Hi, Jerricson here! just a reminder I'm not the best when it comes to explaining stuff so ill do my best and if you have questions about this particular topic please ask Mr. P for better understanding.

Today's lesson was a continuation of transformation but instead we learned Backward Transformation. Basically you have to solve for the original function of the transformed function by doing so you will have to "reverse" the method of transformation and if you still did not understand that here, are some examples of what we learned today.

Example 1: Consider the transformed function y = f(x) + 2 below. What is the graph of y = f(x)

-tips for solving y = f(x), make sure you know how transformation work, anyways back to the example.

To get y = f(x) + 2 from y = f(x) you need to add 2 on the y-values. But since we are doing Backward Transformation and in order to find the original function in this question you need to Subtract 2 on the y-values. like what i mentioned in the first paragraph "Reverse the method of Transformation" 

For more examples please check our unit-2 booklet pages 52-55. and if you don't have the notes ask your friends so you wont fail.

And finally for the Main event, class selfies.

Selfies with Julan and Raphael

Mr.P's sideview

Crazy Bunch

Pre-cal iz Da Best!

THE END

Hi guys! My name is Julan without an I!:)
So today we reviewed the previous topics that we have learned and learned new topics as well. The major topic that we learned today was Inverse of a Function and Restricted Domain. This is a simple definition of how it works... you basically want the inverse of the function f(x)to be a function as well.(given that trhe inverse is not a function already)

This is a picture of how the graph would look like...

                                         Inverse of a fuction and restrained domain ex. #1

For this question first it asked us to graph the function f(x).
Step one: find the x and y-values of f(x)=x^2-2. Those coordinates would be (-2,2),(-1,-1),(0,-2),(1,-1),(2,2) once you have this you can now graph the function.

Step two: graph the inverse of f(x). I hope we all know that all we have to do to get the inverse is to switch the x and y-values. The coordinates would then be (2,-2),(-1,-1),(-2,0),(-1,1),(2,2) now graph the inverse of f(x) using these coordinates.

Step three: describe how the domain of f(x)could be restricted so that the inverse of f(x) is a function. First make sure that you know your domain and range for both f(x) and its inverse. For this graph you would describe it as: By restricting domain of f(x) to {x/x>0,xEr} our inverse will become a function.


These are some of the other work we did in class and some selfies to keep the trend going! :)

 

 

inverse of a function and restrained domain examples

 

                                          selfie with my camera shy friend Manpreet :D

Raphael doing work!

The BEST class in Maples!

 

 

For those that missed class because of the blood donor thingy, I suggest that you go and see Mr.P to catch up on what you have missed.

 

I hope you guys have a good night!



February 24, 2014

Its Ria (Ree-uh). :)) 

     Today we covered a lot of things since we had 2 periods. But I wont talk about everything because it will take up a lot of space and i don't think anyone would want to read a long worded MATH stuff. * Also, just a heads up, i hate putting math into words, so if the context doesn't make sense don't read any further before it confuses you. 

In the morning class, we went through the assignments from our booklet.The most important part that we learned was how to Translate/Reflect/Transform a graph all at once. This one will definitely help you with our hand-in assignment.  

For Example: 

f(x) = 2f(3(x-1))+4 

a = 2, b = 3 , h = -1, k = 4 therefore, both x and y values are affected.

Step 1:   Multiply your y values by 2, then add 4. (Note: You read "k" as is)

Step 2:   Now, you have to multiply the x values by the reciprocal of 3, which becomes 1/3, OR you can just divide it by 3, and then add 1. (Note: "h" is read opposite, from subtracting to adding)

Then you can sketch your graph with your new coordinates! 

** Remember not to mix up your operations. Do the multiplying first, then do the vertical and/or horizontal stretches. Yes, it does make a difference.

I thought this chart would be helpful:

We got a worksheet called "Do you understand the main concepts of Basic Transformations".

We also talked about the Transformation Golf Project. It was a long one so talk to Mr. P if you missed it.

Moving on, during the 2nd period we did a recap on the absolute value of a function. Absolute is when you turn the negative values to a positive, which results a flip or a reflection on the x axis.

We also learned how to write an equation of the transformed graph by analyzing the coordinates of 2 graphs, f(x)/g(x).

Then, we went on how to inverse a relation. I'll make it easier for you. All you have to do is switch/flip your x and y values. 

              example: (-5,1) => (1,-5). 

