Rotation of Conics

We use the equation Ax^2+Bxy+Cy^2+Dx+Ey+F=0 to solve conic sections. When B=0, we obtain the standard forms of equations of circles, parabolas, ellipses, and hyperbolas. Degenerate cases might be represented and that's when two interesting lines, one line, a single point, two parallel lines, or no graph at all appears. When B doesn't equal zero then it is possible to remove the xy-term in equation (1) by a rotation of axes. Therefore it is always possible to select the angle of rotation so that any equation of the form (1) can be transformed into an equation in x' Andy ' with no x'y'-term: A'(x)^2+C'(y')^2. 

Parabola

There are two types of parabolas. A parabola with (x-h)^2=4c(y-k)  form and a parabola with (y-k)^2=4c(x-h). The first parabola has a vertex of (h,k), focus of (h, k+c), directrix of y=k-c and axis of symmetry of x=h. The second parabola has a vertex of (h, k), focus of (h+c, k), directrix of x=h-c, and axis of symmetry of y=k. For both graphs the value of c can be greater than or less than zero. You do the y-intercept if your given an equation of (x-h) and you find the x-intercept if you are given an equation of (y-k). The standard form of the equation of a parabola with focus (0, c), directrix y=-c, c>0, and vertex (0, 0). 

Blog for first week back from winter break

3 things you did well:

The first thing that I did well last semester was staying on top of my homework on webassign and not getting behind on homework. The second thing is studying hard for all my tests and quizzes and asking the teacher for help when I didn't understand something I was studying. The third thing is taking good notes in class and listening to the teacher as much as possible. 

3 goals to improve on: 

The first goal is to get better grades on my tests and quizzes. The second goal is to try different studying styles and to see if they benefit me or not. The third goal is to have more confidence in myself and to not give up when the class becomes difficult. 

Favorite Christmas Story: 

My favorite Christmas story is The Night Before Christmas with the mouses and not a creek of noise.