Precal Blog

So today we were scheduled to have a guest speaker from MIT come to lecture us about matrices. Unfortunately, he couldn’t make it. I guess there was a flight delay during one of the changeovers and the airline had the only other flights completely booked or something. The man was kind enough to lecture us via webcam though. He basically taught us about matrices in equations. So far we haven’t put matrices to much use yet. He started out by making up two equations with two variables: 2x-y=0 and -x+2y=3. We can then put these equations into a matrix equation.

2x-y=0 | 2 -1| |x| = |0|
-x+2y=3 |-1 2| |y| |3|

After doing this, he did some crazy thing with vectors that I did not quite understand. I think he just showed that both equations will intersect at the same point, the answer, when you solve the equations. This one was pretty easy. All you have to do is multiply the y by 2. This will get the first equation to be 2-2=0 and the second equation to be -1+4=3.

For the second part of the video, the man showed us a 3-variable equation. He again made up a set of equations:

2x – y =0
-x+2y – z=-1
-3y+4z=4

Again, this is made into a matrix equation, except this time, x,y, and z are used:

|2 -1 0| |x| | 0 |
|-1 2 -1| |y| = |-1|
|1 3 4| |z| |4 |

So for this equation, the solution was rather simple. Since the z values were the same as the “b” value (the right side of the equation), all we have to do is take 0 of x and 0 of y to get just the z values. So that’s 0x+0y+z.

So there you have it. That was pretty much the gist of the speaker’s lecture. Don’t forget that the homework was 11-2, 1-12. The claims haven’t been announced yet, but there will be 4 of them, so make sure to do them all because we’re presenting tomorrow!

The next scribe will be Air Blair.

Hello everyone. I hope this finds you all on a nice relaxing fun vacation. I am soooo glad that i finally was picked to scribe on an easy day!!

On friday, we got to class and watched the office re-runs while we ate snacks and counted down the final minutes untill we got out for winter vacation. ( which has been so nice by the way!!! I just wish that it could be longer however because i barly had any time to relax with all the hustle and bustle of the holidays.) Anyways, Remember to start to work on your semester portfolios!!!!! and to study so you remember what we were doing in precalculus!!! Just what you wanted to hear right…. REMEMBER TO STUDY MATH OVER BREAK!!

Have a fun last couple of days off!
Katelyn

O yeah, the next scribe will be Kendall

Today we started off class by looking at section 11-1 on our own, then we discussed it with Mr. Bienik. We talked about what happens to a two-dimmensional figure when you perform the same transformation over and over. This process is called iteration. We also talked about fractals. These are generally “rough or fragmented geometric shape that can be subdivided into parts, each of which are (at least approximately) reduced-sized copies of the whole.”

The rest of class we reviewed matrices. We reviewed this so we create fractals, and later even translate, dialate, and rotate matrices. So, first we reviewed what they are.

-A matrix is a way to organize data, kind of like a chart.

-They are labeled with capital letters. (A,B,G…)

-m is used to label the rows. (which go horizantally)

-n is used to label the columns. (which go up and down)

-m and n are the dimensions of a matrix making the order m* n

-The numbers in the matrix are called the elements, and are written `a` with subscripts `i,j` where `i` is the row and j is the column in which `a` appears.

Adding matrices:

`((4,5,0),(3,3,3),(2,3,4),(1,3,5))+((3,3,3),(4,3,2),(5,4,0,),(2,3,4))=((7,8,3),(7,6,5),(7,7,4),(3,6,9))`

It is simple you just add the elements that are in the same row and column, and the sum is in that same spot in the answer. Subtracting is the same way. Also, you need the same number of rows and columns to add and subtract otherwise it wouldn’t work.

Next, multiplying a matrix by a number, which we will call a scalar, works like multiplication always does: repeated addition. So,

`4((-5,2,0),(3,7,1))=((,20,8,0,),(12,28,4))`

Finally, we were given a workheet to do, and look at tomorrow in class.

