Life after high school

There are many majors that I found a lot of interest in but the ones I found the most interesting were in the region of Biological and Biomedical Sciences and also in the region of engineering. I have not really decided what I want to be but know it is between a surgeon and an engineer.

The first major I am interested in is Biomedical Sciences which is basically biology but more in depth. It also deals with how it relates to the body and health, medicine, and disease. People who study this major research for new ways of treating people with various diseases and also analyze technical procedures to improve human human health.

The second major I am interested in is Biological engineering which deals with knowing how to incorporate engineering to helping solve medical problems. People in this field look for new ways to improve machinery to aid in diagnosing and treating certain diseases. I am very interested in majoring in this area particularly because it deals with engineering and medical issues.

The third major I am interested in is Applied Mathematics because it is a more in depth area of mathematics and requires a lot of thinking and reasoning. People in this field use math and statistics to help solve problems in engineering. I am pretty strong in Math so I think I would be very successful in this field.

The first college I am interested in UC Los Angeles because they offer various majors in the area of biology and health. It is also a top university and has a very good medical school. The university is also highly ranked because it has a lot of resources.

The second college that I am interested in is UC Berkely which offers various majors in the area of engineering. It is a large university in a small setting and is very well known around the nation as a top university. It has a lot of facilities in engineering and is popular in giving degrees in the engineering area.

The third college I am interested in is California Institute of Technology. It is a top university and is extremely competitive but offers many degrees in the area of math. Admission is very low because they only accept a small number of students but the university offers many resources and aid.

Tips and Hints

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

I picked up transformations fairly quickly by just seeing how functions shifted and changed. The hardest part for me in learning transformations was to memorize how the parent function looked like. I remember parent functions by picturing a sketch of it on my head for example how it looks like without the axis whether it is curvy or straight. I then just memorize three points one would be a positive value for x a negative value and zero. I know that most functions have some type of particular pattern and would just use that to find other points in the graph. For example I remember the function f(x)=cosx by knowing that it is wavy and the function intersects points

(-pie/2,0), (0,1), (pie/2), and (pie,-1). I then just finish the graph of by seeing the wavy pattern and finding the rest of the points. In terms of knowing how a graph shifts whether it is up, down , side to side I just see in the equation what is being changed whether it is the input or output. I know that if the input is being changed by adding or subtracting the function shifts side to side for the x values and if it is multiplied by negative 1 it flips from side to side,etc. In terms of remembering what happens to the graph when the function is being change I just remember that if you change the input the x values are being affected and the same for the output.

2. Trigonometry is a lot more simple if you remember the basis of it which is the unit circle. To me knowing the unit circle makes trigonometry a lot more simple. I memorized the unit circle by memorizing the values in the first quadrant and knowing how it changes from quadrant to quadrant. For example the tip Ms.Hwang gave us helped me a lot in finding the radian values for the 2nd, 3rd, and 4th quadrants for example in the second quadrant the value is one less the denominator, for the second one more the denominator, and for the fourth two times the denominator subtracted by one.

3. I Still struggle in trying to remember how the graphs for the inverse functions look like and how their domains change. I also sometimes still get confused on whether what changes should i do to a graph when the function is changed.

Logs and Inverses

A major concept that I understood was that a logarithm is the inverse of an exponential function, for example the function 2^x =8 To find the value of x you would need to do the inverse of an exponential function to find the value of the exponent(x). It is basically the same thing as when you are trying to find the value x in the equation x+2=5, To get x by itself you would do the inverse of adding 2 which is subtracting. Now, back to logs in trying to solve the equation 2^x=8 you would take the inverse which is the log base 2 (2^x)=log base 2(8). You would then get x =log base 2 (8),you can simply this more by knowing that 8 is the same thing as 2^3 which would make it log base 2 (2^3) which would make it 3.



Another major concept I understood was the rules of exponents and logarithms for example in logs- log (XY)=log (X) + log (y) , log (X^5)=5log(x), log (X/Y)=log (X)-log(Y). these rules allow you to write and manipulate logs in many different forms. For example you can manipulate log (10) into log (5)+log (2)=log (5x2)=log (10). These rules are very helpful when trying to find a value in an exponential equation or in a logarithm.



A very important concept that I understood about inverses and functions is that f(f^-1(X))=x and f^-1(f(x))=x. This is very helpful in checking whether or not an inverse function is the inverse of a function. I also understood that a function has to pass the vertical line test which states that if a vertical line was passed through a graphed function it should only intersect the function at one point as seen in the image.
The domain of a function would be what the possible x-values for x can be when inserting into a function or graphically. The domain will give you a boundary about what the inputs(x) can be. The range of a function is the possible y-values of a function meaning a boundary of what the outputs(y) will be.

A function that is one-to-one is any function that passes the horizontal and vertical line test which means it is any function that has an inverse. Passing a horizontal line test on a function is the same thing as passing a vertical line test on its inverse function. Graphically when a function is graphed its inverse will be symmetrical to line y=x.

I had some trouble in trying to figure out an answer to a math problem we got in our homework pg 44 number 35 & 36. I am not really sure whether there are different rules when solving an exponential function that has addition on it.

EVEN AND ODD FUNCTIONS

EVEN FUNCTION

The formula for an even function is defined as f(-x)=f(x) for a function of y=f(x). Graphically an even function would be symmetric about the y- axis as seen in the image.

Since y=f(x) and according to the formula for an even function f(-x)=f(x) then for an even function y=f(-x). For an even function it does not matter whether the input for x is negative or positive because the outcome (y) be the same, which is why there is symmetry about the y-axis. As seen in the example for an even function f(x)=x^2 the input can be negative or positive but the outcome will be the same. f(2)=2^2=4 , f(-2)=(-2)^2=4.

ODD FUNCTION

The formula for an odd function is f(-x)=-f(x) for a function of y=f(x). An Odd function if graphed will have symmetry about the origin as seen in the image. The formula basically says that if a negative value is imputed into a function the outcome(y) will be the negative value of the function for a positive value f(-x)=-y. For example in the odd function f(x)=x^3 if -2 is imputed the result would be f(-2)=(-2)^3, which is f(-2)=-8.

If a positive value is imputed the outcome will be positive. From the formula we can see this

f(-x)=-f(x)in seeing what the value for a positive number would be -f(-x)=f(x),-(-y)=f(x), y=f(x).

For example in the odd function f(x)=x^3 if two is imputed the value would be f(2)=2^3,

f(2)=8