Should we write $y=mx+c$ as $c + mx$?

Question

This may seem like a really pointless question but bear in mind I am thinking from the perspective of a school maths teacher.

I am recently thinking it would be more intuitive to have a convention of writing the straight line equation in the form $y = c + mx$ rather than $y = mx + c$.

This is because it more clearly describes the fact that you have something growing outwards by m per 1 x, while adding that we know where it is growing from.

y = c + mx suggests start at point c, and then to get to anywhere on the line, add some multiple (x) of m...

This feels more like what a straight line is.

Any opinions on this convention? Or wider intuition on how to think of a straight line equation?

Answer 1

While both notations are equivalent due to the commutative property of addition, it seems like the equation is written as $y = mx + c$ because this is the canonical polynomial form with respect to $x$. For example, you could write it as $y = mx^1 + cx^0$. If it were a quadratic equation, it would be written as $y = m_1x^2+m_2x+c$.

Answer 2

This is because it more clearly describes the fact that you have something growing outwards by m per 1 x, while adding that we know where it is growing from.

This is not wrong and you have a point.

But why do we care what it's growing from? Isn't the rate of growth more important than where the line began, which is just an offset?

In the end it doesn't matter. After all addition is commutative and it is actually bad practice in math to associate emphasis of an equation on where something occurs. But more importantly the argument as to how important the initial value is, versus the heavy mechanics and shifting an moving, is subjective.

And I dare say I disagree and think the heavy mechanics and shifting is more important. I certainly think if I had a polynomial, I'd want it written as $a_nx^n + .... + a_1 x + a_0$ rather than $a_0 + a_1x + .... +a_nx^n$ because the $a_nx^n$ is more "influential" than the $a_0$. And I'd say the $mx$ is more influential then that $c$.

But 1) it doesn't really matter and 2) It's pretty subjective.

Answer 3

It is the same function, order is immaterial even if we write

$$ y= c+ mx $$

which gives initial intercept $c$ and slope $m.$

By the same token a parabola of focal length $f$ can be expressed

$$ y= c+ mx + \frac{x^2}{4f}$$

that has constants $(c,m,f)$ relates as initial intercept, initial slope and initial radius of curvature.