Parabolas

Parabolas are essentially graphed as "U"s. Any given point on a parabola is equally as far from the focus as it is the directrix. All parabolas are symmetric about their bisector or axis of symmetry. The equation of a parabola contains a y^2 and an x, or the other way around. The vertex of a parabola is the point at the very top of the bend. The can either be the highest, lowest, farthest left, or farthest right point of the graph, depending if it's positive or nevagtive and vertical or horizontal. Parabolas are used greatly in physics, such that they display the natural tendencies of gravity. 

Tower of Hanoi

When playing the tower of Hanoi puzzle, I noticed that to finish, one had to utilize all three pillars. I did not really find any specific rhythms or strategies for finding the fasted solution, but I did notice that one had to go against their instinct to complete it. Mathematical induction is seen in this puzzle through the reoccurring steps that include using all three towers. This proves that to get all the rings on the last tower, the middle tower must always be used. This is an inevitable  pattern that occurs whether there are 2 rings or 6. 

Graphing systems of inequalities

Linear inequalities are graphed as lines. There are usually two to four sets of linear inequalities in a graph. When shading in parts of the graph, one must plug in a single x-y coordinate to each equation in the set. If the result is true, then you shade the side of the line that contains the point. If it is not true, then you shade the opposite side.  Each line must go through this shading process.  Whichever potion of the plane contains shaded areas from both the lines is the "answer". 

Cramer's rule

Cramer's rule is used to solve for the varisbles in a system of linear equations. It starts by finding the determinate of a three count set. Once the determinate is found, you replace the collumn of whatever variable you are trying to find with the answers of each equation.  You then find the determinate of the new equation. If one was solving for x, they would put the outcome of Dx/x, or the determinate of x over the determinate of the original set. To solve for all,the variables, you just repeat this process for y, z, and so on. You write the answer in an ordered pair, such as (4,5,3). 

Systems of equations

Linear equations graph as straight lines. The simplest of linear equations have two variables and two equations. These quotations can be soved in one of three ways, substitution, elimination, or cramer's rule. Each linear equation has what is called a determinate. They are tested by plugging in an x-y coordinate, and if it is true, then that side is shaded. I it is not, however, then the other side is shaded.

 

Rotating conic sections

Conic sections are second degree polynomials. The equation of an expanded conic section is Ax^2+Bxy+Cy^2+Dx+Ey+F=0. If B is zero, no rotation takes place. If, however, it is not zero, then it is rotated about the axises. The center of the graph will never change. It can rotate up to 360 degrees. Comics can be parabolas, hyperbolas, or ellipses. 

Graphs of Polar equations

Like most other types of equations, there are many graph shapes that fall under the category of polar equations. The simplest of these are the line through the orgin or just a normal circle. Others that fall under this category are spirals, cardioids, lemniscates, limacons, and rose curves. An element that all do these graphs have in common is the use of theta, with the exception of the circle at the orgin. While the appearance of circles, lines, and spirals may seem obvious, the other four are a little more complex. Cardioids are essentially heart-shaped graphs that go up and down for sin and left and right for cos. Limacons are graphs with either an interior loop, a climpled side, or just convex, depending on the value of a/b. Rose curves are basically graphs that look like flowers. Their number of pedals is dependent on the value of n. Lemniscates look like an infinity sign, and are either on the x-axis for cos or diagonal for sin.