Blogging Tools

Graphs of Functions:If you want to include graphs in your blogs, go to http://www.walterzorn.com/grapher/grapher_e.htm . Here, you can enter the function you want to graph, and click on print preview on the menu to the right. You should be able to copy the image with the Print Screen key on the top right of your keyboard (It should be above the insert key). Go to paint or any other image progam, paste the graph and and save the it as a JPG image, and you're ready to upload. The grapher in the link above allows you to graph several functions at the same time, and adjust the graph window.

Uses of Logarithms

Logarithms are mostly used in calculus and physics.

In organic chemistry, in twelfth grade, you will use logarithms to find pH. In physics, noise, earthquakes, star brightness, sensation, etc, are measured using logarithmic scales. Computers also use logarithms.

All of the scales mentioned above, however, use base-10 logarithms. The most common logarithm you will come across that is not base-10 is the natural log or ln, which is base-e. Ln will be used a lot when integrating, but you don’t have to worry about that until your second semester senior year.

The only time I’ve used logarithms based some other number different than 10 or e, was when looking for the inverse of functions. For instance, the inverse of 3^x is log base-3.

Still, there are different types of logarithms used in science and engineering. There even are imaginary-based logarithms, but I doubt any of us will encounter those in the years to come.

Reflecting Functions

Reflections are quite simple. When you reflect something, you are basically flipping it across a given line. There are many types of reflections, depending across the line you are reflecting across of. However, there are two types of basic reflections: about the x-axis, and about the y-axis.

Reflecting about the x-axis means you are rotating the function across the x-axis. This is done by multiplying the function by -1, so that you are left with –f(x).
For example,
The reflection of f(x)=x/2-3 will be the same as multiplying f(x)=x/2-3 by -1, which is –f(x)=-x/2+3
Graphically, it would look like this:

f(x)=x/2-3

–f(x)=-x/2+3Another example:

f(x)=x^2-3 -f(x)=-(x^2-3)

To understand this type of reflection better, think of it the following way. By reflecting about the x-axis, you are multiplying each value of y by -1. That means that the reflection about the x-axis of the point (1,1) would be (1,-1). Using variables, this is the same as saying that the reflection about the x-axis of the point (x,y) is (x,-y). You can prove this using the graph above. Each value of y gets multiplied by -1 while x stays the same. Therefore, if the original function is f(x), the reflected function about the x-axis will be -f(x)

Reflecting about the y-axis means you are rotating the function across the y-axis. This is done by replacing x by -x, so that you are left with f(-x).
For example,
The reflection of f(x)=x/2-3 will be the same as solving for f(-x), which is f(-x)=-x/2-3
Graphically, it would look like this:

f(x)=x/2-3

f(-x)=-x/2-3

Another example:

f(x)=(x-3)^2 f(-x)=(-x-3)^2

To understand this type of reflection better, think of it the following way. By reflecting about the y-axis, you are multiplying each value of x by -1. That means that the reflection about the y-axis of the point (3,0) would be (-3,0). Using variables, this is the same as saying that the reflection about the y-axis of the point (x,y) is (-x,y). You can prove this using the graph above. Each value of x gets multiplied by -1 while y stays the same. Therefore, if the original function is f(x), the reflected function about the y-axis will be f(-x)

Review:
If f(x) is a function, then
-f(x) will give you its reflection about the x-axis, and
f(-x) will give you its reflection about the y-axis

The following link contains other examples: http://www.themathpage.com/aPreCalc/reflections.htm

If you would like some help on other types of reflections (about the line y=x, y=-x, etc), please let me know.

Factorizing Parabolas

Factoring parabolas comes especially useful later on in Calculus. Right now, factoring parabolas is used, primarily, to look for the solutions of the parabola, or where the parabola intercepts the x-axis. There are certain word problems that will require you to look for these values. However, this process will become more useful in Calculus when you learn optimization. Factorization, in general, is extremely helpful in calculus in order to simplify problems.

Trig Functions in the Real Life

There are many concepts you learn during Pre-Calculus that you will not know what they are used for at first. Pre-Calculus is taught so that you learn different concepts that will be useful later on in Calculus and Physics.

Trig functions, like any other math concept, are used to explain the world around us. I don’t know if Mr Alcantara gave you a problem in which you had to find the distance, with respect to time, that a toy train going around a circular track was from its starting point, or another problem in which you had to find the distance, with respect to time, Mr Alcantara was from Julian while he swung him on a swing. This was all done using trig functions.

Trig functions are used in physics to describe sound, frequency, vibrations, waves, tides, light, pendulums, and more. Anything that has a repeated pattern of motion can be described using a trigonometric function. Also, they are used to find coefficients of friction between two bodies. Trig functions are also important in geometry when working with triangles. Economists also use modified trigonometric functions to describe how different variables like offer and demand, behave in a given market. Trigonometric functions are used in all fields of study.

The following link has a video of a funny looking professor giving some examples of how trig functions are applied to real life. I didn’t watch it all but you should check it out:
http://www.coolschool.ca/lor/PMA12/unit4/U04L05.htm