Monday, September 29, 2008

Today in class we went over Homework 12. Then we took notes on solving equations, either algebraically or graphically on the worksheet page 380 on the Section Exercises 5.4:

• For question 1d. in Homework 12, one possible solution Mr. Marchetti showed us was the function
  

3. By graphing each side of this problem on the calculator with the first half of the problem before the equal sign as y1 and the 8 as y2 we were able to tell that these two equations intersected at x=.1621 and x=1.46, which is the solution



Unfortunately, the graphing method only gives approximations so it is better used as a way to check your answers.



6. We also solved this problem graphically in class to find intersections at x=2 and x=-7
If this problem were an inequality like this , the solution would be x<-7 Then we learned how to solve a problem algebraically: 2. To solve this problem, both sides have to be multiplied by (x-4) to eliminate the fraction




So then the equation looks like:




To finish solving, bring the 5x to one side and the 3 to the other:



-4x=-23 Then divide both sides by -4 so







10. The first step would be to factor wherever possible, so the new equation would be:


The next step would be to multiply each side by (x+4)(x-1)





After distribution, all like terms are gathered.





Since this equation is quadratic, 15 can be subtracted from both sides to make one side zero and the other side can be factored and solved from there.





To make one of the sides zero so the equation will be equal to zero,


or





The first answer to this problem is not relevant unfortunately, because the x=1 in the second fraction would be:

Which doesn't make sense because it is not possible to divide by zero. This problem can be checked by graphing the system which shows that at x=1, there is a vertical asymptote so it is not a possible anyswer, which is called an extraneous solution.



Homework:

• Worksheet page 380 choose 2 problems from Section Exercises 5.4 from questions 1-8 and choose 3 problems from questions 9-19. Solve each algebraically and then check them graphically.


• PoW 8 due any day from Thursday to Tuesday


• There is a quiz over functions on Thursday 10/9

The next scribe will be Maddy.

September 25, 2008

Today in class, we were first greeted with Mr. Marchetti telling us the proper way to add an equation to the blog posts. First, you click the 'add equation' button, then after you select a 'math' you edit the equation in the red font below the box, then you save it and copy and paste it into the blog post.
Then, we worked on POW 8 - A Spin on Transitivity, and discussed the transitive properties we explored earlier when we stated that if Keith is taller than Mr. Marchetti and Mr. Marchetti is taller than Camilla, than Keith must be taller than Camilla. Or, in mathematic terms: if a=b and b=c, then a=c. By the way, the black spots on the wheels on the POW are 1 for Betty's, 4 for Al's and 7 for Carlos. The POW will (tentatively) be due Thursday, but check your calendars and tell Mr. Marchetti if you have a conflict with turning it in on Thursday and he'll change the due date.
After we worked on the POW, Mr. Marchetti checked homeworks 9, 10, 11 and the packet work.
Later, we discussed 'End Behavior' which mainly deals with the horizontal asympototes of functions. We then found out that every function has two ends, negative infinity and positive infinity. This concept was further investigated on pages 288 and 289.
Our homework due Monday is Homework 12, and to do more problems in the packet.
The next scribe will be Ali Follett.

Today we took at quiz on tables, story sketches, algebra, and reciprocal functions.

Notes

There were no notes taken today

Homework

Revisions for Homework 8 are due tom (thursday)

There will be a homework check on Thursday and Monday from Homeworks 9-11 also including the problems that you have done from the packet

Keep doing problems in the packet that you have trouble with

Next Scribe is Emma

Scribe Post Per 8 9/23

Today
1.) Review HW
2.) Equations & Inequalitlies

HW
1.) Quiz tomorrow
- Up to Ch 11
- Rule of 4
-tables
-story sketches
- Reciprocal Family
- Algebra

Notes on Board
y=kx: linear function; direct variations
y=k/x: reciprocal function; inverse variations

domain: what X must be to make the function undefined (make the denomenator 0)

