Calc1131W9T1 - Sketching a rational function

Tuesday, 1 May 2018

8:29 PM

Machine generated alternative text:
Identify the graphs:
—4
—8
-6

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—8
-6
—4

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—8
-6
Maple plot
—4

$Machine generated alternative text: 3x-1 ax+b Note: as these can all be expressed in the form they are examples of fractional linear transformations - with cc+d the exception of —1$

$Machine generated alternative text: To graph we observe that f has a singularity at -210/273 240 C+149 273 ' while the limiting behaviour of f is O and 240/273 horizontal These determine the respective positions of the vertical Note: give your numeric answers as fractions. O asymptotes.$

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Consider the function f@)
. The graph of this function is shown below
—8
-6
—4
We factor the denominator to find the set of singularities
. These determine
vertical
O asymptotes.
The horizontal
O asymptote is y —
Note: the Maple syntax for the set {1, 2} is {1, 2}.

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Maple plot

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—8
-6
-4
-2

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0.4
-0.4
-0.8

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Maple plot
-4
0.4
-3
-0.4
-0.8

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Asymptotes of a rational function can be horizontal, vertical, or something in between. Consider the function
+52X+43
—1
h@)
The graph of this function is shown below in blue. The non-horizontal/vertical asymptote is shown in red.
y
-10
-20
To calculate the non-horizontal/vertical asymptote, notice
and
Therefore
h@)
2
1
1
-2
10
x
20
—1
lim (4 x + 7)
x+±oo

Machine generated alternative text:
and
—1
lim (h@)
lim (4 + 7)
c+±oo
x+±oo
O is an asymptote of h@) , because
So the linear function y =
h@) never equals the asymptote, and
• the difference between the two goes to zero as —+ ±oo.
The non-horizontal/vertical asymptote of:
—1
has equation y =
42 +62
has equation y =
90 x +29
has equation y =