Going backwards with transformations. (:

Greetings beautiful people! My name is Aron. I don't know what I'm supposed to do here. I guess I'll explain going backwards with transformations and stuff. 

What do you do when you go backwards with transformations? Well, you go backwards. You start where you're supposed to end and end where you're supposed to start. 

Remember this:
Since your going the opposite direction  with your transforming, you do the opposite of you're normally supposed to do.


I don't know how to make graphs on my crappy laptop so I'm gonna need you guys to use your imagination


Remember that since you're doing things backwards you do the addition parts (translations) before the multiplication parts (stretches)!

Normally you do the "opposite" operation to x-values and you do what it says for the y-values but not this time! Now you do the "opposite thing for y-values!

EXAMPLES!!!

Imagine the function y= 2f (x-1) had  the following point:

(6,-4)

You were then asked to find the values of  these points at f(x)

To get that you need to first ask yourself what you have to do first.

What I do first I look for what I'm supposed to do to the x and y-values.

(REMEMBER THAT YOU'RE GOING BACKWARDS)

For X 

1. You subtract 1 to the x-values. 

For Y

1. You multiply 1/2 to the y-values.

You will end up doing this:

(6,-4) =>       ((6-1),(-4/2)) =>        (5,-2)

That means that f(x) = (5,-2)

For another example lets say you had the function y = -2f(2(x+2))+2

(REMEMBER THAT YOU'RE GOING BACKWARDS)

For the x-values you look at this part of the function -2f(2(x+2))+2 then we do this:

Step 1: Add 2 to all x-values

Step 2: Multiply 2 to all x-values

For the y-values you look at this part of the function -2f(2(x+2))+2

Step 1: Subtract 2 from all y-values

Step 2: Divide all y-values by -1/2 

(Don't forget to make the factor by which you divide negative for the reflection)

Step 3: Enjoy! You're done.

_________________________________________________________________________________

It's now time for my final thoughts.

First of all I know that my scribe thing is bad compared to what everyone else has done, I'm sorry.

Second, I like doughnuts. I believe that we can get doughnuts next week.

Third, the key to being good at math is practice. Don't study! Practice! They're very different. Studying is boring, practicing can be fun!

_________________________________________________________________

Don't practice until you get thing right, practice until you don't get it wrong."  -HALT O'CARRICK

Inverse of a Function and Restricted Domain

Hello Class! Navjot here!

Today we learned Restricted Domain and Inverse of a Function!  What does this mean? Well.... sometimes we have a function whose inverse is not a function. In this case you would restrict the domain of f(x) so that f^-1(x) is also a function.  

Example:

• Find the inverse of y = x² + 1

From the graph you can clearly tell that the inverse of this function would not be a function, since it would not pass the vertical line test. In order to make this a function we would have to restrict the domain.

 In this graph the domain has been restricted to include only numbers that are less than 0. By doing this, the inverse of this function becomes a function. 

The domain for the original function is x < 0 and the range would be 1 < y, therefore the domain for the inverse function would be 1 < x and the range would be y < 0.

 

REMEMBER: The domain of the original function becomes the range for the inverse and the range of the original function becomes the domain of the inverse!

I hope that this make sense and that you all learned something! And now I'll leave you with a joke:

Why do they never serve beer at a math party?

Because you can't drink and derive... 

 

 

Also,

 

Sources: http://www.purplemath.com/modules/invrsfcn3.htm#previous

Restricting the Domain

Bonjour Class! It's Melanie :)

Glad you found your way to this awesome page. Let's get started.

Something new that we learned today in class is called Restricting the Domain.

Sometimes, we are given a function which does not have an inverse. 

Thankfully we can generate an inverse that is a function over a specific interval, by effectively restricting the domain. In laymen's terms, this just means that you control or restrict the domain; all the values that are plugged into the function.

For example: Let's say your function is f(x) = x² 

If we square a negative number, our inverse will not match the original function, therefore if we restrict the domain, we can create an inverse for the original function. 

A Few Helpful Reminders:

• Remember the inverse of a relation can be created by switching the x values and the y values of the graph.
• The inverse of a relation is reflected in the line y = x
• In the inverse of a relation, the domain of the original function becomes the range, and the range of the original function becomes the domain.
• One may verify graphically or algebraically whether two functions are inverses of one another.