Easy right? Same goes for the Domain and Range.

Ex:          D: {x | -5 ≤ x ≤ 4, xER}

               R: {y | 1 ≤ x ≤ 5, yER}

                           to

                D: {x | 1 ≤ x ≤ 5, xER}

                R: {y | -5 ≤ x ≤ 4, yER

At the end of class we got our TESTS back. :/ But before we got those tests back we did our daily dose of selfies/selvies with Mr. P. HAHA Its the new trend by KENNETH! So keep the trend going, next blogger! 

I never realized putting numbers and formulas into context can be this hard. Pardon the errors you spot! *fingers crossed*. 

Thats all. :)) Have a wonderful day!

February 19, 2014

Hello everyone! I'm Kenneth Pineda, most of you probably know me, if not I sit right in front of the class.

Here are some selfies with Mr. P

Mr. P thinking bout the next quiz

Anywayyy.....

The first thing we did in class is give out hand-outs and worksheets from the people who were absent yesterday. It took a lot of our class time, so next time you miss a class make sure to see Mr. P beforehand. 

We also did another quiz today, I hope we all did well on that one.

After the quiz we had a quick review from yesterday's class.

We started a new topic, which is REFLECTION.

• Reflection over x-axis

                         - make y values negative

                         - x axis stay the same

• Reflection over y-axis

                         - make x values negative

                         - y values stay the same

We also went over different examples to better understand the concept of the new lesson.

Reflection over x-axis

Reflection over y-axis

Reflection over x-axis then y values are multiplied by -2

Reflection over both x-axis and y-axis

That's it!! Yay!

Don't forget about the Permutation, Combination and Binomial Theorem Unit Test on Friday...

XOXO Kenneth Pineda

Function Transformations

Hi. It's Brook here. Sorry for the late post.

Yesterday we learned about Basic Transformations and how to graph functions.
Graphing any quadratic function is moving (translating), flipping (reflecting), and stretching the base of the graph.

The base of the graph is y=x²

In the function f(x)=a(x-h)²+k, we learned that the variables (a, h and k) are an important part of graphing.

In the first function we have f(x)= 2(x+3)² -1. These numbers have changed the way the transformation was graphed.
The values H and K will tell us if the graph must be moved left or right or up and down but I'll get into that later.

The number 2 tells us if the graph will be wider, narrower or the normal u-shape.

*Remember:
 If the a-value is < 1, it is narrower
 If the a-value is > 1, it is wider
 If the a-value is = 1, it is the normal u-shape.


We used the following tables to represent the ordered pairs on a graph.

These tables show how the function y=x² has changed to y=2(x+3)²-1.
The first table represents the normal function.

In the second table, the y-values were multiplied by 2 but the x-values had stayed the same.

In the third table, 1 was subtracted from the y-values and the x-values still stayed the same.

In the fourth table, 3 was subtracted from the x-values and the y-values stayed the same.



We also learned about how to graph horizontal and vertical translations.


*Note: Read h values as opposite and k values as is!

Vertical translation, given f(x), k > 0:
  y=f(x) + k ----> This tells us that the graph will move k-units up because k is positive.
y=f(x) - k -----> This tells us that the graph will move k-units down because k is negative.

Horizontal translation, given f(x), h > 0:
 y=f(x - h) -----> This tells us that the function must shift h-units to the right.
y=f(x + h) ----> This tells us that the function must shift h-units the the left.

Using the graph of f(x), we had to graph f(x)+1.

How we graphed f(x)+1 was by adding 1 the the y-values.

*  y = k
    k = +1







And that's pretty much it.

February 17, 2014

Today, we learned NOTHING! You know, because today there was no school.

Yay!

February 14, 2014

Permutation and Combinations
Hi Guys! Mydee here, I hope you all had a wonderful long weekend. Last Friday we did examples using both permutation and combinations on the last pages of the booklet. (Sorry for posting late.)

Reminder:
February 18 (Tuesday) - Assignment Hand-In
February 21 (Friday) - Unit Test (Good Luck)

2C2 -> Two people are selected together, 11C5 -> remaining members to be on the committee

In Group A, 5 members can be selected from 20 students, In Group B, 8 members can be selected from the remaining 15 students ( 20-5=15), and the remaining 7 students will be part of Group C.

This question is a permutation, the dash method is used. *Don't forget to divide it by 2 because there is 2 repetition of E's.

