Systems of Equations #6

An inconsistent equation means there is no solution and the lines are parallel. A consistent equation means that there is an answer and if there is one solution, then it is independent, but if there are infinite solutions then it is dependent. There are two methods to solving equations; substitution of elimination. The first step for substitution is to solve one equation for one variable. Then,substitute that variable into the other equation. Next, solve for the variable. Lastly, go back and substitute to find the second variable. The first step for elimination is to interchange any two equations. Next, multiply by a constant number. Finally, add one equation to the other to eliminate a variable. 

Graphs of Polar Equations #5

To find the vertices of an ellipse or parabola, use theta=0,pi. To find the vertex of a parabola, use either theta=0 or theta=pi. These are for the horizontal axis, cos. For the vertical, sin, in order to find the vertices of an ellipse or parabola, plug in theta=pi/2 and 3pi/2. To find the vertex of a parabola, use either theta=pi/2 or theta=3pi/2. Also find x or y intercepts. 

Polar Coordinates #4

In the coordinate (r, theta), r stands for the radius. If theta>0, then you rotate counterclockwise. If theta<0, then you rotate clockwise. When converting from polar (r, theta) to rectangular (x,y), substitute x=rcostheta and y=rsintheta. When converting from rectangular to polar, substitute r^2=x^2+y^2 and tan(theta)=y/x. For both of these conversions, solve for x or r. Trig identities are also key to this process.

Rotating Conic Sections #3

The starting point for rotating conic sections is to use the formula Ax^2+Bxy+Cy^2+Dx+Ey+F=0. Identify all parts in order to being step 1. The first step is to find the angle using cot2(theta)=(A-C)/B. Step two is to replace x and y with either x=x'cos(theta)-y'sin(theta) or y=x'sin(theta)+y'cos(theta). The final step is to use algebra and simplify! A helpful hint is to use the different trig identities used previously so it will be easier to replace things. Rotating conic sections can be tricky, but the trickiest part is to not mess up on the basic algebra steps.

Parabolas #2

Parabolas are U shaped graphs. They have a focus, which is right above the vertex and that distance is called c and is (h,k+c) . The vertex is the center of the parabola such is (h,k). The directrix is y=k-c and is the line below the vertex. The parts described are for a graph that has a c>0, or positive. If the c<0, or negative, then the parts are reversed. The basic equation for a positive parabola is (x-h)^2=4c(y-k) and the basic equation for a negative parabola is (y-k)^2=4c(x-h). 

2nd Semester Goals

Three things I did well this semester were that I kept all my notes neat and color coordinated, studied hard and effectively (I hope) for the final, and asked for help when needed. Three things that I need to improve on are studying harder for all tests and quizzes complete all my WebAssign and blogs on time, and ask for more clarifying help. Last semester was probably the most difficult math time for me. Whenever I studied a lot for something and I think I understand it, the results didn't show it. I know I don't have the best study habits, but it is frustrating when I actually study, but don't get the grade I am aiming for. This semester, I am really going to try to push myself to ask more questions and get more help, even if I think I understand the concept.

This break, I went to San Diego with my basketball team. We took second place in our bracket and only lost by 2! 

Law of Sines and Cosines #15

To use the law of sines, a triangle must either have 2 sides and an opposite angle or 2 angles and an opposite side. The equations to use are sinA/a = sinB/b = sinC/c. When approached with the situation of sin being attached to a letter, take the inverse of both sides. A sin and its number are not allowed to be separated because they are one pair and can only be apart it the inverse is taken. The law of cosines is only used once, then use the law of sines. To use this law, a triangle must be given 2 sides and an included x or 3 sides. The formulas for the law of cosines are a^2=b^2+c^2-2bccosA, b^2=a^2+c^2-2accosA and c^2=a^2+b^2-2abcosC.