Majors and Colleges ")

3 Majors

1. Biomedical Engineering: This major interested me since I love Biology/Anatomy as well as Math and it has both of them integrated! Well in this major, people try to find ways to "fix" people whenever they need to be "fixed". They use engineering to help solve problems related to health (Yay). Their job is to invent new vaccines, invent new technology, etc. that will benefit those who need it in order to be "fixed". With tons of math and science, this major really caught my attention.
2. Biochemistry: As I was reading the description, I knew exactly what they were talking about! I love learning about proteins and enzymes as well as about the chemical reactions that go on inside organisms. In AP Biology, even though many people hated learning about the detailed process of photosynthesis which included the light reactions and about what occurs in mitochondria to create energy, I really enjoyed it and got into it. In the major, I will also learn about how living things function. The creation of DNA is also very intriguing to me, and I hope to study something that I will never get tired of since I will be doing it for the rest of my life.
3. Accounting: I will learn to help businesses so that they can make the best financial decisions. By looking at financial records, I will be able to predicit what the company should do financially and what they should not do. I will "advise"in order to try to guarantee the success of a business. I'll also learn how to prepare financial statements. OK I LOVE MATH! There are many careers that can be obtaind by this major such as becoming a budget analyst (ive liked this since 9th grade!). I think all of this is fun and I would love to be able to do this!

3 Colleges

1. University of California Los Angeles: UCLA is a public university with only 23% of applicants admitted. It is located in a very large city in an Urban setting. 92% of students had a GPA of 3.75 percent or higher, which makes acceptance a lot more difficult. The SAT Reasoning Test or ACT (plus writing) are required as well as the SAT Subject Tests. 70% of Scholarships/Grants are awarded to students as well.
2. Massachusetts Institute of Technology: MIT is located in Cambridge Massachusetts and only 12% of applicants are admitted into the school. This makes it difficult to get in because there is a lot of competition to attend. It offers a Bachelor's, Master's, and Doctoral degree and it is a Private University. It is located in a small city with an urban setting and 97% of 1st year students are in the top 10th of graduating class.
3. University of Southern California: USC is a private university in a very large city in an Urban setting. It costs about $39,274 to attend but 77% of Financial Aid/ Scholarships are awarded to undergraduates and 23% as loans/jobs. Only 22% of applicants are admitted.

Im still looking around and investigating,but these 3 colleges caught my attention. Im hoping to find other choices as well. As for the majors, these are just some I have liked so far, but there are THOUSANDS out there that I still need to look into. Everyday I have new likes and dislikes, and my choices in majors and colleges may even be different by tomorrow :]

Tips and Hints 8]

1. Share how you remember transformations. Do you have any tips or hints that help you to remember?

In order to remember transformations, I first have to memorize what occurs when i change the parent function.

• If I add or subtract a number to the x, then i know that the graph should go either left or right. For example, f(x)=sin(x+1)...Moves ONE unit to the LEFT. I would instinctly believe that the graph would move one unit to the right because of the addition sign, but in reality, it has to move to the left. If you have f(x)=sin(x-1)... then the graph moves ONE unit to the RIGHT (NOT TO THE LEFT!). In order to remember it, I just remember that it moves OPPOSITE to the addition/subtraction sign.


• If I multiply x by a number, then I know that the graph has to shrink horizontally. For example, if I have f(x)=sin2x... then it looks like the graph goes (2x) "faster", compared to its parent function (sinx). It will also compress horizontally because the PERIOD changes to pi instead of the original 2pi. (TO FIND THE PERIOD, REMEMBER THAT YOU DIVIDE THE PERIOD OF THE GRAPH BY THE NUMBER IN FRONT OF THE X). If the number in front of the x is a fraction, then the graph will stretch horizontally instead, since the PERIOD will increase.
  

• If the output has been multiplied by a number, f(x)=2sinx, then the graph stretches vertically. The period still stays the same (if the function has a period). The graph will not be any "faster". If the number in front of the equation is a fraction, then the graph will compress vertically.
• If the number in front of the equation is negative, you just flip the graph. The output has been made negative. You reflect the parent graph across the x axis.

