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

Class Notes:

--------------------------------------

Multiplying/Dividing Fractions:

-REMEMBER: FACTOR FIRST

Multiplication example:

X+3/7 * 14/2X+6

2(X+3) <------ X+3 cancels

=1

Division Example:

7x-7y/4y / 14x-14y/3y

7x-7y/4y * 3y/14x-14y

7(x-y)/4y * 3y/14(x-y)= 3/8

Adding and Subtracting:

1. Find Lowest Common Denominator (L.C.D.)

2. Rename fractions (change denominator)

3. Add/Subtract

4. Simplify

Classwork (following notes):

-Pick 5 from numbers 39-52 (rationale packet)

-Pick 4 from numbers 53-62 (rationale packet)

-#28-38 odds

Homework:

-Quiz (next Wednesday)

-Rationale Packet

-Finish Classwork

Submitted: Matthew Sattler

September 17, 2008

Next Scribe: Erin Murray

World of Functions, Simplifying- Scribe for 9/17/08

Today was a short period so we just took notes and did some practice problems. 

Class notes: 

1. Simplifying: Remember to factor first! 

A. This concept is based in the basic simplifying fractions concept: keep dividing the numerator and denominator. 

**example: 24/72 --> 12/36 --> 4/12 --> 1/3 

B. To factor

** example: 24/72=(2) (2) (2) (3)/(2)(2)(2)(3)(3) 

**In this equation, you cross out the numbers that cancel out to be one to end with 1/3

2. How this applies to factoring equations

**example: x^2-10x+25 --> (x-5) (x-5) because you know that x^2= (x) and (x). You also know that -5 * -5 = 25 and -5+-5= -10

**Methods: The area model 

3. Steps to simplifying more difficult equations: 

a. Factor first!

b. Find common denominator 

c. Rename fractions

d. add or subtract 

e. simplify 

After these class notes we had time to practice some problems in the prerequisite chapter 42 practice packet. (This was given to us Monday in class.) 

Homework: In the prerequisite packet with the practice problems: 

-Choose 5 from 25-38

-39-49 odd

-Choose 4 from 53 to 62

Announcements: There will be a test next Wednesday the 24th. 

Scribe_Post

Briggs Buckley

World of Functions

Next Scribe - Matt Sattler

Homework Due –

• Homework 10 – Difficult Denominators
  
• Homework 11 – An Average Drive
  

Homework Assigned –

• Numbers 25 – 38 odds in Handout Packet
  

Class Agenda

1. Finish & Review “Return of the Shadow”
   
2. Go over Homework 10 & 11
   
3. Begin Rational Functions
   

Class Notes

1. Reviewed how to solve Return of the Shadow:
   
2. Notes – Domain: The set of all inputs that make sense
   

Comes from two places: 1. The function, and 2. The situation

Homework 10:

• Noticed Verticle asymptotes (A place where the function is undefined). Observations from the class concluded that the functions could get close, but would never cross the vertical line.
  

Homework 11:

• Class noticed both Vertical & Horizontal asymptotes.
  
• Noticed (through a horizontal asymptote graph, that as x gets bigger, y gets closer and closer to 25.
• The Class came up with two equivalent equations to help solve the homework:
  

y = 100 / 4 - (100/x) and y= 100x / 4x-100

Rational Functions:

• f(x) = P(x) / Q(x)
  
• Both the P(x) and Q(x) are polynomial
  

We Talked about:

• How to simplify
  
• The Domain / Range
  
• How to Add/Subtract/Multiply/Divide
  
• Graphs
  
• Find Common denominator when adding / subtracting
  
• Multiply straight across
  
• Stay, dot, flop when dividing
  

FACTOR FIRST!!!!!!! – Mr. Marchetti

Other Notes About the Day:

1. Zolla came in & made a call using a students cell phone
   
2. Joe & Zolla got into another heated politics debate (Zolla walked out)
   
3. Mr. Marchetti told everyone that he was 34 (there was discussion as to how young he looked… Mostly by Erin M.)
   

Friday, September 12, Period 8

Scribe_Post

Marcus P.

World of Functions

Next Scribe- Briggs Buckley

Homework Due-

• Homework 9: Bigger Means Smaller

Homework Assigned-

• Homework 10: Difficult Denominators
• Homework 11: An Average Drive
• POW 7: One Mile at a Time

Class Agenda

1. Go Over Homework 9: Bigger Means Smaller

• Family of Reciprocal Functions

2. Don't Divide That!

• Why doesn't dividing by zero make any sense?
• Look at the traits of the graphs, reciprocal functions

3. "Return of the Shadow"

*Hints

• Draw a diagram of the situation
• Use cross multiplication to find the missing component, the length of the shadow.

