UNIT VI
UNDERSTANDING
CONIC SECTIONS 

Working
With Distance and Midpoint Formulas. This is a short presentation, but a very important concept. In fact, further in the following lessons you will see that each one of the equations for the conics is obtained by means of using the distance formula and the definition of the geometric space for each of the conics. Lesson that teaches
how to use the distance formula, how to find the midpoint with two
endpoints, and endpoint of a segment given the midpoint and one
endpoint.

 Parabola:
Formula and
Graphing. You have the Standard Equation of the parabola, and the General Equation for Conics. You may go from the graph to any of the equations, or from any of the equations to the graph. Learn how to use the vertex and general formulas of horizontal and
vertical parabolas, and how to find the equation given the graph. The development of the lesson makes an extensive display of the minor details when you expand the binomials, or when you complete the square to get the actual binomials. These are required depending on whether you are going from the equation to the graph, or from the graph to the equation.

 Parabola: Deriving Standard Equation Proof. Using the distance formula and the definition for geometric space for a parabola; we obtain the equation in Standard Form for the parabolas. Both vertical and horizontal. You will find very helpful the way the lesson presents the graph with all the relevant parts highlighted to be used in the proof. Then as the proofs moves on you see those changes in the graph. Once the initial equation is obtained, you will see all the intermediate steps in working until the Standard Equation of the parabola centered in the origin is obtained. From there then in a visual and very explicit way you see how the parabola is translated to all the four quadrants using a translation vector with components in "x" and "y", and how it is reflected in both the axis "x" and "y". It finalizes with a summary of the formulas and a video to show how the parameters in the formula affect the shape and orientation of the graph.

 Circles:
Formula and Graphing. For this particular lesson as well as the ones for the ellipse, and hyperbola you would benefit from reviewing in past sections of this website how to expand perfect square binomials, and how to complete the square to get the binomials themselves. Inside it, both processes are work to the detail. Mainly, this lesson is aimed at teaching how to use the standard and
general formulas of a circle, and how to use the distance and midpoint
formulas to find the equation of a circle.

 Ellipses:
Formula and
Graphing. Learn the formal definition of an ellipse, and how
to use the general and standard formulas of vertical and horizontal
ellipses, and how to determine the equation from the graph. You will be taken from how to graph the equation when the values of the parameters are given, to how to get the parameters when the graph is the one given to you. The lesson takes great pains in showing you the laborious process of expanding binomials, or completing the square to get the binomials themselves depending of whether you are going from the graph to the general equation for conics, or from the general equation for conics to the graph.

 Ellipse: Deriving Standard Equation Proof. For simplicity the equation is obtained by centering the graph in the origin. Then it is very clearly presented how to translate it to each one of the four quadrants in the coordinate plane by means of a translation vector with components in "x" and "y". This is accomplished using the distance formula and the definition for geometric space for ellipses; we obtain the equation in Standard Form for ellipses. Both vertical and horizontal.


Hyperbolas:
Formula and
Graphing. Equation For Conics: It teaches how to graph the
standard and general forms of horizontal and vertical hyperbolas, and
how to get the equation of the graph of a hyperbola and its asymptotes
using the pointslope formula. It finalizes teaching how to determine
from the general equation for conics, which conic a given equation
represents.

 Hyperbola: Deriving Standard Equation Proof. Using the distance formula and the definition for geometric space for a hyperbola; we obtain the equation in Standard Form for hyperbolas. Both vertical and horizontal. The lesson goes through the process of setting up the hyperbola and the parameters to use during the proof properly labeled in the graph. Then it goes to the proof itself. It takes special care in visually highlighting the fact that the equation is first obtained centered in the origin for simplicity of the proof and then how it is possible to get it in any of the four quadrants of the coordinate plane using a translation vector with components in "x" and "y".


Solving Systems of Equations and Inequalities That Involve Conics. All the lessons given before presented to you how to work with each one of the individual conics. From the circle all the way to the hyperbola. In this case you will learn how to solve systems that involve circles, parabolas, ellipses
and hyperbolas in equations and inequalities. This is accomplished algebraically and graphically verified.


Conics Generator Learn how circles, parabolas, ellipses
and hyperbolas are generated from transversal sections of a right cone, after this is generated by rotating the generator 360 degrees.


General Equation for conics: Videos One of the difficulties that you may experience when learning how to manipulate conics is the laborious process to get each different instance when you change a value in any of the parameters and try to see the effect reflected in the graph. If you don't have a graphing calculator it implies doing the computations, and then plotting the points in graph paper. Even with the calculator, you have to enter the values in the calculator, and most of the ones in the market, while very helpful, still lag in terms of making this “cumbersomefree.” This needs to take place over and over for each change in the parameter that controls the conic equation in question. This is streamlined with the advent of the applets that allow to generate in an almost instantaneous way different instances by playing with sliding rulers, or clicking at check boxes. The changes are reflected on the graph in seconds. A big difference with the described process. Due to technology limitations that don't allow to play the actual applets for you to interact directly with them. You have available for you a set of videos of these applets when they are manipulated to accomplish these same results. Learn about circles, parabolas, ellipses
and hyperbolas with those animated applets captured in videos
