Odd/even extensions

Odd/even extensions

Postby Jaffacake » Thu Feb 06, 2020 6:25 pm

Good evening all,

I have this question,

f(t) is defined on the interval 0 ≤ t < 2 by f(t) = t(2 − t).
a) sketch the odd extension of f for -6≤ t ≤ 6, and state the fundamental period of this extension.
b) sketch the even extension for -6≤ t ≤ 6 and state the fundamental period of this extension.

The graph I have ended up with doesn't seem right, as a t=0, it is 0, t=1 it is 1, t=2 it is 0, and then as I plot up to t=6, i end up getting -24, is this completely wrong? should I just repeat the 0,1,0 pattern? I also can't figure out my fundamental periods.


Hope someone can help.

Thank you.
Jaffacake
 
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Re: Odd/even extensions

Postby HallsofIvy » Sun Feb 09, 2020 10:07 pm

First, can you graph f(x)= x(2- x) for x between 0 and 2? (It is a parabola opening downward and 0 at x= 0 and x= 2 with vertex at (1, 10).)

An "even" extension just copies that piece of parabola for x= 2 to 4, x= 4 to 6, etc. as well as for x= -2 to 0, x= -4 to -2, etc. Do you see that it has period 2?

An "odd extension" flips that parabola upside down for x= 2 to 4, back upright for x= 4 to 6, upside down again for x= 6 to 8, etc. And "upside down" for x= -2 to 0, back upright for x= -4 to -2, upside down again for x= -6 to -4. Do you see that this has period 4?

HallsofIvy
 
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