We began class by taking a look at Elazar's majestically written Scribe post. After Mr. O'Brien made a slight adjustment, we finished going over O'B's version of problem 77 on the IW. Afterwards, the warm up was put up on the board. Our instructions were to sketch f(x), and to find certain limits of the equation without using technology (shown below):
a) Sketch f(x), showing all features
b))
c))
d))
Once the class began to understand the task ahead of them, whispers of panic began to emanate across the room. "How do we factor cubics?" asked Connor, "Without technology?!" whimpered Margaux and Weston. The class struggled to remember how to find the domain of a function, and it seemed as if panic was about to set. Soon, however, answers began appearing on the board with the help of a little technology. Once we had established the correct answers, Mr. O'Brien showed the class how to find these without technology. We knew that it was necessary to factor f(x) to do this, but it wasn't until Sarah suggested factoring by grouping that we were able to do it. After a little magic, our equation looked much simpler:
We realized that the graph of this function could be sketched by finding asymptotes in the function, holes that it may have and x and y intercepts. After canceling, we simply plugged in -1 for x into our equation to find a hole at (-1, 2/3). In addition to this, we knew that the function had 3 asymptotes, y= 0, x= -2 and x= 2. All of this information led us to find this graph:
In red is f(x), and the asymptotes at x= 2 and x=-2 are also shown. There is also hole at (-1, 2/3) not shown in Geogebra
We patted ourselves on the back for a job well done, and found the answers to the remaining questions without the use of technology: b) 2/3 , c) D.N.E, d) -∞ and e) 0. We discussed our semi-new topics, which were limits that approached infinity and writing limits with infinity. We then went over a few examples:
ex/
As we can see from the graph of f(x) as x approaches 4 from the positive side, the limit of the function approaches ∞.
This example is the same as the previous one except we are approaching 4 from the negative side, the function is going up the other way on the asymptote so the limit is -∞.
The easiest way to determine what the value of this equation is as it approaches infinity is to make an end behavior model. As x gets infinitely large, the +1 and the -5 become irrelevant. Our model then becomes 4x/7x, and the x's cancel leaving 4/7. As we all know, this equation flatlines, so the answer to this problem is = 4/7
Similarly to the example above, the easiest way to solve for the limit as it approaches ∞ is to make an end behavior model. The +1 once again becomes irrelevant, and the rest of the equation simplifies to x, so as x reaches infinity, the equation reaches = -∞
For a very quick reminder of what end behavior is, and some pointers on how to determine the end behavior of a polynomial, click this link! http://www.mathwords.com/e/end_behavior.htm
As we were running out of time in class Mr. O'B pointed us in the direction of some Limit Rules (p.71) and an exploration (p.72)
In the last couple minutes of class, Mr. O'brien went over the exploration on p. 72. This exploration basically consisted of showing that two functions whose limits approached infinity individually might not exist, but when multiplied, divided added or subtracted to each other can actually change the limit.
The Homework for Friday's class is pg. 76/ 3,7,15,27,35-38. 53. 55. 59. 61.
Lexi will be the next scribe!
Update! I stumbled upon a cool website with a review of a lot of the things we have reviewed this year. I found some of these very useful when reviewing old topics, or understanding new ones.
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