Here's a funny comic I found entitled "The Awkward Function"

and to finish off this post, a hue of inspiration. As Albert Einstein once said:

"Pure mathematics is, in its way, the poetry of logical ideas."

Hope y'all enjoyed this post! Gotta love pre-cal. ♥

Yours Truly,

Melanie ♥

Sources: 

http://www.mathsisfun.com/sets/function-inverse.html

http://spikedmath.com/488.html

http://www.brainyquote.com/quotes/keywords/mathematics.html

 Hello. I dont know what to say.

Hi 

First Of All I'm Allen forget the John. K? \/^.^\/

Well today... nevermind.. yesterday we study about.. oh yea it's today

we study about Inversions. its easy like EZZZZZ

lets say you have

 in this picture the points of the original functions are

(-2,4)

(-1,1)

(0,0)

(1,1)

(2,4)

 all you have to do to get the inverse function is to switch the x-values and the y-values

which will give us

(4,-2)

(1,-1)

(0,0)

(1,1)

(2,4)

orrrrrr in this case

y=3x+2

to get the inverse function we need to switch x and y so it will be...

x=3y+2 

andd thennn we need to isolate the y variable

x-2=3y

x-2/3=y

so the inverse function will be

y=1/3x-2/3   or   y=x-2 over 3

TO LEARN MORE. WATCH THIS OLD GUY IN YOUTUBE

http://www.youtube.com/watch?v=zY31abWIQS8

OH! I passed my test <3 I hope guys do too. 

because..."you are a smart bunch" -Mr. Piatek 2014

"Insert Mr. P's picture here"

********************  ...AND PLEASE... ******************

"Don't FAILerino so we can get some DOUGHNUTerino"

REFERENCE

http://www.regentsprep.org/Regents/math/algtrig/ATP8/applesson.htm

How to write the Equation of a Transformed Graph

Bonjour tout le monde! :)

This is Vera, (btw it's VEH-RA not VEE-RA) and I sit in the front table between Allen & Macy and across Jim & Maye. 

          Last Friday we had our Unit Test about Permutations, Combinations and Binomal Theorem and hopefully we'll get the results tomorrow... 

          Also, we learned  "How to Write the Equation of a Transformed Graph" 

Here's an example:

 Absolute Value Function

                  To determine the equation of the of the g(x):

• First, let us write down the coordinates of the f(x) function:

                     y = f(x)                

                      (1, 1)          

                      (2,2)            

                      (3,3)           

                      (-1,1)              

                      (-2,2)      

•      Next, is to determine first if there are reflections, stretches and translations. 

                    

• If we look closely in the graph of g(x) we could see that there are no reflections and there are no translations because the graph did not move to the left or right, and up or down. so the only transformation we have in this graph are stretches.

• We could see that the graph was compressed vertically by a factor of (a) which is 3 units, meaning that the x- values will be multiplied by the reciprocal of 3 which is 1/3.

                                                 y = a f (b(x + c)) + d

                                                                Let us substitute 3 in to the equation

                                  y = a f (3(x + c)) + d

• Since there are no other transformations in this graph the final equation will be: g(x) = 3|x|
• Using mapping notation we could get all the y-values of the function g(x)= 3|x| by multiplying all the y-values of the function f(x) by 1/3.
• All x-values remain the same!

                      f(x)                                        = g(x)

                       y values                                   x, y

                      (1 x 1/3 = 3)                          = (1,3)

                      (2 x 1/3 = 6)                          = (2,6)

                      (3 x 1/3 = 9)                          = ( 3,9)    

                      (1 x 1/3 = 3)                         =  (-1,3)        

                      (2 x 1/3 = 6)                         =  (-2,6)

If you look closely, the coordinates we got from our equation was exactly the same as the points in the given graph, therefore our equation is correct!

That's how you get write an equation from a given graph! I hope you'll learn something. :) 

TBH: I don't get this topic as much as the other topics we had and I'm very unlucky to write about a topic I don't understand really well :< 

THAT'S ALL FOLKS! 

