Monday, February 26, 2007

Lost in translation...and stretching.

ScribeBadge11Hey everybody! I'm going to be scribing instead of Kadeem today since he could not do so. And since he told me he was going to choose me tomorrow, I offered to switch with him so that things would work out efficiently. So I'm saying it now, and I will say it again later, tomorrow's scribe is Kadeem.

Friday Morning:
I was not in class on Friday morning, as I had to go to "Doors Open," an extracurricular program for history class. But despite my absence, I asked around and determined that we didn't learn anything in the morning. All we were told was that exercise nine was assigned for homework over the weekend, but we weren't taught about stretches yet so we could not complete the exercise. So basically, the entire class was spent working on other subject work or "working" on the smart board. Nothing new was learned in this class.

Friday Afternoon:
I was present for the afternoon class, but then again there really wasn't any point to be there. We again didn't learn anything or accomplish any work, so we were either completing work concerning other subjects or (as the majority of the class did) played hangman about "math" on the board. Yeah...that was the main topic...math. Well, that was what the afternoon pretty much consisted of. But now on to where the most recent information that he taught us.

Weekend:
I know it's odd that we actually learned something on the weekend, but it's ultimately true. Late Saturday night, Mr. K posted on the blog a post which contains links to helpful sites, a java tutorial applet and a video lesson he created for us. If you did not get to see it, here's a direct link to his post here, or just scroll down (I think the latter requires more effort =p).

A summary of the movie:
In the video he taught all about compressions, stretches and reflections (all of which were elaborated upon in today's class). He began by stating that we've went over that any function, y=f(x) is where "the output is something done to the input." But if we alter the output, such as by multiplying y=2f(x), we alter the y-coordinates of the function. He later elaborated upon this in the video, and in class, and I too will later elaborate upon this as well. After he showed how this works he went more into detail about how vertical stretch will occur if the value of that number is greater than 1, and will vertically compress the graph is less than 1. Then we moved onto altering the input of the function, such as y=f(2x). This alters the x-coordinates of the function, but not by the value given, but by the reciprocal of that. In the function y=f(2x), the graph is not stretched horizontally, but is compressed horizontally, as it makes things speed up. This was later explained in today's class. Once he was done explaining vertical and horizontal stretches/compressions, he went into reflections:

y=-f(x) <-- This function will be reflected over the x-axis, as the negative value is changing the y-values of the function. That negative sign is -1, so if you multiply all y-values by -1 the graph will appear to be flipped over the x-axis.

y=f(-x) <-- The same fits for this function, but it will be reflected over the y-axis. The negative 1 is multiplied by all the x-values of the function, causing them to be flipped horizontally over the y-axis.

He then ended it by saying that inverse functions will be explained in class, which was later proved to be true.

Monday's Class:
Today's class, we learned a lot useful and very interesting information about our new unit. I came into class and I saw that Mr. K had the aforementioned video up on the smart board, and I naturally assumed that we were going to go over it. This was true. So we began the class by watching the video, as much of the class did not get to watch it over the weekend. He paused the video at certain points and elaborated upon and fixed what he said in the video. The first time he paused it was to explain stretches and compressions.
When the function y=f(x) is altered by adding a value to multiply the output, in the case he showed us, y=2f(x), we can find the new y-values of the function easily. The 2 in front of the function alters y-coordinates, so all we must do is multiply the y values in the original graph by 2 to find the new functions y-coordinates. This causes a vertical stretch, and he showed this stretch by using the following graph:


In this graph, the black line represents y=sin(x).
The red line represents y=2sin(x).
The graph shows how once the graph was transformed by multiplying all y-values by 2, the graph was stretched vertically. But if the value is decreased, say to less than 1, such as 1/2, how will the graph look?

In the graph, the transformation is compressed vertically, which shows what happens to the y-values of the original function when the output is multiplied by a number less than 1. The graph becomes compressed, and the y-values are multiplied by, in this case, 1/2, which causes the new function to have lower y-values, causing it's compression. But this is not the only type of stretch that he showed us, as he also showed us horizontal stretch.

