Tuesday, February 27, 2007

The Time Has Come...

Yes everyone, the time has come. im finally blogging ( for the first time in my entire life). Since grade 10 i've always been able to just avoid doing this blogging thing, but, not this semester. So please be nice if it's awful. Oh yeah the scribe for tomorrow will be...


...anounced after my scribe


Morning Class:
Our class started off without me there unfortunately, but none the less it went on anyway :'(. Mr. K and the rest of the class was praising Chris' amazing scribe, ( thanks for setting the bar really high for the people who have never done a scribe before). Also Danny had been choosen to be scribe for the day, and i heard he was crying. luckily KADEEM SAVES THE DAY! and arrives late, after Sandy and Albert, but prior to Miles.

Back to business. Mr. K started us off with 10 questions, with the first 5 we were given a function y=f(x) and a point on the graph (-2.-3) and told to find the coordinates of it's image on the graph.So basically what your doing is finding the X or Y coordinates using f(x).

Here's a few examples:
y=f(x) + 2 --> shows this function being translated up 2 units, so the original point (-2,-3) is moved up 2 units, giving us a new point at (-2,-1)


y= f(x+3) --> shows the function being shifted to 3 units to the left (8)"to the left, to the left"(8). Ha ha. so you use the point (-2,-3) // (x,y) and put it int he function giving us the point (-5,-3)

If your
not understanding that lets just put it this way:

***REMEMBER STRETCHES BEFORE TRANSLATIONS***

OK let's say the original value was (-2, -3) for example. Ha ha ha. First you add the X value to the coordinate than you do the same to the Y value and you get your answer. Remember ADBC. (Stretch, translation, stretch, translation). First you check if f(x) has any stretches and multiply accordingly, then you add/subtract to figure out the location of the x/y coordinate.



The next set of questions Mr. K assigned we were given the final point of the functions prior and told to find the original function.

For example:

Function: y=f(x+3) Final Coordinate: (-3,7) Original Coordinate: (0,7)

When your trying to figure out what the original coordinates were, first of all you have to add/subtract the stretches or else the answer will be wrong, then you multiply/divide to figure out what the original coordinates are. For example if you have the coordinates (3, -2) and are using y= f(2 ( (x+1) + 2 you first have to use translations to find the X and Y values before the stretches, in this example that would be (4, 0) and then you use stretches dividing/multiplying to figure out what the final value is which is (2, 0).

The Next thing we learned was:

Even and Odd Functions



Even Functions

A function is "even" its graph is symmetrical about the origin
or
IFF (If and only if) ( f(-x) = f(x) )














f(x) = x²
f(-x) = (-x)²
f(-x) = x²
Since f(-x) = f(x) the function is even






Odd Functions

A function is odd if you are able to rotate it 180 degrees and it would be the same graph
or
IFF f(-x) = -f(x)



ex.
f(x) = x³ - x
f(-x) = (-x)³ - (-x)
f(x) = -x³ + x
Not an even function

-f(x) = -(-x³ - x)
-f(x) = x³ + x
Since f(-x) = -f(x) the function is ODD!


There is a few more examples of these even and odd examples in Mr. K's slide posted today and in our exercise ( go over them and practice it). So that's was the end of our morning class... all i can remember was my head spinning.


Afternoon Class:
The afternoon class, i knew what was expected of this class already since Mr. K said he would be leaving half way through the class, but not until he taught us this reciprocal thing, and you know what that meant (rush and cram) but fun, and this is what we learned:

We found the reciprocals of the following numbers:




DOCTOR SUESS' PRE CALCULUS CLASS
1 2 4 10 100 1000 1000000 0.5 0.25 0.1 0.01 0.0001 0.0000001
then we found the reciprocals of...
-1 -2 -4 -10 -100 -1000 -1000000 -0.5 -0.25 -0.1 -0.01 -0.0001 -0.0000001



And because Mr. K always reminds us that mathematics is the sign of patterns, we observed the following:

  • As the positive numbers got bigger their reciprocals got smaller, and as the numbers got smaller their reciprocals got bigger
  • As the numbers were "biggering negatively" their reciprocals were "smallering negatively" and as the numbers were "smallering negatively" their reciprocals were "biggering negatively"




The Graph shown above is (x+2) and its reciprocal 1/(x+2)
the purple circles indicate the invariant points. The invariant points are points that don't move on the graph,the points where the function and its reciprocal meet/touch.
As you can see if the function is biggering negatively, the reciprocal is smallering negatively and vice-verse.


And last but not least how to draw the reciprocal when you know the function already, you can see it here.


So all though we've come, to the end of the road, i am so very excited to let go. I'm sorry my scribe went up so late this honestly was the hardest thing I've ever done and it's still pretty bad because i am now half asleep haven't even gotten to study for my AP Chemistry test tomorrow. With that said, tomorrows scribe will be...



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1 comment:

Ricky said...

YAY.. Great Job KADEEM!!! You did well for your first scribe. There's a bit of minor graphical errors but all is well in Scribeville. Ha Ha Ha. You posted it really well and you did a great job summerising the classes. Excellent work. =D