Friday, September 29, 2006

2-3: Sine and Cosine Functions

Ok everyone, I was a little confused by this lesson, but I think I understand it now... Let's all cross our fingers here.

Periodic functions: a function whose values repeat at regular intervals. This is an example of a periodic function. The definition of a periodic function is: The function f is a periodic function of x if and only if there is a number p for which f(x-p) = f(x) for all values of x in the domain.

A cycle on the periodic function is the actual part of the function that repeats (so on this example the loop-de-loop)
A period on the periodic function is the horizontal distance between the cycles. (so the distance between the highest points on two of the "ups" of this graph) (if that made any sense)

Sine and Cosine functions are both periodic functions.
This is an image of both the sine and cosine functions. The sine is red and the cosine is blue!

Sine and cosine can also be used in right triangles. We use SOH CAH TOA to remember how to find the sine, cosine or tangent. (SOH = sine: opposite/hypotenuse; CAH = cosine: adjacent/hypotenuse; TOA = tangent: opposite/adjacent).
But now we're using it on our UV coordinate plane. So if we were to try to find the sine of 120 degrees, we would first draw that on our coordinate plane, like so:
The definitions of the sine and cosine of any angle (in this case 120) are : Let (u,v) be a point r units from the origin on the terminal side of a rotating ray. If x is the angle in standard position to the ray, then sinx=v/r and cosx=u/r. Sinx will be positive in quadrants I and II, but it will be negative in quadrants III and IV (v is positive in 1st and 2nd quadrants and negative in the 3rd and 4th quadrants, divided by r which is always positive). Cosx will be the opposite. It will be negative in quadrants I and II and negative in quadrants III and IV. (u is negative in the 1st and 2nd quadrants and positive in the 3rd and 4th quadrants/r which is always positive)
As the ray goes around the coordinate plane, the value of r is changing. At v=0, sinx=0 and at v=90, where r is equal to v , sinx=1. Also, sin180=0 and sin270=-1. As the ray circles the coordinate plane, the values are repeated, creating the periodic function.

Example problem: Draw a 147 degree angle in standard position. Mark the reference angle and find its measure.
Then find cos147 and cosx(ref). Explain the relationship between the two cosine values. x = 180 - 147 = 33 degrees equals the reference angle measure. Cos33 = 0.838 and cos147 = -0.838. Both of the cosine values are the same, except cos147 is negative. This is because u/r is a negative value (the x value in the 2nd quadrant) over a positive value (r which is always positive).

For additional help go to: http://home.alltel.net/okrebs/page72.html

Amanda!!!!!!!! it's your turn next!!!!!!!!

If you wanna see something totally awesome go to http://www.youtube.com/
then in the search box type "i'm glad I hitched my apple wagon to your star". This music video is by CLAIRE!

If that doesn't work, here's a "magic eye" illusion:





Thursday, September 28, 2006

2-2 Measurement of Rotation


Cartesian plane












-Angles in the cartesian plane can continue past 360
-Clockwise rotations are measured with positive degrees
-Counterclockwise rotations are measured with negative degrees

The same position on a cartesian plane has more than one way of looking at it. You can measure it normally, measure it counterclockwise to get a negative degree, or you can go multiple times around the circle.
















Coterminal angles have the same side.
angles 135 and -225 are coterminal

A reference angle is the smaller angle make with the x axis (always measured counterclockwise).
180-135=45
45 is the reference angle


example:

What is the measure of the reference angle of the angle 5328?
5328/360=14.8
5328 makes 14 whole revolutions plus 0.8 revolutions
0.8(360)=288
288 is the number of degrees left after 14 whole revolutions
To find the reference angle 360-288=72
The reference angle is 72 degrees

look here for more help
http://www.analyzemath.com/Angle/reference_angle.html


natalie you're next, don't forgeeeeeeeeeetttttttttttt!!



look for the man in the coffee beans








Apparently, if you can find the man within 3 seconds, your right brain is functioning better than most. It's functioning normally if between 3 seconds and one minute. You need more protein if it takes from 1 to 3 minutes, and your right brain is an absolute mess if it takes longer. The only advice at that point is to do more of these types of exercises in order to strengthen the right side of the brain.


