Translations

Translations

Today, we learned more about Translations. 

Translations are horizontal and vertical movements (shifts) using basic shape. 

• The word Translation means to slide a graph. 

There are some rules when it comes to shifting. Stated below are the rules. 

VERTICAL TRANSLATIONS

* We call k a vertical translation. 

y=f(x) + k          - The entire graph shifts UP k units. 

Why? Because k is positive, which means we move up. 

y=f(x) - k          - The entire graph shifts DOWN k units

The position of the graph is influenced by k. The statement above says that k is a negative. Therefore, the graph will move down. The value of x does not change.

NOTE: h values are read as OPPOSITE, while k values are to remain the way they are. 

HORIZONTAL TRANSLATIONS

* We call h a horizontal translation. 

y=f(x-h)          - The entire graph shifts h units to the RIGHT.

y=f(x+h)         - The entire graph shifts h units to the LEFT.

The position of the graph is also influenced by h. The value of y does not change. 

For example:

If you are asked to sketch a graph of the function f(x - 9) - 3 on a Cartesian plane, 

with k= -3 and h=9, you will have to move 3 units down, since our k is a negative, and 9 units to the right, since we read h as its opposite (In this situation, -9 will be 9. It will become positive)

The Binomial Theorem

On Tuesday, we learned about The Binomial Theorem. We can determine the number of terms in a binomial by looking at the exponent (n) and adding it by one. 

For example: ( x + y )^4 would have 5 terms.

                      = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4  

                           1          2             3               4          5

There are patterns that work the same in every binomial expansion which are:

• The sum of the exponents in each term is equal to the power of the binomial expansion (n).
• The first x-variable will always equal to n then decreases by one in the next term. The y-variable appears on the second term then increases by one until it matches the value of n.
• The coefficients are the combinations of n that starts with nCo and ends with nCn. 

          Note: n > 0

        

          (a + b)^n = nCo a^n b^0 + nC1 a^n-1 + nC2 a^n-2b^2 + ...nCn a^0b^n

• Another way to determine the coefficients in each term is by using Pascal's Triangle. To use the triangle, simply look at the number of terms the binomial expansion would have, that value would be the number of the row in the triangle. For example: ( x + y )^4 would have 5 terms so look at the 5th row in the triangle. 

We can find a specific term in a binomial expansion without expanding it by using the formula:

                                                           tk+1 = nCk a^n-k b^k

The value of k = # of term - 1 

Example: Find the 4th term of (x - 2y)^10 

n = 10     a = x     b = 2y     k = 3

t4 = 10C3 * (x)^10-3 * (2y)^3

    = 120x^7 * - 8y^3

    = - 960x^7y^3 

Combinations

Today in class, we learned about combinations. We first determined the differences between Combinations and Permutations.
Basically,
Permutations                                        Combinations 
1.) Select                                                1.) Select
2.) Arrange

There is also certain words/hints that you can look for in the question that can help you determine which method to use to solve.

For example;

with Permutations, you might find the words arranged or different. when dealing with Combinations, you might find the words select or choose.

you must use the formula when dealing with Combination - can not use the dash method. The formula will look like this: nCr =     n!       
                                  r! (n-r)!

An Example of a Combination question:
A student has a penny, a nickel, a dime, a quarter, and a half dollar and wishes to leave a tip of exactly 3 coins. How many different tips are possible?
You can tell it's a Combination question because the arrangement of the coins is not necessary. you are simply selecting the coins. 

Solution:
n = 5            
r = 3
5C3 =   5!        
            3! (5-3)!

        =   5!       
           3! 2!
     
        =   5x4x3!          - At this point,3! cancels out 3! and 2! simplifys the 4 into a 2. 
             3! 2!

       =  5x2x3!
            3! 2! 

       = 10 

IMPORTANT PROVINCIAL EXAM NOTE: 

nCx = nCy 
n = x + y 

Permutations with Case Restrictions

09/08/15

In today's class we continued our lesson on permutations. Our main focus in this lesson of permutations was on equations involving only numbers. We were also introduced to equations with not only one but several limitations.

Example:
Using the numbers 2, 4, 5. 6 and 8, how many four digit numbers can be formed if the number is to be less than 5000 and divisible by 5?

2 x 5 x 5 x 1 = 50      or      1 x 5 x 5 x 1 = 25
2 = 5,4                                 1 x 5 x 5 x 1 = 25
1 = 5                                    25 + 25 = 50

The answer is 50 four digit numbers divisible by 5 and less than 5000 can be formed.

Permutations with Repetitions and Restrictions

09/17/15

Today's lesson is about permutations that includes repetitions and restrictions. We cannot use the permutation formula since we are dealing word problems that can have either repetition or a restriction.We can only use the dash method because it all works in all situations.

Example:
How many 5 letter word can be formed using the letters m, a, g, i, c allowing for repetition of the letters?

Since we need 5 locations
_____  •   _____  •  _____  •   ____  •  _____ (dash method)

There are 5 letters that can be used to fill the first location. Repetition is allowed, the same 5 letters can be used. 
__5___  •   __5___  •  __5___ •__5___  •   __5___  = 3, 125 arrangements 

Permutations

09/15/15

 Today we were introduced to permutations and how to permute a set of objects. A permutation is all the possible arrangements of a set of objects where order is important. In order to use the permutation formula thenquestion must not allow restrictions or repetitions. If there are restrictions and/or repetitions you may use the dash method.

The formula for permutations: nPr = n!/ (n-r)!
reminder: ! is the factorial symbol
                     n= total amount of objects to arrange
                     r= amount of objects taken at a given time

Example

Q: A class of 8 people that were consisted of 4 male students and 4 female students were lined up to be photographed. How many arrangements or different photographs can be taken?

A: n=8  r=8

8P8
=8! / (8-8)!
=8!
=40,320

Fundamental Counting Principle

Today in class we learned about "The Fundamental Counting Principal," and "Factorial Notation."
For Fundamental Principal, Mr. Piatek showed us 2 ways of getting to the answer.
-tree diagram for smaller numbers (helpful for students who are more visual)
-multiplication (to get answers for big numbers)
For Factorial Notation, Mr. Piatek made it very clear to the class how Exclamation mark works. Example- (n-1)(n-2)(n-3)! can be said as (n-1)! as they both have ! at the end.

Welcome

Welcome to our blog. This space is designed for students of the Maples Collegiate, attending the Pre-Calculus 40S class, section D, 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.