The Video Sampler

9.27.2006

The Myth of Shifting

September 27, 2006 Wednesday 9:11 PM The Myth of Shifting

I don't know if you were awake at the time...
but somewhere in Pre Calc someone might have mentioned something
about shifting a graph. 
Were you paying attention?
Well don't feel bad about it. 
I've thought an awful lot about shifting graphs to 
quote unquote 'move them around' but it's not real.
Really. A line is a function in analytical algebra.
y = mx + b  and if you add a value to the right hand side
you move the line up. And if you subtract a value from the 
right hand side you move it down. 
Simple? 
Now, someone points out the possibility of subtracting or 
adding a value to the x. This is suppose to 
shift the graph over right or left depending on the sign.
...
And it seems to work doesn't it? Indeed, behold if I 
write y = m(x-h) + b I move the line to the right.
If I write y = m(x+h)+b I move the line to the left. 

It's shifting... isn't it?

No. Not really. Think about it your changing the 
right hand side of the equation, and further more
it's either a higher value or a lower value... like
before. 
So why is it that it 'looks' like the line is shifting?

Shifts are a myth. They don't exist. 
And further more I can tell you shirting and adding
k are equivalent by this equation -mh = k
Which means that if you had k that's like a shift of
h = k/-m  See? No such thing as a shift it's merely
adding a value with a ratio in front of it. 

So I knew this. I 'knew' this now. (the thinking dates back
to August when I scribbled out the equations on the first page
of my class notebook) 

... But now...
I began to think about it again. I started thinking in circles.
Circles also obey the mythical shift law. You can move them 
all over the place and it really 'looks' like they are shifting.
but what is going on?
I had an explanation for the reason lines looked as though they 
were shifting but circles? Now that was a whole new question.

But was it?

Today, while siting with Liz in the EOP place, I preformed
a similar calculation.. only this time it was about circles.
"interesting" I muttered out loud. Can you picture me there with
that pad of equations in front of me and going, "interesting"
to myself. It's a common thing but not that common.

Anyway, I had just noticed a similarity. It looked as though
the circle shifting depended on the intersection of 2 lines
with h and k as their parameters.
I had a headache this evening but decided to go about the graphing
these lines and the corresponding circle regardless.
And so I took a picture.



The thing about it is that the lines don't cross through the 
center of the circle.. but the point of intersection of the 
lines IS related to the circles center. ... well somehow.
I know this because I tried different values to see what would
happen. 

But the lines... the tell tale lines were showing me something.
I reasoned that shifting doesn't exist in the lines. 
But they still 'look' like they shift when you move them because
they are made up of infinitely many points. In other words, you don't
notice that they haven't shifted because there are always more points
to move into the spot you 'think' a line segment shifted to.
That's confusing and hard to explain here. (in fact I don't fully 
understand it) But it's something. Something I could think more
about and explain better someday. 

Here, the intersection between 2 lines was moving. Although each
line can only go up and down... the intersection between them could
be made to move right and left. 
Fantastic! Here it was a literal shifting and not just that 'fake'
shifting that lines or supposedly doing.

The circle looks like it's moving because it's related to these
lines somehow.... so how does that 'somehow' work?

9:37 PM

9:42 PM

Now. (shakes off the mathematics viewpoint)
What exactly is going on in regular life? 
Headaches. in short. But I hope things get better soon.
It's hard. I walk through the day with a headache and I 
wish for relief. 
Maybe I'll relax again. The way I used to.


2 comments:

Anonymous said...

"The way I used to."
And how's that?

Sirhc Senots said...

I used to be more free spirited. I used to ignore all evidence of me actually being a part of the social world. Life was good back then. Easy to forget about the 'mean people' and get into my work. The real work of thinking and figuring out things on my way to my 'suggested personal achievements' So, if I could put everything aside once more. And retreat back into that realm of thought and sanctuary I could once again... relax. ... The way I used to.

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