VERA,

REFLECTION AND STRETCHES

Hey guys!! I'm Macy and I sit at the table in front of the class with my 4 other "friends". First off, I hope we all did well on the quiz we did yesterday. We also learned 2 new topics yesterday, REFLECTION and STRETCHES!

REFLECTION

STRETCHES

• has 2 types: Vertical and Horizontal stretches

1. Vertical Stretches

          -  the stretching of the graph away from the x-axis.
        - if the original (parent) function is y = f (x), the vertical stretching or compressing of           the function is the function a f(x).

• if 0 < a < 1 (a fraction), the graph iscompressed vertically by a factor
  of a units.
• if a > 1, the graph is stretched vertically by a factor of a units.

  

2. Horizontal Stretches

        - the stretching of the graph away from the y-axis. 

- if the original (parent) function is y = f (x), the horizontal stretching or            compressing of the function is the function f (ax).

if 0 < a < 1 (a fraction), the graph isstretched horizontally by a factor of a units.

if a > 1, the graph is compressed horizontally by a factor of a units. Here is a link that will help you understand REFLECTIONS and STRETCHES more:  Transformations Summary: Stretches, Reflections, Translations I hope I helped you understand more how to deal with Reflections and Stretches. If I wasn't able to help then I suggest you to ask Mr. Piatek. :) Oh and since we're all great mathematicians.. Here's a (lame) math joke to end my blog post: Q: What is the definition of a polar bear? A: A rectangular bear after a coordinate transformation.

Introduction to Basic Transformations: Translation of Functions

Hello friends! I hope you all enjoyed your weekend—don’t you think it was too short? Hah. Anyway, this is Kristiana and I shall attempt to summarize what we've learned in class today. 

In case you were falling asleep (not that I was) in the middle of the lecture, Mr. P was talking about Polynomial Functions. If you were really paying attention, you'll have realized by now that I was only kidding. 

Today we talked about Basic Transformation, which marks that beginning of the second unit. Basically, it's a recap of what we've done during the previous grade. I hope you all remember the meaning of function, which is for every x value, there is only one y value (...or something similar to that). In other words, y = f(x). 

Moving on, one technique that we've learned in order to figure out if something is a function or not is the use of table of values. We can also use a vertical line test. If the line intersects the graph twice, then it is not a function. 

y = x²

This graph can be considered a function because the lines only intersect once. 


This table of values could also be used. Notice that y can be repeated multuple times but x can only be used once. 

x	y
-2	4
-1	1
0	0
1	1
2	4

Mr. P also went on about stretching—not our bodies, but instead graphs! Transformation is the act of moving, flipping, stretching or compressing a base graph. 

When we graph quadratic equations, we are really translating (moving), flipping (reflecting), and stretching or compressing it. Don't worry if it doesn't make sense just yet, because today we only had time to discuss translation.

Translations are basically the horizontal and vertical movements (or shifts) using the basic shape of the parabola. Easy, right? 

Here are a couple of rules to remember:

• Horizontal Translation: affects the x values only
  y = f(x-h) : the graph shifts h units to the right (towards the positive side)
  y = f(x+h) : the graph shifts h units to the left (towards the negative side)
• Vertical Translation: affects the y values only
  y = f(x) + k : the graph shifts k units upwards (towards the positive side)
  y = f(x) + k : the graph shifts k units downwards (towards the negative side)

*IMPORTANT NOTE: Read h values as opposite and k values as they are. 
 e.g. y = f(x-3)
        h = +3 
        y = f(x) - 6
        k = -6

This is an example from the booklet we did in class:

Example 1b) implies vertical translation (since the format is f(x) + k), which means that only the value of y changes. Therefore all the points in the basic parabola will shift upwards by three units. In the table of values (for those of you that prefer using it), you will add 3 to the y-value: 

4 + 3 = 7, 1 + 3 = 4, ...etc. 

Example 1c) suggests horizontal translation, so only the values of x change. Since f(x-3) contains h, and remember, we read it as the opposite value. The basic parabola will then shift three units to the right (towards the positive side). And of course in the table of values, you will add 3 to the x-value: 

-2 + 3 = 1, -1 + 3 = 2, ...etc. 

That's pretty much it, I suppose? The topic today was sort-of mundane because it's very basic and general, but it's important to get the main idea because it'll play a big role from now on until the end of this course. If you have any questions or if you're still confused, feel free to either look over your notes from Grade 11 Pre-Calculus, ask me, or ask Mr. P (probably the best choice). 

