Chapter 9 Section 5: Probabilities of Various Combinations

Section 5 is about Permutations and Combinations. In a given situation, it is important that you realize whether the situation is one where the order of elements matters or does not matter. This can be explained by the definitions of permutations and combinations.

Important Definitions:

Combination: A subset of elements where the order of the elements does not matter.

Computation of Number of Combinations:

Number of combinations = Total number of permutations / Number of permutations of each one combination

nCr = number of different combinations of n elements taken r at a time.

n = how many elements in a group.
r = how many elements we are selecting.

Remember that last section we learned about Permutations.
A permutation of a set of objects is an arrangement in a definite order of some or all of the elements in that set. So basically, order matters with permutations.

nPr = number of different permutations of n elements taken r at a time.

n = how many elements in a group.
r = how many elements we are selecting.

(To access these keys on your calculator: MATH -> PRB -> nPr or nCr )

When I was talking about the order mattering versus the order not mattering I meant that, for example, if you have the numbers 1, 2, 3, and 4 and you are making three digit numbers using these four numbers, in a permutation, order matters. So, if you said 123 and also 321 that would count as two separate numbers that you could make. With a combination, order does not matter, so when you use the numbers 123 that is the only time you will count it as a number. If you decided to put them in the order 321, that would not count as a separate number that you made. When the numbers 1, 2, and 3 are used in the same number for a combination, you do not count the different ways that the numbers can be arranged as separate numbers. So, with the numbers 1, 2, 3, 4, there are 24 different permutations, but only 4 different combinations.
To prove this, put in your calculator:

For the permutation:
4 nPr 3 and you should get 24.

For the combination:
4 nCr 3 and you should get 4.

Calculation of Number of Permutations of Combinations:

nPr = (n!)/((n-r)!)and nCr = (n!)/(r!(n-r)!)

Remember: ! is the symbol for a factorial.
For example (8!) = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 = 40,320


Example Problem:

#1.) 15 C 5 = ?

This is a combination, so nCr = (n!)/(r!(n-r)!)

15C5 = (15! )/( 5! (15-5)! )
15C 5 = (15!) / (5! 10! )
15 C 5 = 3003

Note that the numerator is the same number as the number that precedes the "C." The 5 and the 10 (the numbers in the denominator) add up to the numeral in the numerator.

To solve this on the calculator, you would type in: 15 nCr 5 which equals 3003.

#2.) You have 15 CD's that you absolutely love, but, sadly, you only have room for 5 of them in your car. In how many different ways could the group of 5 CD's be chosen?

The word "group" tells you that a number of combinations is being asked for.
You can solve this problem two ways, either manually, or on your calculator. It is useful to know how to solve both ways, because we have both calculator and non-calculator portions to our tests and quizzes.

On the calculator:

15 nCr 5 = 3003

So, there are 3003 different ways to choose the group of 5 CD's.

By hand:

nCr = (n!)/(r!(n-r)!)

15 C 5 = (15!)/ (5! (15-5)!)

15 C 5 = 3003

Link:

http://www.saliu.com/oddslotto.html

This site is to help calculate lotto odds, but it is actually a good explanation of combinations.



This is a photo of a bay in Greece at night. I used this photo as the basis for a painting.

"Poppies at Argenteuil", Claude Monet 1873
I started to paint this picture after Monet awhile ago but I have been so busy that I have had no time to work on it. The unfinished painting is hanging in my living room. Hopefully I can get some time to finish...maybe after SATs.

Tara! You’re next!

Friday's Quiz Topics

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

Precalculus Quiz 9.1-5 Topics
Definition of terms
Probability – rolling a die (and/or events)
Calculating outcomes – and/or
Permutations vs. combinations
Calculating permutations and combinations
Interpretation of combinations
Variations of combinations and permutations – calculating outcomes and probabilities
Repeated elements in calculations

That’s it! The format’s the same as always – ½ Non-calculator, ½ Calculator. There are 9 questions on the Non-calculator portion and 14 questions on the Calculator section. I’ll be around after school on Wednesday, and hopefully accessible by email on Thursday. Mr. Frost should be able to answer any questions you have during class.