The graph now shows y=sin(x) compared to y=sin(2x). The graph is compressed, and Mr. K went in depth into his explanation of why this is. He explained that:

  • In this case, you're doubling the input before it was calculated by the sin function. Before you do this, you multiply the x value by 2, doubling the original value before you find the sin.
  • This makes the function happen faster, as you're multiplying the value before it is input into the function.
  • To determine the x-values of the transformed function, you multiply all x-values by the recipricol of the value. In this case, to find any x-coordinate in the new function, you must multiply the original functions value by 1/2.

This graph shows what happens to the function y=sin(x) when you transform it to y=sin((1/2)x). In this graph, everything is slowed down, and if you take a look at the x-values of the original function, they appear to have been doubled. The graph is stretched horizontally, and as detailed before, to find the new x-coordinates of the transformed function you must multiply the original values by the recipricol of the value.

Each of the above graphs help portray how functions are stretched and compressed and then once we were shown all of these graphs, we were given a general equation to represent these parameters. In general we see:

y=af(bx)

The value of a represents the vertical stretch of the functions transformation. The value of a is used to multiply the y-values of the original function to find the new function. A alters the y-values of a function.

The value of b represents the horizontal stretch of the functions transformation. Though, the value of b is not the value used to determine the new function's coordinates, we must use the reciprocal of b. We must do this since b alters the values being input into the equation before it is actually calculated. For example, in the function y=sin(x), the transformation of y=sin(2x) causes the values of x to be doubled before it is calculated into the function of sine. In order to counteract this, we must use the reciprocal so that the values will end up equivalent before going into the sine function. In this case, if x=Π/2.
y=sin(x)
y=sin(Π/2)
y=1
But when the graph is transformed to y=sin(2x), our y value will end up being 0 instead of 1. This is because the value of b multiplied the x value before it went into the sine function. So if we take the reciprocal of 2, which is 1/2, and multiply x by the reciprocal, then our answer will balance out before going into the sine function.
y=sin(2x)
y=sin(2(1/2(Π/2)))
y=sin(2(Π/4))
y=sin(2Π/4)
y=sin(Π/2)
y=1
So remember, the value of b is not what we're multiplying our x-values of the original function by, we're multiplying by the reciprocal.

Then we partially reviewed translations from previous classes:

y=f(x-a)+b

The value of a represents the horizontal slide of the function.

  • If a is greater than 0 then the function is shifted to the right a units. In this case, the x-coordinates increase.
  • If a is less than 0 then the function is shifted to the left a units. In this case, the x-coordinates decrease.
[Don't forget to pay attention to the sign of (x-a)]

The value of b represents the vertical slide of the function.


  • If b is greater than 0 then the function is shifted up b units. In this case, the y-coordinates increase.
  • If b is less than 0 then the function is shifted down b units. In this case, the y-coordinates decrease.

After we finished reviewing translations, we then worked on graphing some of these newly taught stretches and compressions, as well as translations. He showed us a graph on the smart board, and then we had to alter it accordingly. These questions can be found on slide 2 here. The last question was composed of both stretches and translations, and from this question he taught us an important rule:

[Stretches come before translations]

Always work out stretches before translations. Well, now that all that was done, he quickly taught us about inverse functions.

He began by outright telling us that an inverse undoes whatever has been done to something. He then contrasted this to the "inverse" relationship between his daughter and his wife and himself. Every morning, his daughter goes into her clean room, then she transforms the room into a messy room. At night, the super inverse parents come into the messy room to clean up the mess, and trasnform the room back into a clean room. He showed a diagram of this:


Then we went more into the work involving the inverse function. The -1 above the inverse function's F does not mean it's to the exponent -1, that would be written as [f(x)]-1 . An inverse function "undoes" the transformations of a function through a routine, and the function of a regular function switches y and x values.

f --> f-1

(a,b) --> (b,a)

If you consider Mr. K's example, the messy room is the domain of f(x), and the clean room is the range. In other words, in the function f(x), the messy room represents all the x-coordinates, while the clean room represents all the y-coordinates. In the inverse of the function, however, we switch the messy and clean rooms around. By doing this, we're switching the domain and range around, which is exactly what happens in an inverse function.