• Since there are 2E in the word TEACHER and we are creating a 4 letter word, multiple steps are needed to find the answers for this question. 
• For the first 4 letter word,  2E's can be used but remember to permute it by mutlipying 4!, since we are only using 4 letters from a 7 letter word and divide by 2 since there are repetition.
•  For the second 4 letter word, only 1E will be used. 2C1 => means that one out of the two E's will be used. 1C1 => automatically means that there is only 1E . We don't have to divide it since there is no repetition and because only one E is used. 
• For the third letter word, there will be no E. FOr this one we could just write it as, 5C4, which means that we are choosing 4 letters out of the 5 letters. 
• For the last step, add up all the answers from the previous equations.

Mr. P gave these examples last Friday at the end of the class. 

~ In the first example, FEBRUARY has 2 R's and that is why we have to divide it by 2! at the end. We need to create a 4 letter word, 2C2 => 2 out of the 2 R's are used. 6C2 => selecting 2 more letters out of six letters to create the 4 letter word. Multiply by 4! to permute and then divide by 2! because of R repetition. 

 ~ For the second one, only one out of the two R's will be used. Two methods can be use. 

-  1C1 => this automatically means that you will only used 1 out of the 2 R's,   

-  2C1 => means you are choosing only 1 out of 2 R, this can also be used, but remember to divide it by 2! since are repetition. 

Example # 2: Create 4 letters word using the word "EVERYONE". 

The first method is the same as the one above. The second one is Mr. P's method.

~ We all know that 3C3, 2C2 and 1C1 is equals to 1 and so we don't really need to put it. But DON'T FORGET TO PERMUTE it to the number of letter using and to DIVIDE BY THE NUMBER OF THE REPETITION.

Reminder: If you use this method, make sure that you understand the equation that you are using and that you got the number right. 

Example #3: Create 5 letters word using the word "NOTEBOOK".
~For any of the examples above any method from the above can be used. Use the method that you are most comfortable with and make sure that you understand the steps on how and why the number in each equations are used. 


AND THAT'S ALL FOLKS!! WE ARE NOW DONE WITH UNIT ONE!!! YEHHEEYYY!!?!? THAT MEANS 7 UNITS LEFT! ;)

Eighth Day

Hi everyone! It was truly a winter wonderland this morning, but fortunately almost (if not) everyone were there in class. I am Rowsell by the way and I am going to summarize everything we learned in Pre- Cal today *i apologize if I'm not very specific (i'm more into numbers than words)*...
Firstly, in our morning class we continued discussing about Binomial Theorem and learned the formula, which is in the picture below along with the explanation:

*All the examples are those we did in class
Example 1: Expand and simplify using the binomial expansion:   

a) (3x - y) ⁵

Step #1: list the values Step #2: use the formula and plugged in values Step #3: Simplify Step #4: the answer

*Make sure you check everything before moving on, especially the + and – signs, because they can be tricky sometimes ;)

*And remember (-3x) ⁰  isn’t the same as -3x⁰, because (-3x) ⁰ = 1 and -3x⁰= -3 (the brackets make a difference)

We also learned is to find a specific term within a binomial expansion we can expand the binomial until we locate the specific term or to make things a little simpler we can use the formula shown on the below:

Example 2: Using the formula for calculating specific term,find the following terms

 a) the 4^th term of (x - 2y)¹⁰

Step #1: list the values Step #2: use the formula and plugged in values Step #3: Simplify Step #4: the answer

Thankfully, we did not have any quiz in the afternoon and continued on where we left off in the morning...

Example 3: Find the coefficient of the term containing x¹⁸ if (2x - x²)¹¹ 

Step #1: list the values Step #2: Solve/ Look for k Step #3: use the formula and plugged in values Step #4: Simplify Step #5: the answer

Lastly, we learned about finding the error in the binomial theorem:

Solve for the right answer, then look closely in the "SOLUTION WITH ERRORS" to find the error that was made

Near the end of class, we received an assignment consisting all the things we did/ learned in class about Binomial Theorem

Seventh Day!

Hi guys it's me Gaby! I hope you all did an awesome job on your quiz! If not then that's okay! Keep trying and studying so you do a great job on the next! The quiz was about solving for n. We first had to put it into the combination formula, we then expand the n!, we cancel out the same terms from the numerator and the denominator, then move all terms on one side equaling it to 0, then finally you factor the polynomial. You should get the roots of n=-8 and n=9. Then check by plugging in the roots and your answer should be n=9. Remember negative roots have to be rejected!