In any collection of data, the figure most obviously correct,
beyond all need of checking, is the mistake

Corollaries:
(1) Nobody whom you ask for help will see it.
(2) The first person who stops by, whose advice you really
don't want to hear, will see it immediately.

And on another note - look for the simple solution!

9.4 Probabilities of Various Permutations

Hey everybody!

Here is some info about PERMUTATIONS!

permutations are a way to count the number of outcomes when the ORDER is important.

A really simple way of doing these types of problems is with factorials, ! .
For instance: 9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362,880.
To do this on the calculator, either multiply the entire system out or go to MATH, PRB, 4: !

Some problems that use this are...

If you have 7 people trying to share 3 spots, you have to divided 7! by 4!. It is 4! because you subtract the number of spaces from the total number of people. When you do this, the 4 x 3 x 2 x 1 cancels out, leaving 7 x 6 x 5 = 210.

Repeated Elements: If there is a word or something with repeating figures: take the factorial of the total number of figures/ factorial of repeating figures. NEEMATOAD (frog-like swamp monster): 9 total letters, 2 repeated A's and 2 repeated E's, so 9!/(2! x 2!) = 90,720 total distinct combinations.

Restrictions: If you have 10 people who need to fill 10 spots but 3 are restricted to specific persons, the number of possibilites is 7! x 1 x 1 x 1 = 5040. The three 1's show that there are no additional outcome because of the 3 restricted spots.

Practice Problem!

A. On the Winnie the Pooh ride at Disneyland each cart has enough room for 4 people. Danny and Michelle are dating, and must therefore sit together, but there are 6 people with them who can sit any where. Within the two carts how many possible seating arrangements are there?

B. While waiting in line, Dieter, Danny's friend, was wondering how many distinct combinations the letters in the ride name (including "the") could be made into. Find this number.

Solutions!

A. To find this number, take the number total, 8!, and divide by 4, the number of possibilities for where they can sit. The answer is 10,080.

B. First you need to count the total number of letters, 13. Then count the number of repeats. There are two pairs. The number is 13!/(2! x 2!) = 1,556,755,200.

For more help, check out

http://mathforum.org/dr.math/faq/faq.comb.perm.html

I hope this was helpful. And this has been fun with Permutations!

here is a video with stop motion, drums, and piano!!! IT'S AWESOME
http://youtube.com/watch?v=hVG_esC-rgA
it actually sounds good too...

MARICLARE you're next!!

9.3 Two Counting Principles

9.3 Two Counting Principles
Hello everyone!
What we learned in class today had a lot to do with terminology and ways to interpret what the question's trying to ask. So let's get started!

• Independent Events--> The way one event occurs doesn't affect the way the other could occur.
  
  In class, we used the example of a 3 day summer camp. On the first day, the students were allowed to pick one Outdoor Activity to do in the morning and one Indoor Activity to do at night. What they chose in morning did not affect what they did at night. Therefore, this situation is an example of independent events.
  
  Another example is what the campers eat for meals. What they ate for breakfast won't affect what they eat for lunch or dinner (unless they're sick of the same old thing).
  


• Dependent Events--> The way one event occurs affects the way the other could occur.
  
  Next day at camp, it rained so none of the outdoor activites were available. The campers could pick any indoor activity to do but what they did in the morning could not be repeated in the afternoon.
  
  Another example is usage of showers. If the campers were only allowed to take one shower a day, then a person who showered in the morning will not be able to take one later at night. (Oh no! Surprised?)
  


• Mutually Exclusive Events--> The occurrence of one event excludes the possibility the other will occur. (choose one, can't do the others)
  
  The last day at camp is only a half day so the campers can only pick one activity to do. Once they pick the activity, the other choices are gone.
  
  Another example is choosing an outfit for the day. Once a camper makes a decision between flip flops and converse, she's not going to be able to wear the other pair of shoes also. Therefore, this is a mututally exclusive event.

