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Bending Asymptotes and Bouncing Off Infinity ...Nspired
from "PreCalculus Transformed" by Harrow & Merchant (copyright 2010 Venture Publishing, Andover , MA)
www.vent­pub.com
Nurfatimah Merchant
The Westminster Schools
Atlanta, GA
nurfatimahm@westminster.net
http://casmusings.wordpress.com
Oct 16­8:03 AM
I want to graph the reciprocal of y = x and have no clue as to what it should look like ... what might I do?
Oct 20­11:50 AM
1
The Reciprocal Transformation.
•
Parts of domain obliterated:
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Invariant points:
•
Specific changes:
Oct 10­4:38 PM
Using the Reciprocal Transformation, graph Compare the two vertical asymptotes on this graph:
Types of Vertical Asymptotes:
Created by root of odd multiplicity:
Created by root of even multiplicity:
Nspiring Change...
Oct 10­4:41 PM
2
To determine the local behavior of a graph around its vertical asymptote, we can now rely purely on the “multiplicity” of each vertical asymptote.
Putting this together with an understanding of end behavior and intercepts, it is very easy to graph any "proper" rational function (without having to think through the entire reciprocal transformation).
Easy problem: Graph Domain restrictions:
End Behavior:
(Very large values of abs(x))
Intercepts:
Oct 10­4:46 PM
Oct 21­8:36 AM
3
Similar in appearance to a prior problem, but worth the 30 seconds...
Graph Domain restrictions:
End Behavior:
Intercepts:
Types of x­intercepts:
Note on overall end behavior...
?
Playing with End Behavior... How would we graph Slide to it...for some REAL Nspiration...
Oct 10­5:08 PM
... and bend it ... (like Beckham? Or maybe better...)
Now think about Different forms of a function tell different stories about the function ... Any rational function can be expressed in two forms, one being what we have seen so far.
How far off the function is from its model
(Connections to data analysis)
is always a "proper" fraction, and can be graphed using its end behavior pattern, intercepts and multiplicities of VA’s. Then simply “bend” the x­axis into the shape of the new End Behavior Asymptote (EBA) or Model.
Notes: • When "bending", the original x­intercepts are revealed to be EBA­intercepts (zero error: meaning function IS model at these points).
• The actual zeros of the final graph cannot be "read off" this form of the equation. Oct 10­5:12 PM
4
What if the equation for the function is not in this form, but in the other (single fraction) form?
For instance, What IS it that I want the students to get? What's the point?
CAS capabilities are a HUGE help here. Use polynomial division to get f into "mixed" fraction form ("propfrac" command on the Nspire). Can also use CAS to factor where necessary.
Final form of equation: 1. Domain restrictions (VA's and more...): 2. End behavior:
EBA: How the function approaches its EBA (error term analysis):
3. EBA­intercept(s):
Oct 11­3:02 PM
Fun on the test
I woke up this morning thinking about a rational function that “wiggled” through its cubic end behavior asymptote at a hole, but otherwise had no discontinuities and never had “zero error”. Find a graph and an equation of such a function to convince me that it wasn't just a wishful dream. Justify your claims. Checklist:
1.
2.
3.
4.
5. Most missed by students:
Oct 11­3:32 PM
5
... and beyond ...???
"how obnoxious would you like to make this?"
Moving beyond rational functions ... could an EBA be anything we wanted it to be?
Oct 16­8:13 AM
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