....yeah that guy's kind of creepy...

Tuesday, September 26, 2006

Quiz 1.4-5 Topics


Here’s a list of topics for Wednesday’s quiz:

Precalculus Quiz 1.4-5 Topics
Domain and Range of functions
Calculate composite values from a graph
Plot composite values
Determine domain and range from a graph
Inverse functions – relationships of x and y
Sketching inverse functions given image of original function
Visual proof of functions and inverse functions
Determining domains of composite functions algebraically
Plotting inverse functions with calculator (function mode)
Plotting inverse functions with calculator (parametric mode)
Determining equations for composite functions algebraically
Plotting functions parametrically

I should be online tonight after 9pm for any last minute questions, and in my classroom tomorrow morning...

Good luck studying! If you need a break, check out these "criminals."

Sunday, September 24, 2006

Section 1-6: Reflections, Absolute Values, and Other Transformations

Section 1-6: Reflections, Absolute Values, and Other Transformations

Yay! Section 6!

Main Concepts:

Reflections Across the Coordinate Axes:
- g(x)=-f(x) is a vertical reflection of function across the x-axis.
- g(x)=f(-x) is a horizontal reflection of across the y-axis.


Absolute Value Transformations:
The transformation
g(x) = f(x) (The absolute value bars should be around f(x) but no matter what I do, they will not save on the blog).
- Reflects f across the x-axis if f(x) is negative. (negative values become positive) No y-values will be below the x-axis.
- Leaves f unchanged if f(x) is nonnegative. (positive values stay the same)
The transformation g(x) =f(x) (The absolute value bars should be around the x of f(x) but they will not save on the blog, p.s. please tell me how to do it if you figure it out).
- Leaves f unchanged for nonnegative values of x.
- Reflects the part of the graph for positive values of x to the corresponding negative values of x.
- Eliminates the part of f for negative values of x.

An example of this:

If you have the equation y=x^3-2 and you have y(2), you would get y(2) = 2^3 -2 which equals 6. If you had y(-2), would would get y(-2) = -2^3 -2 which equals -10.

Now if you made the equation y = x(with absolute value bars around it)^3 -2, for y(2), you would still get 6. For y(-2), things would be different. The absolute value of -2 is 2, so 2^3 is 8 and 8-2 equals 6. This exemplifies the fact that it does not matter if you put in 2 or -2, you will get the same y value if it is positive or negative.

Even and Odd Functions:
Even Functions:
- f(x)=f(-x)
- Symmetric on y-axis
Odd Function:
- f(x)=-f(-x)
- When a graph is rotated 180 degrees, you end up with the same picture. - Symmetric around the origin.


Example Problem:

Reflection across the x-axis:

f (x) = x^2 - 9 is our original equation. We will write an equation for the reflection of this pre-image function across the x-axis.

g(x) = - f (x) will give a vertical reflection across the x-axis, so
y1(x) = -f (x)

y1(x) = -f (x) = -(x^2-9) = -x^2 + 9


Additional resources for further study of this section:

http://libraryofmath.com/Precalculus.html#loc
—scroll down until you get to the heading “Graphs and Transformation of Functions”

Also see the site

http://cda.morris.umn.edu/~mcquarrb/Precalculus/Resources/Lecture1.5.pdf


Reminder: Debby! You are responsible for the next blog entry.




Sunday, September 10, 2006

Blog Postings


As a reminder, the requirements for each blog posting will consist of:
  • A review of the main point of the class lecture/demonstration.  This summary should highlight any relevant formulas and/or graphs and communicate your interpretation of the concept covered in class. (15 pts)  For this part of the posting, I am looking for quality, not quantity.