Properties of the Events

1. Let A and B be two events that occur in a sequence.

n(A+B)= n(A) * n(B/A) for dependent events

n(A+B)= n(A) * n(B) for independent events (there's no overlap!)

'A' is the number of ways an event can occure while 'B' is the number of ways an event can occur if 'A' occurs.

2. Overlapping Events: When choices in an event overlap or repeat.

Suzy is trying to decide what kind of frozen yogurt she should eat and what topping to put on. She can choose between chocolate, vanilla, and regular frozen yogurt with choice of granola, blueberries, strawberries, pineapple pieces, reese's, or nothing as a topping. How many choices does she have for either vanilla frozen yogurt or toppings of blueberries or granola? Let's make a table!

As you can see from my paint picture, there is an overlap in the toppings (in bold) and vanilla yogurt (in italics). Two x's are bold and italics. Therefore, to figure out this problems, you must do this:

n (A or B)= n(A) + n(B) -n (A~B)
~=it's supposed to look like a bump or a concave down mini parabola but this is the closest I could get. It means "and" or the intersection of A and B.

so 3+6-2=7 There's 7 choices Suzy has.

If you're still confused about all this stuff, you can check out this website (but I think there's more than we learned on this website lol):
http://www.richland.edu/james/lecture/m116/sequences/counting.html

Before I go, here's a picture of my sword dance team that performed at the Dorothy Chandler Pavilion on Christmas Eve. The Christmas program we were a part of airs on PBS or whatever channel you get Sesame Street on every Christmas and Christmas Eve. I was part of an interview but sadly running behind schedule made them cut it from the show. Oh well. This year, we're doing 3 drum dance (which I am also going to do for JPD!):

Claire, you're up next!!!

9-2 Probability!

Hey guys! So this section is all about probability.



Here are the terms and definitions:



Random Experiment: The act of doing something

Trial: Each time you do something

Simple Event/Outcome: What you get

Sample Space: All the possible outcomes

Probability of an Event: number of ways it can occur / number of possible outcomes

OR: P(E)=n(E)/n(S)



Now try it out:



You have 16 individual socks in a drawer and 9 of them are white and 7 of them are black. If you draw one at random, what is the probability that you will draw a white sock? What is the probability that you will draw a second white sock?



Since there are 16 total and 9 of them are white, you have a 9 in 16 chance of drawing a white sock, so you have a probability of 9/16.

Now there are only 15 socks in the drawer, 8 of which are white and 7 are black. You have a 8/15 chance of drawing a white sock this time.



For more information try:

http://www.k111.k12.il.us/king/math.htm#Probability

and

www.mathgoodies.com/lessons/vol6/intro_probability.html



Ellen you are next

Thursday's Test Topics

Here’s a list of topics for Thursday’s test:

Precalculus Chapter 7 Test Topics:
General equations (no translations) for all the functions
Sketch graphs of all the functions
Determine equations from graphs
Logarithmic properties
Exponential to logarithm and logarithm to exponential
Properties of exponents
Patterns for all functions
Determine points given types of functions
Demonstrate patterns from data tables
Find equations from data tables
Concavity
Find additional points given a data table
Demonstrate pattern given equation
Determine x-values algebraically from equations given y-values
Logistic functions – description, soutions, point of inflection

That’s it! The format is as expected – ½ calculator, ½ non-calculator. I’ll be in my classroom on Wednesday after 3:30 and before school on Thursday. I’ll also be available online later on Wednesday evening, after the girl’s basketball game. If you have specific questions Wednesday night, email me!

See you in class!