Anyways today we learned about the Binomial Theorem, or Pascal's Triangle. Honestly at first it was confusing, but after a while, hopefully everyone has a good understanding of it. 

Here's some methods for you so you can decipher with coefficient is the correct on by expanding different binomials:

The Tipi Method -  This basically shows a tipi shape by simply starting by #1 all the way down, diagonally to #9.

The Hockey Stick Method - Like in the picture above, the pink lines are shaped as a hockey stick. 1+3+6+10+15+21+28=84. All of those numbers going diagonally down towards the left, if you add them all together you get the sum of the number diagonally down towards the right. 

The Triangle Method - By choosing 2 numbers beside each other, you add them together and it should equal to the number right under the 2 numbers you added. 

Fifth Day!

Hey guys, this is Rachel! So, today we had another quiz for solving "n". After the quiz, Mr. P made a brief explanation for both quizzes, (there were 2 different ones) to the whole class. Then he continued our lesson to "Permutations with Repetitions and Restrictions" and "Permutations with Case Restrictions"  


Permutations with Repetitions and Restrictions AHEM *make sure you know this because this is apparently popular on tests and exams* :) Well, first of all, repetitions means you can have repeats. So,you may use numbers more than once when permuting. If we are working with repetitions we are NOT able to use the formula. Just a little reminder. :P Examples are in our notes. I am just explaining how we got answer our answers on some.

Ex #2 Consider the digits 2, 4, 5, 6 and 8. If repetitions are allowed, find...
a) How many four digit numbers can be formed?
Well, since there are 5 numbers in total and repetitions are allowed, we are able to have, =5x5x5x5 or 5^4 ("4" represents the digits needed) =625

Ex #3 a) How many four digit numbers can be formed? (this time, no repetitions are allowed) Since there are 5 different numbers and there are only 4 digits available, we must think in 5P4. ("4" represents the digits needed). =5x4x3x2 =120 We can also use the formula because REPETITIONS ARE NOT allowed. =5!/(5-4)! =120

Ex #4 A family consisting of 7 of the parents and 5 kids are going to be arranged in the photo. Calculate the number of permutations if all 5 children must be seated together. Since all the children must sit together, they are known to be 1 entity. They can also, be moved around even though they are in one group. for example, k= kid. k1 k2 k3 k4 k5 but can be k2 k3 k4 k5 k1 and all the other permutations. since there are 5 children they will represent as 5! Since the mother and father can be in any place without disrupting the group of their kids they are known to be 2 entities or 2!. Since the group of the kids are 1 entity they are represented as 1!. So the total of entities is equal to 3! =5!x3! =720  

Permutations with Case Restrictions
Ex #1 c) How many 6 letter "words" are possible using the letters in GADGET? Since there are 6 letters in the word and we need 6 letters to make up "words" we can represent this as 6P6 or 6!. But make sure to keep in mind that there are 2 G's in this word, meaning we must divide by 2. =(6!)/2 =720/2 =360

Ex #2 Three sets of books are being arrancged on the shelf. The first set has 5 volumes, the second set has 3, and the third set has 2. In how many ways can the books be arranged if the volumes of each set are to be kept together? Think of sets. first set = (5!) second set =(3!) third set = (2!) *don't forget that they are 3 different entities so (3!) When we put all those together it comes to, =(5!)(3!)(2!)(3!) =8640 *To end this, just make sure to read the word problems carefully. Because it's very easy to screw up a little thing. LOL* BYE Y'ALL :D

Fourth Day

        Hi it's Daniel,  on Friday we got our first quiz back, hopefully everyone did well, then we continued onto Permutations part one and two.  In part one, we learned how to permute (rearrange) a set of objects to find out all possible ways of rearranging them with restrictions, and that we CANNOT use the formula nPr = n! / (n-r) when there no restrictions , and that we use the dashed method _ . _ . We learned how to solve for n when the value of nPr is given. At the end of part one we checked our understanding by doing six questions that were given in the lesson booklet.