Family of Reciprocal Functions Notes

K=constant

Y = K/X

• y is inversely proportional to x, inverse variation

Y = KX

• y is directly proportional to x, direct variation

Monday, September 15, 2008

Agenda:

• Return of the Shadow
• Go over Homework 10 & 11
• Rational Functions

We had previously established that in order to solve this problem, we would use a pair of similar triangles from a diagram of the situation. We also found that as the height of the person increases, the shadow length does as well (i.e. height and shadow length are directly proportional.)

Next we set up proportions of the similar triangles to find the length of the shadow.

20/h= 12 + l / l (cross multiply. *note, cross multiplication can only be used when there
is an = sign!)

20l = h(12+l) (distribute the h)

20l = 12h+hl (subtract hl)

20l – hl = 12h (rearrange the equation into an equivalent situation)

l (20-h) = 12h --> l = 12h / 20-h (yay!!)

Domain Restrictions
Set of all x’s (inputs) that make sense in the function.
i.e. h cannot equal 20
h must be greater than or equal to 0
h must be less than 20

" 1 / damn big = damn small" -Carl Hostnik

Then we went over Homework 10 and reviewed asymptotes. We discovered that horozontal asymptotes can exist as well.

We reviewed Homework 11 next. We talked about an equation that applies to the homework and came up with two equivalent equations:

y = 100 / 4 - (100/x) and y= 100x / 4x-100

A horozontal asymptote occurs in this graph, as x gets really big, y gets closer to 25.

Last, we talked about Rational Functions!

• f(x) = P(x) / Q(x)
• P(x) and Q(x) are polynomial
• y= x*x+3x+4, or y= 4 / 9x
• a series of terms with a variable to a power of a whole number

We worked on some algebra, reviewing how to add/subtract, multiply and divide fractions.

Remember!

• Common denominator when adding/subtracting
• Multiply across
• Stay, dot, flop when dividing

Last, we worked on factoring rational functions.

ex. (x+3 / 7) * (14 / 2x+ 6) *2x+6 can be factored into 2(x+3)

2 * (x+3 / 7) * (14 / 2*x+3) = 1

• Next Scribe: ClaireH
• Homework: Practice Rational Functions worksheet, numbers 39-49 odds
• Quiz next Wednesday!!! Study!!

Tuesday, September 9th, Period 8

Scribe Post
-Lauren McDavid

-----------------------------------------------------------------

Announcements: None.
Next Scribe: Marcus Parry.

Homework Due: Homework 8.
Homework Assigned: Homework 9 (due Friday), POW Revisions (if needed, due Monday).

-----------------------------------------------------------------

Class Agenda:
I. Warm up (see below).
II. How to turn a table into a function notes.
III. Homework completion check. (Homework 1-8, except 5)
IV. Brake! Revisited (complete).
V. Learned about regressions (a calculator function).

-----------------------------------------------------------------

Warm Up:
1. (a+3b)2
a2 + 6ab + 9b2


2. (km+5k)2
km2 + 10k2m + 25k2


3. What kind of table is this?
X Y
2 11
4 31
6 59
8 95
10 139
12 191

Quadratic

-----------------------------------------------------------------

Table to a Function Notes:

1. Identify the family.
-->Common differences.

-->Common ratios.

2. Curve fitting.

-->Guess and check.

3. Systems of Equations

-->Quadratic, Cubic, etc.

4. Regressions.

-->Calculator.

5. Algebra.

6. Exponenial-ratio.

Scribe

Notes

Class Warm up:

We expanded equations.

I. E: (a+3b)^2

(a+3b) (a+3b)

From here, use F.O.I.L to fully expand the formula

F=First.

(a)(a)=a^2

O=Outer

(a) (3b)=3ab

I=Inner

(3b) (a)=3ab

L=Last

(3b) (3b)=9b



a^2+3b+3b+9b

a^2+6b+9b



We discussed discovering the family of certain tables. The example given was quadradic because the second differences were the same. Once we knew that the table was quadradic, we were supposed to find a formula for the table. We knew that the table would follow a quadradic function. We plugged the x values in for a the y values in for be and c always remained 1. We did this for three inputs and thus were given a system of equations. We solved the system and got the formula.



Homework

Finish Brake! Revisited and do homework 9.



ANNOUNCEMENTS

Revisions for POW 7, one mile at a time, are due monday. Remember to include the revised work as well as an explanation of what you did wrong and what you did to fix it.



Next Scribe

Camila S

Scribe Posts

One of your assignments this year is to be the scribe for the day. The scribe is responsible for adding a post to the blog that includes all of the information that was important for the day. Scribe posts should include:

• Notes
• Homework assignment
• Important dates and announcements

In addition you should label every post with three labels:

• Your name
• Kind of assignment (in this case scribe_post)
• Period (4 or 8)
• Unit (World of Functions)

Please comment on posts with questions, additions, or corrections. If you are absent check the blog to see what happened in class. Scribe posts are due by the next block. Failure to post will not only effect your responsibility grade, but your classmates as well.