At New York's Kennedy Airport today, an individual, later discovered to be a public school teacher, was arrested trying to board a flight while in possession of a ruler, a protractor, a set square, and a calculator. Attorney General John Ashcroft believes the man is a member of the notorious Al-Gebra movement. He is being charged with carrying weapons of math instruction.
Al-Gebra is a very fearsome cult, indeed.They desire average solutions by means and extremes, and sometimes go off on a tangent in a search of absolute value. They consist of quite shadowy figures, with names like "x" and "y", and, although they are frequently referred to as "unknowns", we know they really belong to a common denominator and are part of the axis of medieval with coordinates in every country. As the great Greek philanderer Isosceles used to say, there are 3 sides to every angle, and if God had wanted us to have better weapons of math instruction, He would have given us more fingers and toes.
Therefore, I'm extremely grateful that our government has given us a sine that it is intent on protracting us from these math-dogs who are so willing to disintegrate us with calculus disregard.
These statistic bastards love to inflict plane on every sphere of influence. Under the circumferences, it's time we differentiated their root, made our point, and drew the line. These weapons of math instruction have the potential to decimal everything in their math on a scalar never before seen unless we become exponents of a Higher Power and begin to appreciate the random facts of vertex.
As our Great Leader would say, "Read my ellipse". Here is one principle he is uncertainty of---though they continue to multiply, their days are numbered and sooner or later the hypotenuse will tighten around their necks.

7.6: Logistic Functions

Explanation:

Logistic functions are characterized by when something grows rapidly but then tapers off because it reaches a maximum or limit. The logistic funciton is a distorted form of the exponential function. A graph of a logistic function is concave up AND concave down. There is a point of inflection between the two stages. There are asymptotes above and below the graph. The high asymptote is labeled "c" and is in the equation for a logistic function. The general forms for a logistic function's equation are:

y=c/(1+a(b^-x)) and y=c/(1+a(e^bx))

The graphs of a logistic function look like this:












when b is greater than 1 ...................................when b is between 0 and 1

To find the value of the point of inflection, you use y=c/2. The higher asymptote is where the graph will stop, and you know that the point of inflection is exactly between the two asymptotes. If you divide c by 2, you'll get the y-value of the point of inflection.

What kind of situations would you use the logistic function for? Populations mostly. Because the logistic functions have a maximum and a minimum, the situations that would most likely be represented with a logistic graph are ones with limits.

Because the growth of a logistic function is NOT consistent, we cannot assign any given pattern to a set of numerical information. If we are given the data table and the asymptote (or carrying capacity) OR "c", then we can determine the equation. "C" is the higher asymptote of the logistic equation, so in most problems they will give you the maximum of whatever the problem is about.

Sample Problem:

A version of The Grove is constructed in Ohio. Because nobody in Ohio has ever been to something like The Grove, all the people rush in to see it and shop there. The numbers of people visiting the outdoor mall increase rapidly at first. At a certain point, however, the numbers level off because there is no longer a rush to go see it. The maximum number of shoppers in one week is 396,000. Given the data table below, find the particular equation for this logistic function using the second and last data points.

x(weeks).............................y(# of shoppers in thousands)

1............................................................71
10........................................................182
15........................................................239
21........................................................281
30........................................................321
50........................................................370
74.........................................................394

How to solve it:
First plug in the numbers of your two points. The two points are (10, 182,000) and
(74, 394,000). "C" is the maximum number of shoppers (396,000)

y=c/1+ab^-x
182,000=396,000/1+ab^-10
394,000=396,000/1+ab^-74

Then multiply each side by the denominator to get rid of the fraction.

182,000+182,000ab^-10=396,000
394,000+394,000ab^-74=396,000

Then subtract the value without variables from both sides.

182,000ab^-10=214,000
394,000ab^-74=2,000

Now divide the top equation by the bottom equation. The a's will cancel out, and you can combine the b's together. We know that when a value with an exponent is divided by another value with an exponent, we can combine them by subtracting the bottom exponent from the top exponent. We have to be extra careful with logistic functions however, because both exponents will be negative! So, -10 - (-74) = 64.

.46193b^64=107

Now isolate b by dividing both sides by .46193 ...

b^64=231.6374

Now get the b by itself by taking both sides to the 1/64 power...

b=1.088

To get the rest of the equation, all you have to do is plug b back into one of the original equations and solve for a.

