Polar Graphs

Polar coordinates

Polar coordinates are a way to label a point on a plane using the radius and angle, rather than x and y coordinates such as (x,y). Some important terms include: Pole=origin, polar axis=ray from the pole, r=directed distance from O to P, and lastly, Theta=angle couterclockwise from polar axis to segment OP. Unlike normal (x.y) coordinates, polar coordinates can be different yet be at the same point! What do I mean? Well, since an angle can be measured counterclockwise or clockwise, a coordinate like (r,theta) will equal (-r, theta+pi).
You should know the ways to convert coordinates from polar to rectangular. The equations you need are as follows: 

Here is an example of a conversion: 

Graphs of Polar Equations

Polar Equations, as already learned, are different from rectangular equations and useful in their own ways. Polar coordinates correspond with Polar equations, which both use angles and radius. These equations can make much cooler shapes and figures that rectangular equations can not make due to its limitations. Some typical polar equations: 

Polar Equations can also have symmetry and be symmetrical to many axes. One can use the symmetry test to yield what the polar equation/graph is symmetrical to. The symmetry test is as follows: 

One should also use a graphing utility to check your graph or graph extremely complex equations/graphs. 

Ellipses and Hyperbolas Review

Ellipses and Hyperbolas are closely related in both equations and definitions, even though they look nothing alike, so they can be easily learned together. First of all, an ellipse is the set of all points (x, y) the sum of whose distances from two distinct fixed points (foci) is constant. Some terms include: vertices=The line through the foci that intersects the ellipse at two points, major axis=The chord joining the vertices, midpoint=center of the ellipse, minor axis=The chord that is perpendicular to the major axis. Ellipse equations: 

The Hyperbola is the set of all points (x, y) the difference of whose distances from two distinct fixed points (foci) is constant. The terms that were defined with the ellipse also apply to hyperbolas. Hyperbola Equations: 

As you can see, they are similar in many ways such as how their equations are presented. The main differences that can be confusing and need to be memorized are the differing signs in the equations. 

Vectors in Space Review

Vectors, can be associated with an animated villain or cool mathematical concept. Sadly, I will have to explain the math one. Vectors in space, unlike normal vectors, exist in a 3D field, and therefore have three coordinates that denote is position unlike vectors on a plane. A vector in space can be denoted by v= <v1,v2,v3>. This means that there are three axes in all, the x, the y, and the z.The zero vector is denoted by 0 = <0,0,0>. The unit vectors are as follows : i= <1,0,0>, j= <0,1,0>, and k= <0,0,1> in the direction of the positive z-axis. One can obtain the component form of a vector by subtracting the values of the coordinates of the starting point by the values of the coordinates of the terminal point. Equations for vectors in space are as follows: 

and an example of a vector in space: 

Polar Coordinates Review

Joel had taught the class about polar coordinates, a way to label a point on a plane using the radius and angle, rather than x and y coordinates. Some important terms include: Pole=origin, polar axis=ray from the pole, r=directed distance from O to P, and lastly, Theta=angle couterclockwise from polar axis to segment OP. Unlike normal (x.y) coordinates, polar coordinates can be different yet be at the same point! What do I mean? Well, since an angle can be measured counterclockwise or clockwise, a coordinate like (r,theta) will equal (-r, theta+pi).
You should know the ways to convert coordinates from polar to rectangular. The equations you need are as follows: 

Matrices and Systems of equations Review

Miss V's daughter, Kara, followed in her footsteps and taught the ways of Matrices and Systems of equations. The two ways to solve system of equations with matrices, Gaussian Elimination and Gauss-Jordan Elimination. To use these techniques, one must know Elementary Row Operations, the ways of which one can manipulate matrices to get numbers where they want. These operations include: 1) interchange any two rows 2) multiply a row by a non-zero constant 3) add one row to another. 

The Gaussian Elimination with Back-Substitution method steps are as follows:
-Write the augmented matrix of the system of linear equations
-Use elementary row operations to rewrite the augmented matrix in row-echelon form
-Write the system of linear equations corresponding to the matrix in row-echelon form, and use back-substitution to find the solution
Make sure that you have a leading 1 and that the main diagonal is consisted of 1's. Everything under the main diagonal has to be zeros but above the main diagonal there can be any numbers.

The Gauss-Jordan Elimination method steps are as follows:
-Write the augmented matrix of the system of linear equations
-Use elementary row operations to rewrite the augmented matrix in row-echelon form
-Write the system of linear equations corresponding to the matrix in row-echelon form
Make sure that you have a leading 1 and that the main diagonal is consisted of 1's. Everything under and above has to be zeros.

