Conic Section

Conic sections are the curves which can be derived from taking slices of a "double-napped" cone. (A double-napped cone, in regular English, is two cones "nose to nose", with the one cone balanced perfectly on the other.)

If you think of the double-napped cones as being hollow, the curves we refer to as conic sections are what results when you section the cones at various angles.

There are some basic terms that you should know for this topic: 

Center: the point (h, k) at the center of a circle, an ellipse, or an hyperbola. 

Vertex: in the case of a parabola, the point (h, k) at the "end" of a parabola; in the case of an ellipse, an end of the major axis; in the case of an hyperbola, the turning point of a branch of an hyperbola. 

Focus: a point from which distances are measured in forming a conic; a point at which these distance-lines converge, or "focus". 

Directrix: a line from which distances are measured in forming a conic. 

Axis: a line perpendicular to the directrix passing through the vertex of a parabola. 

Major axis: a line segment perpendicular to the directrix of an ellipse and passing through the foci; the line segment terminates on the ellipse at either end; also called the "principal axis of symmetry"; the half of the major axis between the center and the vertex is the semi-major axis. 

Minor axis: a line segment perpendicular to and bisecting the major axis of an ellipse; the segment terminates on the ellipse at either end; the half of the minor axis between the center and the ellipse is the semi-minor axis. 

Each conic has a "typical" equation form, sometimes along the lines of the following:

parabola: Ax² + Dx + Ey = 0
circle: x² + y² + Dx + Ey + F = 0
ellipse: Ax² + Cy² + Dx + Ey + F = 0
hyperbola: Ax² – Cy² + Dx + Ey + F = 0

Parabola

To form a parabola, you would start with a line and a point off to one side. The line is called the "directrix"; the point is called the "focus". The parabola is the curve formed from all the points (x, y) that are equidistant from the directrix and the focus. The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola up the middle) is called the "axis of symmetry". The point on this axis which is exactly midway between the focus and the directrix is the "vertex"; the vertex is the point where the parabola changes direction.

The "vertex" form of a parabola with its vertex at (h, k) is: regular: y = a(x – h)2 + k sideways: x = a(y – k)2 + h
The conics form of the parabola equation (the one you'll find in advanced or older texts) is: regular: 4p(y – k) = (x – h)2 sideways: 4p(x – h) = (y – k)2
The relationship between the "vertex" form of the equation and the "conics" form of the equation is nothing more than a rearrangement: y = a(x – h)2 + k y – k = a(x – h)2 (1/a)(y – k) = (x – h)2 4p(y – k) = (x – h)2

                                                                  1st semester 

3 things done well

I didn't do very well first semester. I did well on memorizing the trigonometric identities. I did well on knowing how to solve majority of the trigonometric identities. But for the majority of math, essentially chapters 2-4, I did not do well on. 

3 Goals to improve on

Whenever I need help, I would to come to Ms. V before a test and especially as soon as possible. I need to study more, if I did not get a chance to ask for help. Also I need to resolve my internet issue quicker. 

Favorite Christmas break story

My dad and his friend kept arguing whether to celebrate and eat on Christmas Eve or on Christmas Day.