182,000=396,000/1+a(1.088)^-10...multiply by the denominator

182,000+182,000a(1.088)^-10=396,000...take 1.088 to the -10 power

182,000+182,000a(.4271)=396,000...multiply 182,000 times .4271

182,000+77,726.67a=396,000...subtract 182,000 from 396,000

77,726.67a=214,000...divide both sides by 77,726.67

a=2.753! (if you got a couple answers a little different than mine, that's because I used the whole decimal, but I wrote them on the blog as if I'd rounded them.)

Your equation to fit this data table is: y=396,000/1+(2.753)(1.088)^-x!!!

Other problems that you may come across will usually ask you to find the x or y value that corresponds to a given x or y value (which is not in the table) that they give you. For this kind of problem, all you have to do plug this number in to your equation.

Other web source:

If you need more help, http://www.wmueller.com/precalculus/families/1_80.html is helpful. It explains logistic functions really well.

Landry, you're up next! Good luck!

Ok this is so cool. Stare at the point in the center of the circles for around 15 seconds. Then move your head forward. NEAT!

7-5 Logarithmic Functions!

Hey guys! Happy Valentine's Day!!!




So here is lesson 7-5 Logarithmic Functions:

Multiply-Add Property of Logarithmic Functions:

• Logarithmic Functions are invereses of exponential functions. With exponential functions the pattern is add-multiply so with Logarithmic functions the pattern is MULTIPLY-ADD.
  
  Example:
  
  
  
  In the above table for a logarithmic function you can see the Multiply-Add pattern. When you multiply the x's by 3 you add one to the y's.
  
  Particular Equation for Logarithmic Functions:
  
  

The General Form for a Logarithmic Function is: y=a+b(logx) or y=a +b(lnx)

• When solving for a and b you plug in two sets of points into the general form similarily to solving for exponential functions. Then, instead of dividing the two equations you subtract one from the other. Then you solve for b and then you plug in the b value into one of the original equations and solve for a. When you have both a and b you can put them back into the general form and you have your equation.
  

Example Problem:

Taking the points (6,1) and (18, 2) from the above table plug them into the general form for x and y to get:

1= a +b(ln6)

and

2= a + b(ln18)

Then subtract the second equation from the first to get :

-1=b(ln6)-b(ln18)

Next you can factor out the b to get:

-1=b(ln6-ln18) --> -1= b(ln 6/18)

b= -1/ln(1/3) which = .910...

Now that we have the b value we can plug it back into either of the original equations for a.

1= a =+.910(ln6)

a=-.631

Now when we plug in the a and b values we get y=-.631 + .910(lnx)

Graphs of Logarithmic Functions:

The Graph of natural log y=ln(x) goes through (1,0) and looks like this:

Now, if we were asked to graph y= 3ln(x-1) we know what it is the above graph with a vertical dilation of 3 and a horizontal translation of 1 (to the right). It would look like this:

Now, the graph of y= -ln(x-3) would be different. Because of the negative sign the graph would reflect over the x axis and it would also shift 3 to the right looking like this:

The graph of log(x^2-1) would look like this (it is reflected around the origin because it is squared):

The domain of this graph is x<-1 or x >1.

For More Help with the graphs go to: http://www.sosmath.com/algebra/logs/log4/log42/log422/log422.html

For help with the equations go to: http://www.sosmath.com/algebra/logs/log4/log47/log47.html

So here is a link to a clip from the movie White Nights where Mikhail Baryshnikov(one of my favorite dancers EVER) does 11 pirouettes. He is amazing and his turns and jumps are crazy!!

http://youtube.com/watch?v=VHbvHoAWKV0

Reminder: Natalie you're next!

Monday's Quiz Topics and New Posting Order

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

Precalculus Quiz 7.1-3 Topics
Identify types of functions from graphs
Describe characteristics of graphs – increasing/decreasing, concavity
Use patterns to determine the effect on y when x is changed by a certain factor, given a specific relationship between x and y (general and specific cases)
Numerical patterns for linear, quadratic, exponential and power functions
Identify a function from a data set
Determine additional points for a data set
Given a data set and type of function, determine an equation using algebraic techniques.
Predict a data point once equation is determined. Identify if the data point is extrapolated or interpolated.
Demonstrate the validity of a pattern given points in a data set.

