Comparing End Behavior of y=sin(x) and y=sin(x/2)

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In summary, the function y=sin(x/2) is oscillatory and its end behavior is the same as y=sin(x). However, the graph of y=sin(x/2) is stretched out by a factor of 2 and has a period of 4π. There are no other points of interest, such as asymptotes or holes, in the graph. To learn more about trigonometric functions like sine, resources such as Mathworld can be helpful.
  • #1
frenkie
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is y=sin(x) the end behavior of y=sin(x/2)?
 
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  • #2
Yep, oscillatory. the function oscillates between -1 and 1.
 
  • #3
I wish i knew what that looks like? is there a picture anywhere? sorry if that's too much trouble.
 
  • #4
and why is it oscillatory?
 
  • #5
frenkie said:
is the end behavior of sin(x/2) = sin(x) because the function settles on that equation...and i have no idea why it is oscillatory? care to explain?


also, are there any interesting points in the graph of sin(x/2)...i think the teacher is asking for asymptotes, holes and etc...which don't exist in sin(x/2)..correct?

sin(x/2) looks just like sin(x), only its squished along the x-axis by a factor of 2.
 
  • #6
also, are there any interesting points in the graph of sin(x/2)...my last question.
 
  • #7
:eek:

The behaviour of trig fns like sine is fundamental!

Have a look on mathworld or such.

(btw: in answer to your last question - zero at 0, 2n\pi, \pi\in\mathbb{Z}, diff to find extrema etc...)
 
  • #8
J77 said:
:eek:

The behaviour of trig fns like sine is fundamental!

Have a look on mathworld or such.

(btw: in answer to your last question - zero at 0, 2n\pi, \pi\in\mathbb{Z}, diff to find extrema etc...)

[tex]sin\left( \frac{x}{2}\right) =0\mbox{ if }x=2n\pi,n\in\mathbb{Z}[/tex]

J77, double click on the equations to see how to typeset in here (we don't use $..$)
 
  • #9
thank you very much guys..appreciate your help...
 
  • #10
And no, there are no other points of interest.
 
  • #11
benorin said:
And no, there are no other points of interest.
:biggrin:

Thanks for the latex thing, benorin.
 
  • #13
frenkie: is y=sin(x) the end behavior of y=sin(x/2)?

benorin: Yep, oscillatory. the function oscillates between -1 and 1.

The question doesn't even make any sense. But I would hesitate before saying, "yep". Yes, they do both oscillate between the same 2 fixed numbers, but the former oscillates twice as rapidly as the latter.

benorin said:
sin(x/2) looks just like sin(x), only its squished along the x-axis by a factor of 2.

No, it is stretched out by a factor of 2. The period of [itex]\sin(x/2)[/itex] is [itex]4\pi[/itex], which is twice as long as the period of [itex]\sin(x)[/itex].
 

1. What is the end behavior of y=sin(x)?

The end behavior of y=sin(x) is that as x approaches positive or negative infinity, the y-values oscillate between -1 and 1. This means that the graph of y=sin(x) does not have a specific end behavior.

2. How does the end behavior of y=sin(x) compare to y=sin(x/2)?

The end behavior of y=sin(x/2) is similar to y=sin(x) in that it also oscillates between -1 and 1 as x approaches infinity. However, the graph of y=sin(x/2) appears to be more compressed and has a shorter period than y=sin(x).

3. Does y=sin(x/2) have a different end behavior than y=sin(x)?

No, both y=sin(x/2) and y=sin(x) have the same end behavior. As x approaches positive or negative infinity, the y-values oscillate between -1 and 1 for both functions.

4. How can the end behavior of y=sin(x/2) be determined without graphing?

The end behavior of y=sin(x/2) can be determined by looking at the parent function, y=sin(x), and its end behavior. Since y=sin(x/2) is just a compressed version of y=sin(x), its end behavior will be the same.

5. How does the amplitude of y=sin(x/2) affect its end behavior?

The amplitude of y=sin(x/2) does not affect its end behavior. The end behavior is determined by the parent function, y=sin(x), and is not affected by any changes in amplitude.

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