        In part two, we continued to further our understanding of using the formula nPr = n! / (n-r) , we learned to use the formula n! / a!b!c!d! when trying to find the number of distinguishable permutations of a word given using all of its letters, like REFERRED. n would be equal to the amount of letters in the word which in this case would be 8, a!b! Etc would be any letter in that word that is repeated at least once which would be E=3 R=3, the equation would look like and be solved like this,
8! / 3!3!, then expand 8!, (8)(7)(6)(5)(4)(3)! / 3!3!, cross out both denominators, then 3! And 6 from the numerators. Then the equation will be left like this to calculate the final answer, (8)(7)(5)(4) = 1120. We were sadly notified that this formula WILL NOT be on the formula sheet for the provincial exam.

      At the end of class we were given a booklet do do questions 1-20 but omit questions 12, 13, and 14. That pretty much sums everything up. Have a good weekend everyone.

Third Day!

Hey everyone. Andrew here. Today we went over yesterday's worksheet on the Fundamental Counting Principle. We also had our first quiz! Yay! After both of those things, we continued with our lesson on Factorial Notation.

Firstly, we learned how to simplify. To do this, you can expand either the numerator or denominator.
For example, say you have a question like this: 5!/4!
You can expand both of them so that you'd get something like this: (5x4x3x2x1)/(4x3x2x1)
Then, you just cancel out like terms so that you end up with just 5.

Another way to do this would be by expanding only one of the two (Numerator or Denominator) so that you end up with a different, lower factorial.
For example: 12!/8!
You can just expand the Numerator: (12x11x10x9x8!)/8!
Then, just cancel out the 8!, so that you are left with: (12x11x10x9)
Finally, just multiply them all together to get 11880.

The final thing we learned today is how to solve for n.
Firstly, isolate n.
Then, cancel out what you can. If you can't, expand something until you can.
Eventually, you should get something that looks like this: ax^2 + bx + c = 0
Then, you can just solve the square. You will get 2 x values.
Finally, plug in the two x values one at a time to check.

*Remember: the x value CANNOT be negative!

That's about all. See ya all later!

First and Second Day!

Hey everyone! My name is Melissa and I hope all of you had a good semester break and that your exams went well for you! I'll be the first to post for our class and on the first day of Pre Cal 40S, we had a slideshow that Mr.P created (and I'm sure I can speak for all of us that his opening credits were pretty awesome). We discovered that Mr.P expects us all to have Apple products but unfortunately he won't supply us with them.... :(  We also had a quick introduction of each other names and an interesting fact about ourselves.
After the ice breaker, we were presented with the topics and course outline that we will be following from now until June. He showed that in order to pass this course and be successful, we need to be on top of your homework and always ask questions if we don't completely comprehend a concept. However, I am sure all of us will be up for the task and finish high school on a great note! Afterwards, we received 2 worksheets of problems that we've seen before in previous years from graphing to factoring to identifying the domain and range and seeing if it was a function or not (keep in mind the vertical line test for that!) And that concludes the first day!
On Thursday morning once we got our textbooks we began with our first lesson but not before we watched the Monkey Business Illusion. Just goes to show Mr.P likes to mess with our heads, luckily some of us caught on with the background tricks. We first learnt about the Fundamental Counting Principle and how we can get the total amount of combinations in a certain scenario without actually having to go through the whole factoring tree which is pretty time consuming. The pictures below with show our first example with the tree. To make things simpler, we learnt that since we need one sandwich, drink and dessert, we can just multiply the amount of choices we have (in this case 2 sandwiches, 3 drinks and 2 desserts) we can get the exact same answer as the tree just without all that drawing and writing. I've also included that paragraph that explains the keywords that we'll see throughout the semester and is a good reminder as to what operation you'll need to use to figure out the equation correctly and efficiently.

Afterwards, we received a quick assignment would be our homework for the night and it went over all the concepts we learnt in the morning class. That concludes the morning of our second day!

In the afternoon we jumped straight into Factorial Notation! We learned the basics of factorial notation, which included learning that factorial is represented by an !, we've been lied to our whole lives, I thought an exclamation point meant when you were exclaiming something exciting or joyful we'd use that....not anymore....We just went over simple equations and what it meant and how we could solve them with and without a calculator. Also finding little tricks like when you have 3! it results in 3(3-1)(3-2) which equals 6. But an easy way to find 4! is just taking the answer from 3! which was 6 and multiplying it by 4, which gets  you the answer of 24. It's a simple yet good tip to keep in mind. And that concludes the first two days of Pre Cal 40S! Hopefully we'll all get to know each other better and I wish you the best of luck in your last semester of high school!