Transformation

Hey kidz!!! I am here to summarize and explain in english what we learned so far.

Vocabularies : 

Transformation - moving the function to a different position keeping its original size, area and line length.

Translation-moving the function in a different position without rotating or flipping.
Reflection- A transformation in which a geometric function is reflected across a line, creating a mirror image. 

I will tell you how the equation affects the function by showing this formula:

y= Af(B (x+C))+D 

A affects the "y -coordinate". Multiply the "y value" to the value of A.(whatever the value is,use it)

D affects the "y-coordinate". Add or subtract the "y value" to the value of D.(whatever the value is, use it)

B affects the "x-coordinate". Multiply the "x value" to the reciprocal of B.(whatever the value is, get its reciprocal and use it)

C affects the "x-coordinate". add or subtract. the "x value" to the reciprocal of C.(if the value is positive, add or subtract it with its negative value. vice versa )

-for Y multiply "A" first then add/subtract D.(if both are given)

-for X multiply B first then add/subtract C.(remember if C is positive, make it negative. if C is negative make it positive.)(if both are given)

Now I will show you how each of them affects the function by giving examples:

1) y=f(x-2)

 definition:function f(x) was horizontally translated 2 units to the right.

 English: the whole function was moved 2 units to the right.

  

2)y=f(x+2)

 definition: function f(x) was horizontally translated 2 units to the left.

 English the whole function was moved 2 units to the left.

NB: if it's f(x-y), the graph will move to the right. if it's f(x+y), the graph will move to the right. opposite right? Deal with it.

3)y=f(x)-2

definition: function f(x) was vertically translated 2 units down.

English: the whole function was moved 2 units down.

4)y=f(x)+2

definition: function f(x) was vertically translated 2 units up.

english: the whole function was moved 2 units up.

5) y=1/2 f(x)
definition: the function f(x) was vertically stretched by a factor of 1/2 about the x-axis.
English: the whole f(x) will open wider.


6) y=2f(x) 
definition: the function f(x)  was vertically stretched by a factor of 2 about the x-axis.
English:the whole function will open narrower.


7) y= f(2x) 
definition: the function f(x) was horizontally stretched by a factor of 2 about the x-axis.
English:the whole function will open narrower.

8) y= f(1/2x)
definition: the function f(x) was horizontally stretched by a factor of 1/2 about the x-axis.
English:the whole function will open wider.



I hope I explained it to you very well in terms we can all understand. GG

Poland - Volleyball World Champions

Poland dethrones Brazil after winning its second World Championship.

Combination Formula

Hey kids. A couple of days ago we learned how to do Combinations and I will try to repeat the lesson here as good as Mr. P teaches.

Permutations and Combinations may look similar, but have different concepts. One thing to know is that permutations is an ordered collection of elements while a combinations consists of unordered collection of elements.

Combination Notation: C(n,r) or nCr  which is also       n!

                                                                                    r! (n-r)!

Remember, we cannot use the dash method for combinations and must only select the elements (no ordering).

Here Is a simple example with the formula: 10C2

*First we write out the formula.

nCr =     n!             *then we substitute          10!

           r!(n-r)!          (n= 10, r=2)                    2!(10-2)!

*We solve the bracket in the denominator so, it equals

   10!       *Now we must cancel the 8! by listing the integers down from 10 until we reach 8

  2!8!            

=  10x9x8!    *cross out common integers             = 10x9!      

      2!8!                                                                                        2!

*now you can solve!

   10x9   =  90 which is 45. We have our answer! Yay!

      2           2

Now, let’s try it out with a word problem.

“A student has a penny, a nickel, a dime, a quarter and a half dollar and wishes to leave a tip consisting of exactly 3 coins. How many *different* tips are possible

nCr =   n!                     n= 5, r=3           *n must be total number and r must be how much is needed

          r!(n-r)!

=        5!                    =   5!

      3!(5-3)!                  3!2!

=  5x4x3  *cancel out 3        = 5x4                 = 10        And that’s basically how you use the formula!!!

      3!2!                                        2

Permutation Formula

Permutations: An arrangement of things where the order is important

Formula: nPr=       n!              n= total # of objects   n≥r
                         (n-r)!           r= # of objects selected out of n 

The factorial (!) sign means to multiply all of the positive integers from n consecutively down to 1.
You can use this formula only when there are no repetitions and restrictions. 
Assume no repetitions are allowed unless otherwise.
Repetition means you are allowed to pick the same item more than once and Restrictions are when you are asked to place an item in a specific location.
Note: nP1=n! |  nP0= 1 | nPn=n! (0!=1)

For example:
How many three letter words composed from the 26 letters in the alphabet are possible? (assume no repetitions)
n=26 r=3
nPr=     n!      =         26!      =      26!      =      26*25*24*23!    
         (n-r)!            (26-3)!           23!                   23!

= 26*25*24= 15600

Repetitions allowed  
You cant use the formula but you can use the dash method 

    26          *         26          *    26           = 17576 
1st Letter         2nd Letter      3rd Letter
can be any       can be any      can be any

Restrictions 
How many 4 letter "words" beginning with A are possible, given the word MAPLES

  1    *     5        *       4        *        3     =  160
Must    5 letters      4 letters       3 letters
Be A     to chose      to chose        to chose
              from           from              from 

Example 2: Evaluate 
7P3
n=7        7P3=     7!       =  7!   =   7*6*5*4!   = 7*6*5 = 210
r=3                   (7-3)!        4!              4! 

Exmaple 3: solve
nP1=20                                                                          
n=n! r=1      
             nP1=20
             n!     = 20
           (n-1)!
            n(n-1)!  = 20
            (n-1)!

              n=20

Permutations

Hello class! This is Charmaine! And I'm going to try to explain what permutations is. When I look for the definition on dictionary.com this is what comes up:
 the act of changing the order of elements arranged in a particular order, as abc into acb, bac, etc., or of arranging a number of elements in groups made up of equal numbers of the elements in different orders, as a and b in ab and ba; a one-to-one transformation of a set with a finite number of elements.
In simpler terms, counting without actually counting (which also is clearly stated in our booklets) so here's an example!

A soccer team is getting new uniforms. The players get a choice of 3 different jerseys, 2 different shorts, and 2 different pairs of socks. How many possible varieties can you come up with for a full uniform? 
You can arrange this in two ways.
a) The tree method (More visual)
 
As you can see, there are 12 different possibilities.
 b) The dash method
 
Which also equals to 12!

So there you go, I hope this helped you understand permutations a bit more if you didn't understand it already!

Welcome

Welcome to our blog. This space is designed for students of the Maples Collegiate, attending the Pre-Calculus 40S class, section C, with Mr.P. We are going to use this space to discuss our daily lessons, ask questions you didn't get a chance to ask in class, and to share your knowledge with other students. Most importantly we will use this blog to reflect on what we're learning.
Have a great semester.