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April 26th, 2013, 07:45 AM   #1
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Linear Variation Proportion

Dear All,

I wonder if someone can help explain.

I understand that proportionality or variation describes a relationship that increases or decreases in accordance to the constant of proportionality.
For example...

y = kx (direct)

or y = k/x (indirect)

These cases are easy to understand with simple analogies.

What about more complicated examples, e.g. y = x^2?

With this function I would say that y increases in proportion to x, however I'm unsure if it can be described as directly proportional.

Also, what about functions like y = sinx. y increases in proportion to x and can be described as a function of x. but I don't think it is directly proportional.

Could someone please confirm or provide more complicated examples of direct variation or proportionality?

Many thanks.
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April 26th, 2013, 09:41 AM   #2
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Re: Linear Variation Proportion

That x can be replaced with f(x), any function of x, or indeed many different functions. For variation, the only thing necessary is some constant to precede the functions. Even a 'simple' formula like F = ma can be considered as a variation formula with k = 1 and you would read it as..

'Force varies jointly with respect to mass and acceleration'

For a more complicated formula like the law of gravitational attraction,



here k = G and you could read this as 'Force varies jointly with respect to m1 and m2 and inversely as the square of the distance'

Usually physicists put more words regarding the distance like 'between the centers of gravity of the 2 masses m1 and m2'

y = ksinx is a valid variation and can be read as 'y varies directly as the sine of x'

Or even if you have something like y = 5z^2cosx , here you have joint variation with k = 5 and you would read it as 'y varies jointly with respect to the square of z and the cosine of x.'

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April 27th, 2013, 11:38 AM   #3
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Re: Linear Variation Proportion

Hi,

These are good examples, but I'm still slightly confused, just to clarify with a few more examples perhaps

based on the reply I am assuming these statements are all correct:

y = kx (y is directly proportional to x, based on k)

y = ksin(x) (y is directly proportional to the sine of x, based on k)

y = kx^2 (y is directly proportional to x squared, based on k)

However it is incorrect to say these statements...

y = sin(x) (y is directly proportional to x)

y = x^2 (y is directly proportional to x)

as these statements do not fully describe the relationship between y and x.

Is this correct?

Many thanks
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April 27th, 2013, 04:14 PM   #4
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Re: Linear Variation Proportion

Yes, everything you wrote in the previous pos is correct IMHO. When interpreting variation statements into equations I personally have never seen the 'based on k' part.

If I analyze the statement 'y varies directly as the square of x'

As soon as I read 'y varies' I write down y = k , then I read the rest of the statement to figure out HOW y varies, 'the square of x' is x^2 so then I complete the equation.

y = kx^2

I think the word 'directly' is not necessary. 'y varies as the square of x' is perfectly fine.

You can see the word 'directly' cause some awkwardness when we have joint variation.

y = kx^2z^3

you can read this as 'y varies jointly with respect to x squared and z cubed' and it is not so easy to put 'directly' in this statement.

There are only 2 possibilities , directly or inversely. If you don't see any of these 2 words it is assumed that everything is direct.

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April 27th, 2013, 06:18 PM   #5
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The word "proportion" has rather limited uses, though not quite as limited as agentredlum suggests. However, "proportion" without qualification is assumed to mean "direct proportion", and thus excludes other relationships such as y = x.
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