as a locus (collection) of points satisfying a geometric property.
Definition of Parabola
The standard form of the equation of a parabola with vertex at (h,k) is an follows.
( x
 h )
2
( y
 k )
2

4 a ( y
 k ), a

4 a ( x
 h ), a

0

0
The focus lies on the axis a units (directed distance) from the vertex. If the vertex is at the origin (0,0) the equation takes one of the following forms. x
2 
4 ay y
2 
4 ax http://www.cut-the-knot.org/ctk/Parabola.shtml

Example 1) Find an equation of the parabola with vertex at (0, 0) and focus at
(4, 0). Graph the equation.

Example 2) Analyze the equation y
2 
8 x .

Example 3) Analyze the equation x
2  
8 y .
Example 4) Find the equation of the parabola with focus at (0, – 12) and directrix the line y = 12.

Example 5)
Find the equaton of a parabola with vertex at (0, 0) if its axis of symmetry is the y-axis and its graph contains the point

1,

Find its focus and directrix, and graph the equation.