Reminders: 

• Always identify if the question is asking for either the h or the k. 
• Remember that the h value is read as the opposite and the k value is read as it is.
• Vertical Translation: affects the y values only
• Horizontal Translation: affects the x values only

Don't forget to do your homework! It's important to keep informations fresh in our minds so that we don't forget them quickly. Oh, and our Permutations, Combinations and Binomial Theorem Assignment is due tomorrow, so make sure y'all don't forget to finish that. We also have a test on that topic this week. Good luck and may the best mathematician thrive. Haha. 

Yours Truly, 

Kristiana 

Binomial Theorem - Finding Specific Terms and Coefficients

Greeting's everyone, and Happy Valentine's Day! I'm Lyka, and if you don't know me, you can find me on this area of the class...

I'll be continuing what Maridel discussed yesterday in her blog about Binomial Theorem! BOOYAH!

Previously, we learned about binomial expansion.

In order to find a specific term, we can use binomial expansion (which would most likely be tedious if we are given a big number for a power), or we can use this formula:

For example, in one of the examples we encountered, we were asked to find the 14th term of (x+y)¹⁶. 

• First, we listed down what we can plug into the formula above! 
• Plug in what you have to the formula and... voilà! You found the 14th term! WOOHOO! 

And that is it! YAY! But, just to remind you that, in the provincial exam, you may encounter questions that ask you to find the coefficient of the term, or what n is, and it may mess you up! So to help you out, keep the following points in mind:

• Read the question. Out loud. Sometimes how the question sounds like in your head does not actually sound that way.
• See how much you could do within the time frame of calculating it. (For example, with calculating the coefficient of a term, you could either write down Pascal's triangle to find it, or use binomial expansion.)
• Be flexible with the formula in finding what they're asking for. YEAH!

WOOT WOOT! You made it to the end of this post!

Remember...

Peace like a river,

Lyka

The Binomial Theorem

Hi everyone! My name is Maridel and today I am going to summarize what we learned in class today which is... 
The Binomial Theorem.

We learned how to expand and simplify using binomial expansion.
 
Firstly, let us expand and simplify the following:








Do you notice any patterns?
As you can see, when you look at the exponent for each binomial and +1, that is the number of terms when the binomial is expanded and simplified. 
So for example, if







 

The exponent in this case is zero. Zero plus one is equal to one and therefor you end up with one term.

Also, when expanded and simplified you will notice that this value of exponents for x decreases as the value of the exponents for y increases.
For example, if

 

The exponents of x decrease each term. As you can see the exponent of x goes from 6, to 5, to 4, to 3, and so on.. While the exponents of y increases each term. The exponent of y starts off at 0. Then it goes to 1, to 2, to 3 and so on..


The coefficients also follow a pattern! If you compare pascal's triangle to 
On the left, is Pascal's triangle. As you can see, the expanded and simplified binomials follow the same pattern. In row 1 we start off with 1. In the next row according to pascal's triangle there are two 1's and the picture on the left justifies pascal's triangle because the coefficient of both x and y are 1. You may also notice the hockey stick pattern! The numbers that are highlighted created a hockey stick shape. When the numbers on the long end of the hockey stick are added together, the sum is equal to the number on the short end!
 
Next, let me introduce the formula:

 

Let's try an example now, shall we?
Expand and Simplify using the binomial expansion:





First things first, start off by listing the values: 
a= 3x       b= -y      n=5 ... n is equal to 5. Just remember that in each term, the exponents must add up to the value of n. And in this case, the exponents in each term must add up to 5! So whether the exponents are x to the power of 1 and y to the power of 4, or x to the power of 2 and y to the power of 3.. as long as they add up to five you are doing perfectly fine!!
*NOTE: Don't forget the negative signs!

 Next, use the formula and plug in the values. By doing this you are expanding.


  
And now, lets Simplify! Remember that anything to the power of 0 is equal to 1!





And we are left with:



I recommend that you always, always, ALWAYS double check your answers.

... And that's it! Hopefully you now all understand the concept of binomial expansion. Have a great day! and don't forget...