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I have recently been confused by the term "Proportional". This started when I came across 2 different websites that seem to contradict one another when it comes to $y = x^2$. The first implies that the two variables are proportional, but the second appears to deny this.

I am sure that I am missing something and/or being overwhelmed by overloaded terms as both are reputable resources. The forms of proportional relationships that I am familiar with are the directly and indirectly proportional equations of $y = k * x$ and $y = k/x$ with '$k$' being a constant. So when I read that equations of the form $y = x^2$ (presumably w/ '$k$' $= 1$) being considered proportional, it throws me off. Also, the exponential form of $y= a * b^x$, where '$a$' is a constant, now has me rethinking its proportional nature . . . which to me has always just been based on the factor '$b$'. If one can say that "$y$ is proportional to $x^2$", then why can't one say "$y$ is proportional to $x+1$" (i.e. $y = 1 * (x + 1)$? I am clearly confused and would surely appreciate being set straight. Many thanks.

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  • $\begingroup$ No, it means 'linearly related to' as in y = ax + b. That's the ONLY meaning. $\endgroup$
    – user117644
    Sep 4, 2015 at 21:31
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    $\begingroup$ You misread; the first link does not suggest that the variables in $y=x^2$ are proportional. It claims that, in the equation $d=kt^2$, $d$ is proportional to the square of $t$. (In $y=x^2$, then, $y$ is proportional to the square of $x$.) $\endgroup$ Sep 4, 2015 at 21:38
  • $\begingroup$ @mistermarko dont you mean 'affinely related to'? $\endgroup$ Sep 4, 2015 at 21:46

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Community wiki answer so the question can be marked as answered:

As columbus8myhw explained in a comment, you misinterpreted the first link. In $y=x^2$, it is not $y$ and $x$ that are proportional, but $y$ and $x^2$. Indeed, as you suggest, $y$ can be proportional to $x+1$, or in fact to any function of $x$.

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  • $\begingroup$ Thanks @joriki. I see that I did not characterize the question accurately. However, I remain unclear. For y = x^2, I am under the impression that the ratio between all x^2 and y for this equation will not have the same ratio. So it is strange to be told that the relationship between y and x^2 is proportional. But, based on your last statement, it seems that you are using the term proportional in y=x^2 with an intended interpretation of something like: the value of y can be determined by squaring the value of x. Please advise. $\endgroup$
    – SaiyaJin95
    Sep 4, 2015 at 23:01
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    $\begingroup$ @AlonzoArcher: No, I mean quite literally that $y$ is proportional to $x^2$. I don't understand what you mean by "the ratio between all $x^2$ and $y$ for this equation will not have the same ratio". If $y=x^2$, then the ratio between $x^2$ and $y$ will always be $1$. $\endgroup$
    – joriki
    Sep 4, 2015 at 23:09
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    $\begingroup$ @AlonzoArcher: This is basically just the same problem over again. An equation or relationship isn't "linear" per se; it's linear in a certain variable or expression. Thus, $z=3xy^2$ is linear in $x$ and in $y^2$ but quadratic in $y$. That article you link to isn't concerned with any of that; it focuses on the fact that "linear" is more general than "proportional" because there can be an additional constant term (against, constant with respect to some variable, function, ...). Thus $y=3x^2+z^3$ is linear in $x^2$ (with constant term $z^3$), whereas in $y=3x^2$, $y$ is proportional to $x^2$. $\endgroup$
    – joriki
    Sep 19, 2015 at 5:10
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    $\begingroup$ @AlonzoArcher: This is what they mean when they say that every proportional relationship is also a linear relationship -- that if $y$ is proportional to some variable, function, ..., e.g. to $x^2$, then that's a special form of linear relationship, namely one without a constant term. $\endgroup$
    – joriki
    Sep 19, 2015 at 5:12
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    $\begingroup$ @AlonzoArcher: You can find some examples e.g. with this Google search, e. g. this first hit (Nr. 104). $\endgroup$
    – joriki
    Sep 20, 2015 at 18:49

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