EX. f(x)= -Sinx

• If a number is added to the equation, such as f(x)=sinx+1 (without parenthesis), then the graph will shift up(in this case, up one unit). If a number is subtracted, f(x)=sinx-1, then the graph will shift down (in this case, down one unit).

The only tip I can give you is to MEMORIZE all this information and remembering how the graph will shift, shrink, or compress depending on how the parent function is manipulated.

2. Share how you remember or understand trigonometry. Do you have any tips or hints that help you remember/memorize all those facts?

To understand trigonometry, you have to know that the coordinates on the unit circle are not just random numbers that were made up. They actually MEAN something. Take for example, pi/6. The angle is 60 degrees, since 180 degrees (half of the unit circle) divided by 3 is 30 degrees.

The coordinates are (sqrt of 3/2, 1/2). The x axis is sqrt of 3, divided by the hypotenuse 2, and you get your x coordinate. Same thing for the y coordinate.

You can also remember this by keeping in mind that the longer side of the triangle is sqrt of 3/2 and that the shorter side is 1/2.

For pi/4, just remember that the sides of the triangle are the same (excluding the hypotenuse) and therefore the x and y coordinates are the same. (sqrt 2/2, sqrt 2/2).

The way I remember which coordinate is which for pi/6 and pi/3, (and any other coordinate at "?/6"&"?/3")...

FOR pi/3, I take into consideration that the "/3" comes second, and therefore the 2nd coordinate (y coordinate) has to be sqrt 3/2... (1/2, sqrt 3/2).They will both come SECOND because they include the number "3". Since pi/6 does not have the 3 at the bottom of the fraction like "/3"(the 3 doesn't come second), then the sqrt of 3/2 cannot be the 2nd coordinate (it cannot come second). So the coordinates for pi/6 is (sqrt 3/2, 1/2). The "3" comes FIRST.

This makes sense in my mind, and I tried my best to explain it. Usually people do not understand what i'm talking about.. haha

I think that you should just have the unit circle MEMORIZED and you should only use the tips if you FORGET some of the coordinates.

TO GRAPH: All I do is remember the main coordinates of the Sin, Cos, Tan graphs (What y is at 0, pi/2, pi, 3pi/2, & 2pi). Then I just continue the graph over again since its a new period and looks the same (For a parent graph). For Tan though, I have to memorize where the asymptotes are at as well. For Csc and Sec, I just graph the sin or cos graphs (sin for csc and cos for sec) and at the top or bottom of the curve is where i draw the parabolas. The asymptotes are where the sin or cos graphs cross the x axis. To remember Cot graphs, I just memorized that the asymptotes are at pi and that the "middle" point is at pi/2. Also i remember that the Cot graphs are the tan graphs flipped horizontally. When transformations are involved, I just follow the rules of transformations that I mentioned above.

3. What still confuses you or worries you about trigonometry?

Well when you shift a graph up, it is difficult to find the zeros of the graph. For example, if the regular graph would have the x intercept at pi/4, etc, and you shift the graph up, I just know that the period will be BETWEEN pi/4 and pi/3. I don't know how to find the exact x intercept without a calculator.

