Vertical Asymptotes on the graph of y=Acsc(Bx-C)+D

[xxMsJojoxx]

In trying to figure out the vertical asymptote for an equation, for example , we were given this information about the vertical asymptote:


What is k? This variable is not in the generic formula y=Acsc(Bx-C)+D.

Thank you.

 

[skeeter]

[MATH]y=6\csc\left(\dfrac{\pi}{3} x + \pi\right)[/MATH] is undefined where

[MATH]\dfrac{\pi}{3} x + \pi = k \pi \text{ where } k \in \mathbb{Z}[/MATH]

 

[xxMsJojoxx]

Please see CORRECTION on the Post below, relating to the information given about the vertical asymptote:

In trying to figure out the vertical asymptote for an equation, for example View attachment 22491, we were given this information about the vertical asymptote:


What is k? This variable is not in the generic formula y=Acsc(Bx-C)+D.

Thank you.

 

[xxMsJojoxx]

[MATH]y=6\csc\left(\dfrac{\pi}{3} x + \pi\right)[/MATH] is undefined where

[MATH]\dfrac{\pi}{3} x + \pi = k \pi \text{ where } k \in \mathbb{Z}[/MATH]

How do I isolate go get k, to plug in to the equation to get the vertical asymptote?



I also tried graphing this, but it doesn't seem to be correct, in reflecting the vertical asymptote?

 

[Dr.Peterson]

As they say, k can be any integer. For example, if you find that x = -3 + 3k, this means that there are asymptotes at:

x = -3 + 3(0) = -3​

x = -3 + 3(1) = 0​

x = -3 + 3(2) = 3​

...​

as well as

x = -3 + 3(-1) = -6​

x = -3 + 3(-2) = -9​

...​

Do you understand? And do you see the asymptotes at these locations on your graph?

 

[xxMsJojoxx]

As they say, k can be any integer. For example, if you find that x = -3 + 3k, this means that there are asymptotes at:

x = -3 + 3(0) = -3​

x = -3 + 3(1) = 0​

x = -3 + 3(2) = 3​

...​

as well as

x = -3 + 3(-1) = -6​

x = -3 + 3(-2) = -9​

...​

Do you understand? And do you see the asymptotes at these locations on your graph?

Thank you, Dr. Peterson. I understand the graph in your example x=-3+3k.

But on my original question, I don't understand why x=3K is the asymptote for the equation .



The graph for x=3k clearly touches and crosses the equation . How can it be the asymptote then?

Re: An asymptote is a line that the equation approaches but never touches, like this below.

 

[Dr.Peterson]

The equation x = 3k is not a line to be graphed on the xy-plane at all, much less an asymptote. The variable k is not y! (I didn't know Desmos was capable of making that mistake, graphing x = 3y when you enter x = 3k.)

I listed some of the asymptotes: x = 0, x = 3, x = 6, ... . These are all vertical lines, each obtained by taking a different integer value for k.

(Also, where did you get that equation x = 3k? The equation I used, x = -3 + 3k, is the one obtained from the formula you quoted.)

 

[xxMsJojoxx]

The equation x = 3k is not a line to be graphed on the xy-plane at all, much less an asymptote. The variable k is not y! (I didn't know Desmos was capable of making that mistake, graphing x = 3y when you enter x = 3k.)

I listed some of the asymptotes: x = 0, x = 3, x = 6, ... . These are all vertical lines, each obtained by taking a different integer value for k.

(Also, where did you get that equation x = 3k? The equation I used, x = -3 + 3k, is the one obtained from the formula you quoted.)

Thank you for clarifying this, Dr. Peterson.

I'm doing a review question (#21) in my textbook, and got the answer given at the back of the book.

 

[Dr.Peterson]

x = 3k and x = -3 + 3k, for any integer k, are equivalent, because -3 + 3k = 3(k-1), so it just obtains the same value for different integers k.

But I'm surprised they would give the answer that way, when the formula they taught you, x = C/B + pi/(2|B|) k, directly yields the answer in the form I gave, x = -3 + 3k. That could certainly confuse some students.