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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Mathematical Methods CAS => Topic started by: FlorianK on October 30, 2012, 12:18:16 pm

Title: Why is this an asymptote?
Post by: FlorianK on October 30, 2012, 12:18:16 pm: if I have the graph f(x)= $\left | \frac{2}{(x-2)^{2}} - 2 \right |$
Why does y=2 count as an asymptote?
The graph shows asymptotic behaviour when x becomes very large or small, but the graph also crosses the asymptote when x=2 $\pm \sqrt{2}$
I'm confused...
Title: Re: Why is this an asymptote?
Post by: paulsterio on October 30, 2012, 12:22:19 pm: A graph can cross a horizontal asymptote, it just needs to display asymptotic behaviour from one side.
Title: Re: Why is this an asymptote?
Post by: aishuwa1995 on October 30, 2012, 12:32:33 pm: Isn't it just a general rule that all truncus graphs have 2 asymptotes (one horizontal and one vertical)?
Title: Re: Why is this an asymptote?
Post by: paulsterio on October 30, 2012, 01:10:54 pm: Quote from: thiskid on October 30, 2012, 12:32:33 pm
Isn't it just a general rule that all truncus graphs have 2 asymptotes (one horizontal and one vertical)?

What's a rule? :P

It is something that can be derived, thus, it is not an axiom (which is what I think you mean by rule) - i.e. it is not defined to be such, it is found out to be such
Title: Re: Why is this an asymptote?
Post by: aishuwa1995 on October 30, 2012, 02:14:36 pm: Quote from: Paul Stereo on October 30, 2012, 01:10:54 pm
Quote from: thiskid on October 30, 2012, 12:32:33 pm
Isn't it just a general rule that all truncus graphs have 2 asymptotes (one horizontal and one vertical)?

What's a rule? :P

It is something that can be derived, thus, it is not an axiom (which is what I think you mean by rule) - i.e. it is not defined to be such, it is found out to be such

oh right..my bad, I totally mis-read the question :P
Title: Re: Why is this an asymptote?
Post by: meemz on October 31, 2012, 08:08:54 pm: Quote from: FlorianK on October 30, 2012, 12:18:16 pm
if I have the graph f(x)= $\left | \frac{2}{(x-2)^{2}} - 2 \right |$
Why does y=2 count as an asymptote?
The graph shows asymptotic behaviour when x becomes very large or small, but the graph also crosses the asymptote when x=2 $\pm \sqrt{2}$
I'm confused...
for the "y" asymptote, i always plot x=infinity
in this case when x approaches infinity, 2/(infinity-2)^2 is 0.. therefore we're only left with |-2| = 2
Title: Re: Why is this an asymptote?
Post by: AsOne on November 01, 2012, 10:51:53 pm: just to expand on what meemz said :

if you look at the fraction part.

NO matter how big x gets, the fraction will never be zero.

if the fraction part will never be zero, then f(x) will never be able to be 2 !

Hence that is why y=2 is an asymptote.

As to why x=2 is an asymptote, it is because the bottom of the fraction will become zero. And you anything divide by 0 is undefined !

Does that help ? =)

Hope that made sense.
Title: Re: Why is this an asymptote?
Post by: Jenny_2108 on November 02, 2012, 01:19:23 pm: Quote from: FlorianK on October 30, 2012, 12:18:16 pm
if I have the graph f(x)= $\left | \frac{2}{(x-2)^{2}} - 2 \right |$
Why does y=2 count as an asymptote?
The graph shows asymptotic behaviour when x becomes very large or small, but the graph also crosses the asymptote when x=2 $\pm \sqrt{2}$
I'm confused...

I think first, when you sketch the graph g(x)= $\frac{2}{(x-2)^{2}} - 2$
You see the horizontal asymptote is -2

Now you sketch f(x)= $\left | \frac{2}{(x-2)^{2}} - 2 \right |$ just like sketching a modulus function by reflecting the negative part of y value in the x-axis
Thus the horizontal asymptote now becomes 2

Its okay to cross the horizontal asymptote, but not for vertical asymptote
Title: Re: Why is this an asymptote?
Post by: Moko on November 02, 2012, 02:44:45 pm: Thanks Ennjy for that...it's not like it was said about 10 times before or anything.
Title: Re: Why is this an asymptote?
Post by: BubbleWrapMan on November 02, 2012, 10:21:15 pm: Don't be mean