It looks like point-slope but it is not. I cannot find the name anywhere althought it seems to used for linear approximation. Can anyone help ?
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Yes, it is point-slope form, the slope being m and the point (x₀,y₀).

Other forms are slope-intercept.

This is a linear representation.

It can be used in precalculus to approximate to the gradient of a curve, by taking two points close together on the curve, joining them together with a straight line, and working out an equation for the gradient around  a point. Sometimes it’s referred to as the tangent line or equation of the tangent. It is a linear approximation to a tangent line and the concept is a stepping stone to learning about differentials, or differential calculus. It is useful for predicting values of a function, and a method, known as Newton’s Method, essentially uses the idea to solve equations that might be difficult to solve in any other way.

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