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A line is the locus of points such that the gradient (slope) between that point and one fixed point in the plane is a constant. Technically, that definition does not include a vertical line because its gradient is not defined! You could get around that this by requiring that either the gradient is a constant or, if it is undefined, then the inverse gradient (dx/dy) is constant.
Y=2x+c where c is arbitery constant
They are straight lines through the origin and their gradient is the constant of proportionality.
Not a constant, but the differential, i.e. gradient, of the equation. It = 0 at maxima and minima, where the curve is at its turning-point(s).
k is the constant of variation and is the gradient (slope) of the relevant graph.