**INTRODUCTION TO CONIC SECTIONS - THE PARABOLA
Copyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to
ingrid.stewart@csn.edu.**

Learning Objectives - This is what you must know after studying the lecture and doing the practice problems!

1. Memorize the characteristics of the focus, the directrix, and the latus rectum.

2. Memorize which standard form pertains to what graphs.

3. Change any equation to standard conic form.

2. Find the coordinates of the focus and the vertex.

3. Find the equation of the directrix.

In the next three lessons, we are getting away entirely from trigonometry, and

very brieflyinvestigate a few concepts in analythic geometry. Namely, we will introduce conic sections.A conic section (or simply a conic) can be described as the intersection of a plane with two cones stacked tip to tip. The most common intersections are the parabola, the hyperbola, the circle, and the ellipse.

Let's discuss the parabola first!

The Conic Section resulting in a Parabola - see #1 through 4 in the "Examples" documentIn algebra, we discussed quadratic functions and their graphs called parabolas. Their standard form is

.y = a(xh)^{2}+ kwhen

and whena < 0a > 0Parabolas can also be given a geometric definition. It is based on a point called

focusand a fixed line calleddirectrix.The focus is a distance, often called

, away from the vertex of the parabola and also lies along the axis of symmetry. The directrix is perpendicular to the axis of symmetry and is also a distancepaway from the vertex, but below it.pGeometrically, a parabola is defined to be the set of all points that are always the same distance away from the

directrixand thefocus. Parabolas resulting form conic sectioning can be open up, down, right or left.

You must know the characteristics of the focus, the directrix, and the latus rectum!

Parabolas Open Up or Open Down:

p > 0 p < 0Note the line sement called

latus rectum. It is is often helpful when graphing a parabola. It passes through the focus, is parallel to the directrix, and has its endpoints on the parabola.Standard Form of the equation of a parabola is . If

the parabolas are open up. Ifp > 0the parabolas are open down.p < 0

You must know that this standard form pertains to parabolas open up and down.

You do not need to memorize the following information, however, you must know how to work with it when given.Coordinates of the Focus are

.(h, k + p)Coordinates of the Vertex are

.(h, k)Equation of the Directrix is

.y = k p

Parabolas Open Right or Open Left:

p > 0 p < 0Standard Form of the equation of a parabola is

. Ifthe parabolas are open right. Ifp > 0the parabolas are open left. down.p < 0

You must know that this standard form pertains to parabolas open right and left.

You do not need to memorize the following information, however, you must know how to work with it when given.Coordinates of the Focus are

.(h + p, k)Coordinates of the Vertex are

.(h, k)Equation of the Directrix is

.x = h p