Conic Sections Review

Parabolas:

The basic equations for a parabola are (x-h)^2 = 4p(y-k) and (y-k)^2 = 4p(x-h).

P is the distance between the vertex and its focus, which is in the same direction as the parabola, and between the vertex and the directrix, or a line directly behind the parabola. (h,k) are the rectangular coordinates for the vertex.

Ellipses:

The basic equations for ellipses are:

[(y-k)^2]/a^2 + [(x-h)^2]/b^2              AND          [(x-h)^2]/a^2 + [(y-k)^2]/b^2       (h,k) are the rectangular coordinates of the ellipse's center. a is half of the major axis (longer "diameter") and b is half of the minor axis (shorter "diameter"). Hyperbolas:  Once again, (h,k) are the coordinates of the center. A and b are the distances from the center to the edge of a rectangle through which the asymptotes of the graph pass diagonally. A will also be the distance from the center to the vertices of the hyperbola

Rotating Conics Review

In this section, we learned how to find the equation and sketch the graph of a conic section that is not perfectly vertical or horizontal, but on tilted axes. This picture shows the three basic equations needed in this chapter:


Steps:

First, you must use the top equation to find the angle of rotation of the axes. You then use that angle to simplify the bottom two equations. The ultimate goal is to replace x and y in the original conic's equation with x' and y', or the new x and y that correspond with the new axes.

Parametric Equations

Parametric equations involve a third variable, usually t. This allows us to depict the passage of time on a graph. The first step is to eliminate the parameter, usually by substituting to equate the x and y variables. Then, graph the regular rectangular equation on a graph. Finally, insert a few values for t in the original equations and label them on the graph. Below are two example problems:

Summations and Limits to Find Area

The following picture shows the six equations used in this lesson:

Here is an example of a problem that utilizes nearly all of the above equations:

Equation #6 is used throughout the first half of this problem. The main strategy is to factor 1/n out of the equation, then putting it outside of the summation. Once 1/n is removed three times so it's 1/n^3, you have to split up the summation into three summations using rule #5. You then use rule #1, #2, and #3, respectively, to simplify each summation. Simplify, then take the limit.

Determinant of a Matrix

There are several strategies to finding the determinant of the matrix. The basic strategy for a 2X2 matrix is to multply the top left number by the bottom right number and subtract the product of the two remaining numbers. Anything bigger than a 3X3 requires a complicated process of finding minors and cofactors that can be seen in action here:

http://www.educreations.com/lesson/view/determinant-of-a-matrix/16850106/?s=ciem9G&ref=link

For 3X3's, you can rewrite the first two columns of the matrix to the right of the matrix and multiply numbers diagonally, then add and subtract. This strategy is also seen in that Educreations video.

Verifying Trigonometric Identities

This section features many problems in which a combination of trig functions must be changed to look like another set of trig functions. These problems are not that difficult now, as we have memorized many trig identities. Strategies on these problems:

• Work with only one side of the equation (choose the more complicated side). 
• Try to convert everything to sines and cosines.
• Simplify complex parts of the problem ASAP using trig identities.