Next scibe will be KatelynW

So I have absolutley no idea what I am supposed to post for the last two days …sorry about that there was alil communication problem…:)and I didnt know that I was scribe ..oops

Anyways…thursday we of course had the test and I’m pretty sure that the majority of us know how that went…and want to forget about it. Friday we had Ms. Hill as a sub. and worked on our review packets all hour…

REMINDER: Ms. Hill said that the review packets are due monday.

oh and a quick question…are we supposed to know how to do the first ones with out our calculator? I’m still a little confused on the decimals that we are supposed to have memorized for the semester exam.

Knowing that everybody is completely stressed about this upcoming week (and knowing how many learning targets I already have to take) my advice for this class is to figure out exactly what you are going to have to take on the final exam and start going over that now so you know what you are going to have to ask Mr. B on monday.

so I have to get back to my Quarter Journal and Semester Portfolio… which I know you are probably all doing right now too….soooo good luck.

The next scribe is going to be Rose

Just wondering, am I the only one that failed hardcore on the test today? I’m predicting something like a 0-1-8 for my score…

So, today I have the extremely difficult job of blogging what we did in class today which was study for our end of quarter two test, tomorrow.

END OF QUARTER TWO LEARNING TARGETS
28. I can transform trigonometric expressions and prove trigonometric identities.
for this learning target the main thing you need to know is your pythagorean, quotient and reciprocal properties.
the pythagorean properties are: sin^2 (x) + cos^2 (x) = 1 : tan^2 (x) + 1 = sec^2 (x) : 1 + cot^2 (x) = csc^2 (x).
the quotient properties are: csc x = 1/sin x and sec x = 1/cos x.
the reciprocal properties are: tan x = sin x/cos x and cot x = cos x/sin x

29. I know the range and domain of the six inverse trigonometric functions.
we received a bunch of worksheets on this learning target, they’re white and one has a large chart which lists everything.

30. I can sketch an accurate graph of the inverse of any of the six trigonometric functions.
this learning target goes hand in hand with the learning target above and these are also on the white sheets we received from mr. b.

31. I can find exact values of inverse trigonometric functions.
for example: sin .5 is 30 degrees : sin 30 degrees is .5 :

32. I understand the composition of a trigonometric function with its inverse.
the example mr. b gave us is : tan (tan^-1 x )= x, x is any real number : tan^-1(tan x)= x, x is (-3.14/2, 3.14/2)

33. I can derive and apply the composite argument properties for sine.
34. I can apply the composite argument properties for cosine.
35. I can derive and apply the double angle properties for sine and cosine.
these learning targets go together. we have about five pages worth of notes in our notebooks and spent a few days on this in class.

36. I understand the relationship between a polar graph and its auxiliary Cartesian graph.
we just did this in class. we have plenty of polar graphs with examples on it.

Good Luck tomorrow everyone!
Tomorrow’s scribe is Meghan. Have fun with that hard blog.

Today continued graphing on the polar grid. the first equation that we graphed was r=2+2cosx. So just like before we started by graphing the auxiliary graph. That would look like this-

Then we start plotting on our polar grid. We knew that the graph went from 2 to 4 to 2 between -90 and 90. SO on the polar graph we plotted (2,90) (2,-90) and (4,0) Then we picked values for theta to find some more points. it theta= 30 the r value was 3.75. You can plot (3.75, 30) and becuase the symmetry for cosine is over the x-axiz you could also plot (3.75,-30). The we did 60, when you plugged 60 in you got 3 so plot (3,60) and (3,-60). Lastly we choose 120, which was 1 in the equation, so you can plot (1,120) and (1, -120). Then we contected all the poitns to form a sideways heart. We learned that this was an example of a cardiod which is a form of a limacon (soe french word) the final polar graph of r=2+2cosx was-

So we learned a cardiod occurs when we have r=a+bcosx when a=b. Then did an example where a wasn’t equal to b, 1+3sinx. Again we started by making the x,y coordinate graph which was-