  • An example problem, including a statement of the problem, the answer, and the solution method.  (For your first post, using an example covered in class is acceptable, for additional postings, original examples will be required.) (10 pts)

  • A link to an additional Internet resource supporting the Topic of The Day. (5 pts)

  • A reminder to the next BlogMaster of their responsibility to post. (5 pts)

  • A “personalization” of your posting.  This personalization can be a comment about the day’s class, an image, a quotation, a question posed for discussion, a joke, or something else that reflects you as a student.  These personalizations must be in good taste!  (5 pts)

In addition to your posting, you will be expected to comment on a minimum of two (2) of your classmates postings during each quarter.  These comments must either further enhance your classmates’ understanding of the posted topic or further a discussion question posed in the original posting.

Additional Notes:
  • Postings will be due within 24 hours of class.  I will post a schedule of class scribes for the first quarter once everyone has joined the class blog.

  • For help with posting equations and graphs, please feel free to come ask me for assistance.

  • Initially, the blogs will be hosted on blogger.com.  As the year progresses, we hope to migrate to an internal website.

  • While we are on blogger.com, there is some software available through the website that allows creation/editing of posts via Microsoft Word.

Tuesday, September 05, 2006

Blog Policies

There are some things I want you to remember about blogging. Many of things have been discussed by other teachers and classes, so I will paraphrase them here and try to give them proper credit:

First of all, our class will not be the only people to view our postings. The Internet is accessible almost everywhere these days, and even if a post is deleted, there’s no guarantee that the posting hasn’t been copied and propagated to other sites or linked to from those sites. This has a couple of implications:

First, privacy. We will only be using first names on the site. If I post pictures or video, no one will be identified, other than “Mr. French’s class”. Do not use pictures of yourself for your profile here. If you want a graphic image associated with your profile, use an “avatar” – a picture of something that represents you but is not you. Here’s a link to a fun image creator.

Second, etiquette, appearance and common sense. Bud the Teacher has these suggestions, among others:

  1. Students using blogs are expected to treat blogspaces as classroom spaces. Speech that is inappropriate for class is not appropriate for our blog. While we encourage you to engage in debate and conversation with other bloggers, we also expect that you will conduct yourself in a manner reflective of a representative of this school.

  2. Never EVER EVER give out or record personal information on our blog. Our blog exists as a public space on the Internet. Don’t share anything that you don’t want the world to know. For your safety, be careful what you say, too. Don’t give out your phone number or home address. This is particularly important to remember if you have a personal online journal or blog elsewhere.

  3. Again, your blog is a public space. And if you put it on the Internet, odds are really good that it will stay on the Internet. Always. That means ten years from now when you are looking for a job, it might be possible for an employer to discover some really hateful and immature things you said when you were younger and more prone to foolish things. Be sure that anything you write you are proud of. It can come back to haunt you if you don’t.

  4. Never link to something you haven’t read. While it isn’t your job to police the Internet, when you link to something, you should make sure it is something that you really want to be associated with. If a link contains material that might be creepy or make some people uncomfortable, you should probably try a different source.
Are there other considerations we should take into account? Use the comment feature to add any others or to clarify/expand on one of the above.

Monday, September 04, 2006

Welcome!


Congratulations! You found our class blog! This is where we as a team will hopefully create a resource to help us conquer any issues that arise during our class this year. This is the place to talk about what’s happening in class; to ask a question you didn’t get to ask in class; to share your knowledge with fellow classmates and any other Internet users who choose to read our notes;…and most importantly it’s a place to reflect on what we’re learning.

A large part of retaining knowledge requires reviewing and discussing new information on a regular basis. This blog is intended to help each of you do just that. Between creating your own posts and commenting on your classmates’ posts, you will have the opportunity to explore each of the topics we cover this year in greater depth. I hope you will use this forum to help yourself and your classmates in whatever ways you can think of.

Blogging Prompt

Occasionally I will include a posting of my own, either to clarify a concept or to generate some further discussion. These postings will have a title similar to the one above this paragraph.

To get things rolling, here’s a question for you to think about and respond: Is God a mathematician? Why or why not?

Don’t forget to email me with the information I requested in class so I can include you on the team!