(a, b) = (messy room, clean room)

(b, a) = (clean room, messy room)

He then showed us the inverse function of f-1(x) through different methods.
Algebraically
The inverse of a function can be determined algebraically. In the equation, switch the x and y variables, then solve for y to receive the naturalform of a function. We must due this so that we can graph it. Consider the function f(x).
f(x) --> y=√2x3-4

f-1(x) --> x=√2y3-4

x2=2y3-4

x2+4=2y3

x2+4=y3

y=√((x2+4)/2) <-- we are now able to graph this.

Conceptually
We can also display the inverse of a function by undoing whatever has been done to the function. Whatever order is taken to transform the function, the inverse will undo these transformations by reversing the order and reversing the operations. An example is shown in here on slide 3.

Graphically
In order to graph the inverse of a function, you must graph the function then exchange all x values with all y values. You can also check here on the third slide to see an example of a graph, or for any of the inverse information.

And that concludes my scribe post for today. To reiterate my earlier statement, tomorrow's scribe is:

KADEEM
I know Craig chose him last time, but me and Kadeem "switched" our scribe posts so thing can work out better, since he was not able to today. Good night everyone, hope everyone understands our unit so far. Good luck in the rest of the unit and see all of you guys tomorrow!



12 comments:

Mr. Kuropatwa said...

SiWwy, you are awesome!

For the level of detail, summarizing not only this class but the "informal weekend class," for the graphics and meaningful use of colour and the well written text I nominate you for the hall of Fame!

WOW!

John D. said...

MrSiWwy,

This is one heck of a scribe post. I loved the amount of detail you put in your scribe post, and the good use of colours to emphasize several points. You got my vote for the Scribe Post Hall of Fame.

e said...

MrSiWwy,

Nice job. I got it all, except for one little thing. I'm racking my brains, but I don't get it. Help, please:
The value of b represents the horizontal stretch of the functions transformation. The value of b is converted to it's recipricol, and then we use this recipricol multipled by the original functions values to find the new functions values. B alters the x-values of a function.
The last sentence doesn't seem to be saying what the second does (which is the confusing one to me). Why would you need to multiply b by its reciprocal? And why do you need to multiply the function values, when you say that b alters the x-values?

Otherwise, I support the nomination.

MrSiwWy said...

Thanks everyone for your input, and I'm glad you liked my post.
And towards e, I'll go back to change that accordingly now. Thanks for the feedback!

Tim_MATH_y said...

Detail, images, explanation, length, everything. I nominate this post to the Scribe Post Hall of Fame.

Great Effort!

KaDeeM said...
This comment has been removed by the author.
Ricky said...

I nominate this post into the hall of fame b/c Mr. k said that it is a prime example of what a scribe should look like and I agree with that proposal indefinitely.

Lani said...

Hi MrSiWwy,

May I add a 6th Hall of Fame worthy?!

Kudos to you for your command of the language and your detailed explanations!

Best,
Lani

VincentR said...

MrSiWwy meticulously depicted the concept of the lesson and enabled the bloggers to comprehend the entire lesson without being in the class or listening to the podcast. GOod job! Indeed, your work deserves to be recognized in the Scribe Post Hall of Fame!

Jojo Rocks said...

I just got around to see the recent posts and I noticed yours Chris. Setting the bar even higher. Best scribe post ever. Someone put this in the scribe hall of fame already!

got MELk said...

Hey silly man, I just wanted to say thank you for making such a good scribe post.I forgot some things from the unit and I'm so thankful that you put detail and effort into your scribe that it helped me review.

Thanks,
Mel

«Craig» said...

Mr. SiWwy,

WOW! This scribe post is amazing, the detail is superb and the use of visuals is to a tee. Hands Down a Hall of Fame inductee. I think you deserve more than just a nomination, maybe a standing ovation. *applause*

Thank You for your excellent effort.