9.2 Graph and Write Equations of Parabolas

Focus

A fixed point that lies on the axis of symmetry of a parabola

Directrix

A line that is perpendicular to the axis of symmetry of a parabola

STANDARD EQUATION OF A PARABOLA WITH VERTEX AT THE ORIGIN

The standard form of the equation of a parabola with vertex at (0, 0) is as follows:

Equation	Focus	Directrix	Axis of Symmetry
x^2 = 4py	(0, p)	y = __-p__	Vertical (__x = 0__)
y^2 = 4px	(p, 0)	x = __-p__	Horizontal (__y = 0__)

Example 1

Graph x = . Identify the focus, directrix, and axis of symmetry.

1. Rewrite the equation in standard form.

^x^=	^Write original equation.
__2x__ = __y^2__	Multiply each side by __2__.

2. Identify the focus, directrix, and axis of symmetry. The equation has the form y² = 4px where P =_{___}. focus is (p, 0), or The directrix is x = -p, or

x =_.

Because y is squared, the axis of symmetry is the __x–axis__ .

1. Draw the parabola by making a table of values and plotting points. Because p _>_ 0, the parabola opens to the __right__. So, use only __positive__ x–values.

x	1	2	3	4	5
y	__±1.41__	__±2__	__±2.45__	__±2.83__	__±3.16__

Example 2

Write an equation of the parabola shown.

Solution

The graph shows that the vertex is__(0, 0)__ and the directrix is y = -p =__3__. Substitute __-3__ for p in the standard form of the equation of a parabola.

x^2 = 4py	Standard form, __vertical__ axis of symmetry
x^2 = 4(-3)y_	Substitute for p.
_x^2 = -12y_	Simplify.