Logs and Inverses :D

1. Recap 4 major concepts that you have understood about either/both of these two topics.

1. Relating to inverses, I believe that I understood the concept of one-to-one rather clearly. I learned that in order for a function and its inverse to be "one-to-one", the graph of the function must pass the horizontal line test, which means that if a horizontal line is passed over the graph of a function, the horizontal line intersects the graph at one point at a time. This horizontal line test determines if the inverse exists as a function. If the graph of the function passes the horizontal line test, then the function is "one-to-one".
2. I understand that the inverses of a parent function are symmetrical to the parent function itself. If you fold the paper between both lines, such as folding the paper of the function f(x)=3 and f^-1(x) at the line y=x, the lines will lay right on top of each other.
3. I also understand how to find the inverse (f^-1(x)) of a function (f(x)) and how to verify that (f o f^-1)(x)=(f^-1 o f)(x) = x. When you multiply the function by its inverse, you should get "x", and vise versa. In order to demonstrate how to find the inverse of a function, I will use the function f(x)=x^2 (x< (or equal to) 0). This function is also equal to y=x^2. First you change x to y, and y to x in the equation. You get x=y^2. The goal is to leave y alone, so you take the square root of both sides of the equation and end up getting sq root of x=y, also known as f^-1(x)=sq root of x. --- To verify that it is the inverse, you plug in sq root of x into the original function and get (f(x)=sq root of x^2). When you solve it by cancelling out the sq root and the ^2, you get f(x)=x. Also if you input x^2 into the inverse function, you get (f^-1(x)=sq root of x^2) also, which is equal to f^-1(x)=x. This proves that f^-1(x)=sq root of x.
4. Today in class (Thursday), as we were solving logarithm equations for x, I believe that I understood how to solve most of them easily. For example, when solving the equation "7-3e^-x=2, when I got to the point of "e^-x=5/3", I knew that I was supposed to take the natural log of both sides so "e" could cancel out and so I could be left with "-x=ln5/3"--> "x= -ln5/3". Even though sometimes I may get confused or forget how to do a step, I know that with a little more practice, I will grasp the concept more fully.

2. Write about what you did NOT understand completely.

• Relating to Logarithms, I can honestly say that I am not so sure how to graph them. I know that their inverses are functions, but I still cannot grasp the concept of graphing them without using a graphing calculator, unless I input the values for x to get the y output. I do not know how to get the inverse function of a logarithm function that is complicated. Sometimes the log functions are kind of complex and.. yeah I need help. :D
• Even though I understand, in general, how to solve the logarithm functions for x, most of the problems on the homework C2 were a little more tricky and I did not understand them. For example, number 35 (e^x+e^-x=3) may look simple at first, but once I tried to solve it, my answer looked nothing like the one in the back of the book. This was "mind-boggling"! I tried to take the ln of both sides to cancel out the e's but I ended up getting "x-x=ln3" --> "0=ln3". Im not sure how to solve this or anything in relation to that...

Even and Odd Functions :]

EVEN Functions

An even function can be defined as f(-x)=f(x) algebraically.

• This means that for every input value of x and its opposite value (x and -x), the output will be the same (y).



• Therefore (x,y) and (-x,y), or (x,-y) and (-x,-y) will be reflections of each other about the y axis if plotted on the same graph.



• Because of this, the graph of an even function will always be symmetrical about about the y axis. (Quadrant 1&2, or Quadrant 3&4).



• Any function with an even power (x^2, x^4, x^6, etc.) or with an absolute value will have these characteristics.

Example: All 3 graphs are symmetrical about the y axis.

In the first graph, x^2, If 1 and -1 (x and -x) are inputted into the equation, the outputs (y) will both be 1.

If 5 and -5 are inputted, the outputs will both be 25.

f(-x)=f(x)

f(-5)=f(5); x= -5, 5

25=25; y=y

ODD Functions

An odd function can be defined as f(-x)= -f(x) algebraically.

• If the point (x,y) or (-x,y) is plotted on a graph, (-x,-y) or (x,-y) must also be a point on the graph, respectively, in order for it to be an odd function.



• The graph of an odd function will always be symmetrical about the origin.



• Symmetry may be on Quadrant 1&3, or Quadrant 2&4 in most, but not all, cases.



• Any function with an odd power (x, x^3, x^5, x^7, etc.) will have these characterisitcs.

Example: All 3 graphs are symmetrical about the origin.

In the first graph, x, if 1 and -1 (x and -x) are inputted, the outputs will be 1 and -1 (y and -y) respectively.

If 5 and -5 are inputted, the outputs will be 5 and -5 respectively.