Sunday, March 4, 2007
____ Well, I'm BOB-ing again! This time, it's for the second unit, TRANSFORMATIONS. In this unit, we meticulously discussed the "translations" and "transformations" of different functions. In the first transformation, we learned that 'a' in the standard form f(x) = af(bx) controls the vertical stretch or compression of the function. It alters the y-values of the coordinates of a point. If a is greater than 1, then the y-values are multiplied by that value and stretches vertically. If a is greater than 0 and less than 1 (or a fraction), the function compresses vertically. 'b' in the standard form f(x) = af(bx) controls the horizontal stretch of the function and alters the x-values of the function. If b is greater than 1, the function compresses horizontally and "speeds up". Mathematically, the x-values are multiplied by the reciprocal of the b values greater than 1. On the other hand, if b is a fraction, the function stretches horizontally, and the function's x-values are multiplied by the reciprocal of those fractions. Therefore, the graph "slows down". The second transformation we learned in this unit is the shifts. 'a' in teh equation y = f(x-a) + b alters the x-coordinates of the function and determines the new coordinates of the function by shifting it a units horizontally. 'b' in the equation y = f(x-a) + b alters the y-coordinates. It determines new coordinates for a point by shifting the fucntion vertically. If it's negative, fucntion shifts b units down. If it's positive, fucntion shifts b units up. The same thing goes for a. If a is positive, function shifts a units to the right, and negative a units to the left (shown as (x+a)). Overall, I believe that this unit is easier than the first unit. Although, there has been a few times where I keep forgetting the phrase, "STRETCHES BEFORE TRANSLATIONS". But now, I know what to do. Also, we learned a few ways on how to transform a graph. Either you start with the x-coordinates first then the y-coordinates OR you can simply do the stretches of x and y coordinates, then their shifts. In short, do whatever is comfortable! So, days pass by our knowledge about transformations increases. We learned how to graph reflections over y and x axes, and inverse functions where the graph is reflected over the y=x line, by switching the x and y values. We covered EVEN FUNCTIONS where the graphs are symmetrical over the y-axis, and ODD FUNCTIONS where the graphs are symmetrical about the origin. Odd Functions will be the same if the graphs are rotated 180 degrees. They are determined algebraically by replacing -x where x exists. Graphing RECIPROCAL FUNCTIONS was a tricky one. This time there are terms that we had to know such as INVARIANT POINTS (points that never move such as 1 bcz the reciprocal of 1 is itself). and ASYMPTOTES (where the function crosses the x-axis; the reciprocal function cannot cross the asymptotes). But, as we did more practice with them, I grasped the concept for sure! Absolute Values Functions was easier! All you have to do is re-draw the negative part of the graph above the x-axis because of the mathematical reason that y = x. Therefore, all y-values or OUTPUT must be positive. Those that are positive already remain where they are, but those that are negative change their signs to positive, thus, reflects themselves over the x-axis. Well, this was fun! HAHA! In general, I think this unit is easy and fun bcause of the moments where we could all draw on the PRO SMARTBOARD! See how technology evolves and aids us in our daily lives! Gotta love it! Pre-test will be on Tuesday? I think and there will be another group work in there too, definitely. This will help us a lot too! All I have to do is do the exercises which I seemed to have neglected! Now, I'm done! GOOOOOOOD LUCK!