Graphing Systems of Inequalities

Hi,

Systems of Inequalities is very interesting, because it is a review topic. Generally we've finished a lot of the systems of inequalities, but now we are touching over it again. When graphing systems of inequalities, graph it without the inequality and fill in the inequality. It's generally pretty easy to do. However we must remember to observe the rule: when dividing by a negative, we must flip the inequality sign. Sometimes it is easy to make careless errors, but most of the errors will be made in the algebra. The solution will be located in the shaded part.

Thanks,
Kaili Chiu

Cramer's Rule

Hello,

Today I will talk about Cramer's rule. Cramer's rule involves using matrices to solve systems of equations. A quick refresher on matrices: finding the determinant of a 2x2 is ad-bc. Finding the determinant of a 3x3 involves making smaller 2x2's and finding the minors or determinants of those. Substitute the solution column for each variable, finding the determinant of the new matrices formed. Dx/D = x value, Dy/D = y value, and Dz/D = z value. This will solve your equation and allow you to find your answer. Not so hard was it?

Thanks,
Kaili Chiu

Sequences and Series

Hi,

Generally sequences and series was pretty annoying. Especially the card problems, they were tough. Overall, there were many formulas to this, and we also had things like permutations and combinations. These were pretty easy, but when combined in word problems, they were really hard. Honestly, I had a little trouble with them. Overall though, after this week I feel confident in sequences and series. The test was a bit tough though. I will definitely need to review them in the future.

Thanks,
Kaili Chiu

Systems of equations.

Hello,

Today I will be covering systems of equations. Systems of equations are a collection of multiple equations with solutions. These systems can be consistent or inconsistent, and if they are consistent it means that they have a solution, and if they are inconsistent, it means that they have no solution. If they are consistent, they have the option be be either dependent or independent. If they are dependent, it means that the solutions are zero, and you must solve for z and substitute z for t. Independent solutions mean that they are real numbers greater or less than 0. We can solve systems of equations using elimination or substitution or even matrices.

Thanks,
Kaili Chiu

Graphs of Polar Equations

Hi,

Graphing Polar Equations is quite a task. We have to first master polar coordinates, which was explained in the previous blog post. Now that we've reviewed that, we can take a look at the graphs. We use the conversions of r^2 = x^2 + y^2. and x=rcos(theta) and y=rsin(theta). A circle could be r=asin(theta). There are limacons, cardioids, rose curves, and lemniscates. We played around with them in class, when we made art to create a picture. Overall it was a pretty cool lesson, and I enjoyed it a lot.

Thanks,
Kaili Chiu

Polar Coordinates

Hello,

Today, I will talk about Polar Coordinates. Polar coordinates are on a different plane than the (x,y) coordinates. (x,y) coordinates are on the Cartesian plane, whereas the Polar coordinates are on a polar plane. Instead of the common (x,y) point, on the polar plane it is (r,θ). θ is the angle at which the axis is rotated in relation to the positive x-axis of the cartesian plane. Although this seems very confusing at first, it is only just tedious. All in all, though intimidating, this lesson was quite easy.

Thanks,
Kaili Chiu

Rotating Conic Sections

Hello,

Conic sections can be rotated around the origin, substituting x prime and y prime for x and y. We can use unit circle measurements to find the angle. Using the unit circle measurements for the angle, we can adjust the entire cartesian plane to rotate the graph. Some useful information on rotating conic sections include knowing basic trigonometry such as x=xprimecos(theta)-yprimesin(theta), and the y point. These allow the student to plug x and y points into the original formula given. The result will be the rotated equation. Simplify and turn in your answer!

Thanks,
Kaili Chiu

Parabolas

Hello,

Today I am going to talk about parabolas. They are quadratic graphs and are shaped like a "u", although if the standard form is negative, it is shaped like an "n". The a value in standard form determines the opening up or down, and if a>0, then it opens up, and if a<0, it opens down. Parabolas have a vertex, a focus, and a directrix. On a regular (x-h)^2 = 4c(y-k) graph, (h,k) is the vertex, (h, k+c) is the focus, and h-c is the directrix. The axis of symmetry is on the same x coordinate as the vertex of a standard parent graph. Parabolas are present in many fields, including physics and engineering.

Thanks,
Kaili Chiu