Tuesday, February 19, 2008

Graphing Sine and Cosine Functions

Hello, Francis here with your daily school scribe. Now I'll introduce you to the beginning of class. Let's bring it back.

Mr. Kuropatwa firstly showed us a new feature available on our blog. It's a translation feature that can translate this blog into one of many different languages. Brought to you by Google.
Mr. Kuropatwa also stated that the circular functions test will be sometime in the middle of next week. Which means that it would be a smart idea to start on your BOB posts.
Starting tomarrow there will be a substitute teacher and same goes for Thursday, because Mr. Kuropatwa won't be showing up for class on those days.

All of these functions that we graphed today are based on patterns and as said by Mr. Kuropatwa "Mathematics is the science of patterns". Okay now with today's lesson which was completely made up of graphing those sine and cosine functions, that are ever so interesting. As seen on slide 1 we started off class by graphing the sine function: y = sinx - 1 and on slide 2 the cosine function: y = 2cosx, then we stepped up to a slightly harder function found on slide 3 and I suggest you see how all that turned out, so check out that slide show. When graphing any of these functions you should remember a simple pattern: "1,2,3,4" where 1 is the maximum point, while 2 is the average point(between the max. and min. points) and 3 is the minimum point and 4 is the average once again. Remember when graphing one of these functions you should put arrows at the end, pointing in the direction the function is about to head to next, so if it's at the max point, and it has nowhere to go but down, you should end the line with an arrow pointing downward. Also remember not to put sharp points when you change direction from up to down or down to up, put curves, because if you put sharp peaks, then be prepared to lose your precious marks.

The general equation for these functions is: f(x) = Acos(x-C) + D or
f(x) = Asin(x-C) + D.
When graphing these functions from these equations remember the word: DABC. This work or phrase is like the BEDMAS of graphing. It's the order in which you should always graph.
Now I will expose what these letters actually mean.

Firstly with D: is the vertical movement of the curve itself, it determines the sinusoidal axis(the axis between the max. points and min. points, its the average value of the curve.

Second is A: This is the amplitude it determines the max. and min. points of the curve on either side of the sinusoidal axis. The graph is stretched if the value of A is: |A| > 1 and its compressed if the value of A is: 0 < |A| < 1.

Third is B: this determines the period of the graph with an equation found on slide 18. If the value of B is negative then the graph curve will start with the pattern being 1,2,3,4. Where 1 is the min. point and 2 is the average point, and 3 is the max. point and lastly 4 being the average point.

Last is C: This shifts the curve left of right (horizontally). This is also called the "Phase Shift". If the value of C is negative(C <> 0) the curve shifts to the right.

For the sake of summarizing I will say this:
D - Lay down the sinusoidal axis.
A - Stretch of the amplitude.
B - Label the x-axis.
C - Look at the scale and slide the curve accordingly.

All of the details are found on slides 17-19. I suggest you check it out.
The sine and cosine curves are quite similar with the difference being that when at zero, cosine = 1 and sine = 0. But if the sine curve is slid to the right by pi/2, then at 0 sine = 1. With this, you can state that cosx = sin(x +(pi/2)) or sinx = cos(x-(pi/2)).
Well thats the 411 of today's class, hope it was helpful. The next helpful person to be voluntold(forced to do it for free, hence "volunteer that is told") is Paul.
That's me, the one and only, Francis! Until next time, I'm out.

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