That’s it! I’ll be around after school on Friday and in early on Monday morning. If you have questions over the weekend, send me an email and I’ll try to respond Sunday evening.

As promised, here's the new randomly assigned posting order:

Debby
Anna
Natalie
Landry
Ellen
Claire
Mariclare
Tara
Lucy
Nick
Arash
Allison
Celene
Annie
Kaori
Scott
Henry
In order to attain the impossible, one must attempt the absurd.
- Miguel de Cervantes

Poor study strategies:

7.3: Identifying Functions from Numerical Patterns

So, before I begin, I'd just like to say that Blogger is very Mac-intolerant. Because I myself am a Mac user, Blogger.com gives me less formatting options (I only have spellcheck and insert a picture). There are several tables throughout this blog, but the only way i could get them to separate is by a series of dots. Please forgive this slight amount of sloppiness.


This lesson is all about identifying the type of function when given a set of data. Each function has a specific pattern between the x and y values which will help you determine the equation.

Chapter Summary

Linear
Linear equations are probably the easiest to identify (but i'm sure you knew that already). If you look at a set of data, such as,

x .......y
1 .......7
2 ...... 9
3 ......11
4 ......13
5 ......15

It can be determined that the following set of data is a linear function due to the fact that both the x and the y values increase in a consistant, addition manner. When each x value increases by one, the y value increases by two. Because both sides increase by addition, this is said to be an ADD-ADD PATTERN.

Power
Both the x and y values in a power function increase raipdly, due to the fact that both values are increased in muliplicational incriments. Here is a set of points in a power function.

x.......... y
3........135
6....... 1080
9 .......3645
12 ......8640

Looking at the x values first, the first increase goes up by an addition rate of +3, but the y values do not share consistency with this pattern. therefore, we must move on to multiplying both values by a certain number. in this situation, when the x value is multiplied by two, the y value is multiplied by a value of eight (or, rather, two cubed.) Because you must multiply both sides, power functions are said to be MULTIPLY-MULTIPLY PATTERNS.

Exponential
Exponential functions are, in a way, a blend of the Linear and Power functions in the sense that, though the x values increase from addition, the y values increase through multiplication.

x .............y
1 ............15
3 ............135
5 ..........1215
7 ........109355

We see here that, while the x values go up by an addition increment of +2, the y values go up by a multiplication value of 9. Because of the addition, there is still some consistancy between the increasing y values. Due to the mix of addition and multiplication, exponential functions are referred to as ADD-MULTIPLY PATTERNS.

Quadratic
Quadratic functions are a bit more complicated; the similar intervals required are not between each y value, but the difference of each interval of the y values (I hope this isn't confusing you more... it's hard to explain, but I'm doing my best... bear with me, people.) Take this set of x and y values, for example.

x............... y
1 ..............15
3 ...............5
5 ..............19
7 ..............57
9 .............119

The first thing that we can quickly notice is that the x values increase in a simple addition matter, each variable being two greater than the previous one. As for the y values, the difference between the first two values listed is a decrease of ten (or an increase of -10). The second and third values listed are fourteen numbers apart from each other. These two values of -10 and 14 are at a value of twenty four apart from each other. the third and fourth y values are 38 apart from each other, while the fourth and fifth values are 64 away from each other. These two values share a difference of 24. The second difference of the values is always constant, so Quadratic functions are said to be CONSTANT SECOND DIFFERENCE PATTERNS. If this information is not enough to get rid of any confusion you have, feel free to ask Mr. French or myself at any time.

NOTE: One clue as to determining whether or not a set of values belong to a quadratic funtion is searhing for a change in the direction of the y values. For instance, in the example given, the first two values are 15 and 5. This data tells us that the funtion is decreasing. The third value, 19, moves is a positive direction from its previous value. Quadratic functions are the only functions in which you will see a change in y value direction.




