Cramer's Rule

In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the vector of right hand sides of the equations.  This section is like a review for me since I can still recall form last year's algebra 2. Cramer’s rule may only be used when the system is square and the coefficient matrix is invertible.

Systems of Equations

A system of equations is a collection of two or more equations with a same set of unknowns. In solving a system of equations, we try to find values for each of the unknowns that will satisfy every equation in the system. There are three types of linear equations: inconsistent, consistent with independent equation and dependent equation. When X = a number, y = a number the system is independent and the equations will have different values of m when both are placed in y = mx + b (slope-intercept) form. When an inconsistent equation, such as 0 = 3,the system is inconsistent and equations will have the same value of m, but different values of b, when both are placed in y = mx + b form. When an identity, such as 5 = 5, the system is dependent and equations will be identical when both are placed in slope-intercept form

Polar coordinate

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction. To convert polar into rectangular form we need to know that x=r*cosine-data and y=r*sine-data. If data is greater than zero then the graph is counterclockwise, if the data is less than zero than the graph is clockwise. I also acknowledged Cardioids: these have a graph that is vaguely heart shaped and always contain the origin and Limacons with an inner loop: these will have an inner loop and will always contain the origin.Limacons without an inner loop: these do not have an inner loop and do not contain the origin.

Rotating Conic Section

We learn rating conic section on chapter7.4. We studied conic sections with equations of the form: Ax^2+Bxy+Cy^2+Dx+Ey+F=0. Step 1: find angle cot2pi=(A-C)/B where pi is greater than 0 but smaller than 90 degree. Step 2: replace x and y: x=xprime*cosine-data-yprime*sine-data. y=xprime sine-data+yprime cosine data. Step 3:plug these into the original equation and simplify to solve. There are three constant equation help us to recognize different graph shape. Parabola:B^2-4AC=0.

Ellipse: B^2-4ac smaller than 0. Hyperbola: B^2-4AC greater than 0.

Parabolas

We did parabolas on chapter 7 of this semester. Through researching on internet, I now can identify "parabolas" as 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". I found this chapter to be relatively interesting since I personally like the graphing section of math a lot. A parabola equation is (x-h)^2=4c(y-k) for vertical shift. (Y-k)^2=(x-h) applies for horizontal shifts. 

My second semester goal

My second semester's goal for this class is that I wish I can be more focus at the lecture contents. I'll do detail review before every quiz or test in order to get a descent grade. I hope I can enjoy the math more and explore more math theories in mathland. I was sometime absent-minded last semester and now looking forward to improve that. For Christmas, my family went on a trip to LAke Tahoe. They had beautiful views but the weather was freezing. In general, my break was quiet peaceful and relaxing. 

My polar equation graph