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April 7th, 2017, 02:39 AM   #1
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help with ideas, explaining graphing rational functions

I'm beginning to tutor some college students in algebra, and I'm looking for help with ideas to explain graphing rational functions.

I know all the procedures and rules for graphing rational functions, and my students have often memorized the rules, too. But they don't make sense to some students. I want to be the kind of tutor who gives them a glimmer of insight into the patterns in the math and doesn't just teach the rules by rote memorization.

So for example, consider the horizontal asymptote in the end behavior of

$ \frac{2x+10}{3x-20} $

The students sometimes already know the rule: the degree of the numerator equals the degree of the denominator, so take the ratio of coefficients on the leading terms of the numerator and denominator.

However, I understand this personally through an intuition (as I'm sure most members of this forum do). I think of $x$ getting very large, and it's clear that the constant terms of each polynomial are dwarfed by the terms of degree greater than 0, so for very large $x$ the polynomial starts to behave like

$ \frac{2x}{3x} = \frac{2}{3} $

But sometimes the students don't respond to this notion of "$x$ going towards infinity." Myself, I think of a graph and imagine running my finger along it to the right or to the left. I imagine going very far. I imagine the polynomials in the numerator or denominator "getting very big" as I "run my finger toward infinity." But I have had students who didn't understand this principle.

Another phenomenon is a vertical asymptote. I imagine the denominator "going to" zero and the fraction blowing up.

I'm just wondering if there is a way to explain this notion of "going to zero" or "going to infinity" that makes sense to beginning algebra students.

Mike

Last edited by skipjack; April 7th, 2017 at 04:21 AM.
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April 7th, 2017, 03:43 AM   #2
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Algebra huh??

You could try making them do the long division and see that the 2/3 comes out naturally with a rapidly diminishing inverse power series in x as a remainder.


$\displaystyle \left( {3x - 20} \right)\mathop{\left){\vphantom{1{2x + 10}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{2x + 10}}}
\limits^{\displaystyle\hspace{61px} {\frac{2}{3} + \frac{70}{9x} + \frac{1400}{29x^2}}}$



Edit

Can anyone tell me why the mathml doesn't work?
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Last edited by skipjack; April 7th, 2017 at 05:12 AM. Reason: to use hspace instead of hfill
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April 7th, 2017, 05:24 AM   #3
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$\dfrac{2x + 10}{3x - 20}=\dfrac{2 + 10/x}{3 - 20/x}$

As $x\to\infty$, the $10/x$ and $20/x$ become arbitrarily small and the fraction becomes arbitrarily close to 2/3.
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April 7th, 2017, 05:37 AM   #4
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Quote:
Originally Posted by mike1127 View Post
I'm beginning to tutor some college students in algebra, and I'm looking for help with ideas to explain graphing rational functions.

I know all the procedures and rules for graphing rational functions, and my students have often memorized the rules, too. But they don't make sense to some students. I want to be the kind of tutor who gives them a glimmer of insight into the patterns in the math and doesn't just teach the rules by rote memorization.

So for example, consider the horizontal asymptote in the end behavior of

$ \frac{2x+10}{3x-20} $

The students sometimes already know the rule: the degree of the numerator equals the degree of the denominator, so take the ratio of coefficients on the leading terms of the numerator and denominator.

However, I understand this personally through an intuition (as I'm sure most members of this forum do). I think of $x$ getting very large, and it's clear that the constant terms of each polynomial are dwarfed by the terms of degree greater than 0, so for very large $x$ the polynomial starts to behave like

$ \frac{2x}{3x} = \frac{2}{3} $

But sometimes the students don't respond to this notion of "$x$ going towards infinity." Myself, I think of a graph and imagine running my finger along it to the right or to the left. I imagine going very far. I imagine the polynomials in the numerator or denominator "getting very big" as I "run my finger toward infinity." But I have had students who didn't understand this principle.

Another phenomenon is a vertical asymptote. I imagine the denominator "going to" zero and the fraction blowing up.

I'm just wondering if there is a way to explain this notion of "going to zero" or "going to infinity" that makes sense to beginning algebra students.

Mike
If you're graphing ratios of two polynomials, then you're already at a stage when students are able to identify particular features of certain curves (like straight lines and quadratics) and sketch those functions based on their features rather than using a brute force table of values and function evaluations. That means you can teach them the general tool-kit for sketching any function. Generally, when sketching any function, you can try and find any of the following:

1. x- and y-intercepts
2. stationary points
3. asymptotes
4. periodic behaviour/single-valued?/general shape (smooth or highly varying/piecemeal)
5. what happens at the extremes (very small values, very large values)


If students are struggling to understand point 5, then try the following:

1. break the function into bits that operate together (e.g. in your example you break the numerator and denominator into separate 'chunks')

2. For each chunk, consider how that individual chunk changes as the numbers get bigger and talk about contribution to a total. E.g. for the 2x+10 chunk, say "when x gets bigger and bigger, the value of 2x gets bigger and bigger whilst the +10 part stays the same. That means that if x gets really big, the +10 part doesn't contribute much at all to the total. In fact, if x gets really, really, really big, the +10 part is negligible and we don't care about it anymore". You should also explain about higher powers. For example, if your chunk was $\displaystyle x^2 +x$ you can say "when x gets bigger and bigger, the $\displaystyle x^2$ part contributes much more to the total than the $\displaystyle +x$ part.

3. If the student still doesn't understand how the proportion changes, get them to calculate some numbers that tell you how important each part is... make a table with six columns:

i) input x-value
ii) 2x
iii) +10
iv) total (2x+10)
v) col. 2/col 4. *100
vi) col. 3/col. 4 * 100

and get the student to fill in the table for input x-values equal to 10, 100, 1000, 10000, 100000, 1000000.
The last two columns are the percentage contributions to the total by the 2x and 10. They should clearly see that the percentage gets closer and closer to 100% for the 2x column and closer and closer to 0% for the +10 column. It should be obvious then that the 2x part is more and more important than the +10.

Last step: if the student is progressing well, ask them to put one also row on the table: $\displaystyle x \rightarrow \infty$ and see if they can fill in the data.
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