You may also be given to f(x)s, and then asked to list the possible outcomes. Use your knowlege of what each function can be described as (add-add, multiply-add, etc.) to determine how the variables are changed in order to receive the following values.

For example...

f(5) = 12
f(10) = 18
f(20) = ?


You must allow yourself to look at these given equations and solve them from various viewpoints. You can say that x increases +5 between the two equations. The change can also be interpreted as x2. Similarily, the y values can be seen as changes of +6 or x1.5. By working out these different possibilities, f(20) can equal several things...

...in a linear equation, f(20) = 30

...in a power function, f(20) = 27

...in an exponential function, f(20) = 40.5



Also, verbal explanations are key in what an equation looks like.

If y varies directly with x,
-if x doubles, y doubles

If y varies inversely with x,
-if x doubles, y is reduced by .5

If y varies directly with the cube of x,
-if x doubles, y is multiplied by eight (which is equivalent to 2 cubed)


Example Problem

Question
Using the given points, determine wether the data is an add-add, add-multiply, multiply-multiply, or constant second differences pattern. Tell what kind of function that has this pattern. What should the next three sets of values be?


x..........f(x)
2...............14
6..............112
18............896
54............7168


Solution

1) First of all, we need to look at the list of x values. There is not any addition consistancy between them. Once multiplying is tried, we find that each value is multiplied by a factor of 3 to get the next value.

2) As for the y values, there too is no addition link. We divide the second value, 112, by the first value, 14, to get 8 (or two cubed). This is also the answer you get when you divide 896 by 112, and 7168 by 896.

3) So we know know that this is a Multiply-Multiply pattern, which tells us that we are dealing with a power function.

4) Since we figured out that when each x value is multiplied by three, the y value is multiplied by 2 cubed, we can determine the next three vaules in this pattern. The first x value is found by multiplying the previous x value, 54, by 3, to get 162. The x value after that is found with the same means, and 162 times 3 is 486, and repeating the process once more gives us the next x value of 1458.

5) As for the y values, we must multiply the last given y value, 7168, by 2 cubed, or 8, in which case we get 57344. Repeating this process twice more gives us the values 458752 and 3670016.

6)
So the final table looks like this...

x..........f(x)
2...............14
6..............112
18............896
54............7168
162.........57344
486........458752
1458.....3670016


If you need help, here is an extremely extensive website with information regarding these concepts.
Concepts Help



Two new bands I've been getting into recently are called "the Eternals" and "Big Sir". For any of you Mars Volta fans out there, Big Sir includes Mars Volta bass player Juan Alderete de la Pena (who I met at a train station) on the fretless bass. Let me know what you think!

The Eternals



Big Sir




Debby! Congrats! It's your turn next!

7-2 Identifying Functions from Graphical Patterns

Overview:

Linear:
general: y=ax+b
parent: y=ax
transformed: y-y1=a(x-x1)


Quadratic:
general: y=ax^2 + bx+c
parent: y=x^2
transformed: y-k = a(x-h)^2


Power:
general: y = ax^b
parent: y=x^b
transformed: y=a(x-c)^b + d


Exponential:
general: y=ab^x
parent: y=b^x
transformed: y=ab^(x-c)+d

If you are given x and y in a quadratic funtion and required to find a, b, and c use matrices.
To find slope its rise/run

Practice Problem:

Find a, b, c for the following quadratic equation if (1,-7) (2,0) (3,9)
use matrices
[1 , 1 , 1 ]-1 [-7 ] [ 1 ]
[4 , 2 , 1] times [ 0 ] = [ 4 ]
[9 , 3 , 1] [ 9 ] [-12] so a=1, b=4, c=-12

y=x^2 +4x -12

Other Material
http://library.thinkquest.org/2647/algebra/functype.htm

This is one of the funniest SNL skits i've seen in a long time. I'm sure everybody has watched SNL at one point or another so I think that this is appropriate.
http://youtube.com/watch?v=1dmVU08zVpA