Then we went to the polar grid. So first we knew that it had to go from 1 to 4 to 1 between 0 and 180. so we could plot (1,0), (4,90), and (1,180). Then we picked points to find the curve. First we used 30, weh you put that it the equation you get 2.5. So you graph (2.5, 30) and becuase for sine the symmery is over the y-axis you can also plot (-2.5,-30). Then we used 60 which was -.5, so you can graph the points (3.4, 60) and (-3.4,-60). Then we drew in that part of the curve. Then on teh x,y graph we know it goes from 1 to o and ends a little over 180 so we drew that in in estimatly they spot ending at 0. tehn the next part goes from 0 to -2 to 0 from a little over 180 to a little less then 360. so again we picked points. first we did 210, whose reference angle is 30,which gave us the points (-.5,210) and its opposite. Then we did 240, whose reference angle is 60, so the points were (-1.4, 240) and it opposite. Then we connected those creating the inner loop. Then from a little less then 360 the graph goes back to one so we estimated and drew tht part completly the graph. This type of graph is also a limacon. When the lal<lbl and inner loop is formed. The final graph looked like-

So thats it, sorry my graphs are a little messy but you can get the basic shapes of them.

Tomorrow’s scribe will be Hannah! =)

After 8 days of a non-bloging siege, I had waved the white flag and posted a belated blog post. What started as a practical joke (on me) ends as a battle for keeping the blog up to date. When I first heard about this missing link of post-evolution, I ran to the help of the intelligent community by bring you (the reader) a post that you should read if you want your life in balance.

Today, Tuesday January 15, 2008, we were brought the continuous mission of the polar graphs. We practices two problems, 2 + 2 cos x and 1 + 3 sin x. Each of these problems had a different outcome. The graphs are posted below. These are both called limaqon, French for snail.

Tomorrow scribe is the only true Cowboys fan in the state of wisconsin, and knows that the COWBOYS LOST TO THE GIANTS!

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

EXAMPLE DONE IN CLASS

Ok the first thing we did today was get another copy of the Quarter 2 Learning Targets, as well as the “How Did I Do On That Test” mid quarter 2 checklist. So if you didn’t get those yet, you should get on that.

Then for the rest of class we went over more graphing on polar graphs. It would be practically impossible for me to explain it right, so I’m not going to.

Tonights homework assignment is:

1.) Make up and graph more examples

r=___sin___(theta)

or

r=___cos___(theta)

*check with calculator

2.) Explore some examples like these:

r=___+sin___(theta)

or

r=___+cos___(theta)

3.) What determines the # of petals?????????

TOMORROW’S SCRIBE WILL BE STEPHANIE.

In today’s class, we started off by going over the claims from Thursday which were 6 a,b and 7 a,b (We didn’t have us go over number 12). For the most part there was relatively no issues with number 6, as long as you remembered your values for the circle diagram it was pretty easy. There were some questions pertaining to number 7 however, especially part b. Many felt that after transforming the rectangular coordinates (0, -4) to polar coordinates which are (-4, 0º) you would start on polar axis, and rotate zero degrees, then -4 which would land on the u-axis between quadrants II and III. After much explanation however, we were told that since the polar and rectangular coordinate systems lie on top of one another, you you simply start at the origin, rotate zero degrees, then -4, which ends up on the v-axis between quadrant III and IV.

(Sorry, I was having trouble with Winplot so no graphs)
After getting through all of the claim problems, we were introduced to graphing equations in polar mode. What we first did is plugged 3cos(3theta) in while in function mode. What you get is a typical cosine graph, with 3 periods between 360 degrees. After doing such, you type in the same equation while in polar mode and you get what looks like a three leaf clover. It is derived by looking at each step in the function form of 3cos(3X), and each section between critical points corresponds to the polar coordinates. Example: In the first section of 3cos(3X) in function mode, the graph goes from 3 to 0, and is between 0 and 30 degrees. On a polar graph, you would then start at 3, draw and arc up to, and touch just once, the 30 degree line, and then finish it off by stopping at 0. The next part, the graph goes from 0 to -3, and is between 30 and 60 degrees. Once again you would sketch it in, continuing from the last part, you’d start at 0 and draw to 3 while staying in between 30 and 60 degrees. Keep sketching section after section on the polar coordinate that corresponds to the original function until you are finished.

The next scribe will be Phil