Horizontal Asymptotes
1.) biggest power of X on bottom means HA: y=0
2.) powers are the same in numerator and denomenator means look at the leading coefficients of the highest powers of X, because they are the HA's
3.) biggest power of X on top means no HA

Scribe Post 9/22/08

Class Notes



We started class today with a warm up: we had to find the common denominator for

[(x+1) / (x^2-5x+6)] - [(3x+11) / (x^2-x-6)]

and simplify

[(X^2-1) / (2x+2)] * [4/(x^2-2x+1)]

Next we went over some problems from the packet and disscussed vertical and horizontal asymptotes. Conclusively it was decided that several rules apply when finding the horizontal asymptote (HA) as x is getting damn big.

1. When the highest power of x is in the denominator the HA: y=0. Example: 1/(x^2+3)


2. When the highest powers of x are equal in the numerator and denominator use the coefficients in front of these values to find the HA. Example: (6x^2)/(3x^2+4) HA: y=6/3 or .5


3. When the highest power of x is in the numerator there is no horizontal asymptote, but there is a slant asymptote. Example: x^3/x^2. as x gets bigger the top and bottom continue to grow, so it just keeps going.

For HAs, however, there are some exceptions. For (x^2-9)/(x+3), a rule 3 case, there is no HA but when x=-3 the value is undefined. The graph of this appears to cross this line where the HA would be but there is actually a "hole," explained by the tern removable discontinuity.*plug this into the y= on your calculator to see the graph* Mr. Marchetti explains this using force fields. VAs are like star trek force fields that cannot be penetrated, while HAs are not. Close to zero they are weaker, but get stronger and eventually impenetrable farther away.

Announcements

Quiz tomorrow, wednesday, and Homework * revisions due thursday

Homework

Study for Quiz: Up to but NOT including rational functions, study tables, story sketches, reciprocal family, and algebra (used for tables) Also, due thursday are more problems from the packet, do some from the sections you need more practice on.

Next Scribe is Julia

gogolplex

For all of you wondering about how much gogolplex is, I found a page about it. ITS HUGE!!
Here's the link
http://www.procrastinators.org/oldsite/googolplex.html

Scribe for 18 Sept 2008

The first thing we did today was get back Homework 8: Mystery Tables, which most of the class got an R on, for lack of explanation. We will be able to revise them to earn points back. Today we also went over our worksheet about factoring with rational functions and all that fun stuff, which I'm not going to put on here because it would take a long time to explain, and there were just a few problems. Class went fairly well otherwise, expcept for the point at which we were sidetracked by talking about big numbers such as gogol and gogolplex and infinity, which prompted Mr. Marchetti to say, "We're talking about a HUGE number that makes damn big look damn small!"

Notes:

y=1/x+5 is undefined @ x=-5
the vertical asymptote (VA) is @ x=-5
the horizontal asymptote (HA) is @ y=0

for a more complicated rational function:
h(x) = 2/(xsquared + 4x +3)
the denominator can be factored as (x+3)(x+1) which can then be set to equal zero to find the vertical asymptotes:
(x+3)(x+1) = 0
x= -3, -1
VA: x= -3
x= -1
HA: y=0

some other notation-type stuff we learned was about infinity, which I don't know how to write on here, but essentially, it is phrased like this:

as x approaches infinity, y approaches 0
as x approaches negative three from the right, y approaches negative infinity
as x approaches negative one from the right, y approaches infinity

then the last thing we talked about was if we had really big numbers being plugged in for x. for example:
y= 5x+3/7x+5
x= a billion

5 billion/7 billion
billions cancel out
we're left with 5/7
so, as x approaches infinity, then y approaches 5/7

we also looked at the next POW 8: A Spin on Transitivity, which we will have a week and a half to do

Announcements:
We will have a test this coming Wednesday, the 24th of September

Homework:
Revise Homework 8 for next Thursday, start on POW 8, and from the rational functions worksheet, pick a few problems from each section of questions:
# 1-10
# 11-18
# 19-24
#25-34
# 41-46