Cross Product Review

The two main ways to find the cross products are using determinants/cofactors, and the quicker method without cofactors. In the determinants/cofactors method, you must use cofactor expansion where the values of the vectors are placed in a determinant and then expanded into cofactors for addition. An example can be seen below: 

Another method is by using the determinant also, but by using the quick method without cofactors, but by copying the first two columns to the right so one can multiply diagonally while adding or subtracting the values. You add when the multiplication is done from a top, left to a down, right while you subtract when it is from a down, left to a top, right. One can learn many things through the dot products such as how lluxvll equals the area of a parallelogram. Another important thing is the triple scalar product, which can only be found out when there are three vectors in space. The triple scalar product can be used to find the volume of the 3D shape the vector forms from one side. 
An example of the shortcut method can be seen below: 

Bionomial Theorem Review

Jacqueline had re-taught the class the binomial theorem/expansion. In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. There are also two main ways to find out the binomial coefficients of an expansion. The main way that we had normally used was with Pascal's triangle, a triangle of numbers that can be used to find the coefficients of a binomial expansion, where the row number correlates with the power. With this pascal triangle, individual terms can also be found easily with the knowledge of the coefficient that corresponds to the specific term. The binomial theorem is as follows. 

The Pascal's triangle: 

Here is an example using the binomial theorem: 

Parametric equations

Parametric equations are basically normal x and y equations with the addition of a third variable, t. With this thrid variable, one can learn another useful item that corresponds eith the change of x and y such as time. To eliminate the parameter, first start with the parametric. Then, solve for the variable t in one of the equations given. After that, substitute the value of t that was found into the second equation. Lastly, move around values to obtain the rectangular equation, or normal x and y equation. To eliminate the parameter, you can also create a graph by creating a table of values that contain the values of all three variables. 

Bernoulli's equation

Finding out the pressure of water in a pipe can be extremely hard and tiring, but it can even harder to find out the pressure of more than one place in the pipe! However one can actually easily do this with an equation! With bernouilli's equation "p+1/2pv^2+pgh", which is constant for any two points in a pipe or any other water filled container, one can easily find out the balance between pressure, velocity, and height, and therefore, easily find out the pressure. With this one equation, one can also learn that as area increases, pressure decreases! This equation and many others that involves physics or other math-related subjects, one can understand the world more deeply. This shows that a single equation can be extremely useful and versatile.

Cross Product of Vectors

There are many ways to find the cross product of vectors. This includes using the determinant form. In this method, you must use cofactor expansion where the values of the vectors are placed in a determinant and then expanded into cofactors for addition. An example can be seen below:

Another method is by using the determinant also, but by using the quick method without cofactors, but by copying the first two columns to the right so one can multiply diagonally while adding or subtracting the values. You add when the multiplication is done from a top, left to a down, right while you subtract when it is from a down, left to a top, right. One can learn many things through the dot products such as how lluxvll equals the area of a parallelogram. Another important thing is the triple scalar product, which can only be found out when there are three vectors in space. The triple scalar product can be used to find the volume of the 3D shape the vector forms from one side. 

Vectors in Space

In mathland, we had reviewed the concept of vectors, but with a twist. With vectors in space, there are three numbers to denote the vector due to the the ammount of directions it goes. A vector can be written as <V1, V2, V3>. This means that there are three axes in all, the x, the y, and the z. One can obtain the component form of a vector by subtracting the values of the coordinates of the starting point by the values of the coordinates of the terminal point. All in all, vectors in space are just normal vectors with one more direction/value. One can use the normal equations for vectors on a plane with the addition of the third value into the equation according to how the Xs and Ys are presented in the equation. These equations include the vector length, unit vector, vector addition, and etc.

Math Genius

A man who was attacked and beat up, suddenly becomes a math genius! How did this happen? Well, due to his brain injury, this man became a savant, an individual who is exceptional in one or a few areas of cognition. This one injury can help everyone learn how they might also become better mathematicians! As shown through measuring his brain waves and activity, math skills resides mostly in the left hemisphere of his brain. This man now sees the world through mathematical concepts such as geometry. I hope that we can figure out soon how we all can quickly and easily increase our math skills. With the way human technology has been progressing, I would not be surprised at all if the use of surgery to increase skills was available in my lifetime. One can learn more here : http://www.huffingtonpost.com/2014/05/06/brain-injury-jason-padgett-math-genius_n_5273609.html

Limits Introduction

In mathland, we had learned about limits, a concept that is used mainly in calculus. However, we still learned the basics of limits. Limits are basically the values that equations that are functions or sequences approach. To find a limit, one can apply graphing or just create a table where the values close to the x value that the function goes to can be known. Through this method with the table, an approximation can be made for the value of the limit. An example of each method is shown below. 

Also, we can easily tell when there is no actual limit. The cases include when the function has two lines that converge but goes to infinite. Another scenario is when there is a gap. Lastly, when there is extreme occilation between two values.

Evaluating limits

Today in Mathland, we learned avout something that will help us later in Calculus. we learned about how to evaluate limits. There are three main steps to it. First, you must direct substitute to see if you get 0/0. If you do not, that will be your limit. Next, you do cancellation or rationalization. You do rationalization if there is a square-root. Next, you would direct substitute to get the answer. Here are a few examples.

There can also be one- sided limits. This is true for when there are gaps in the graph. This can be seen in the following picture. 

After learning all this, we then learned about the difference quotient. The difference quoties is used to find the slope through two points on a graph. This can be seen in the following picture.