# How can I create an equation for a curve segment to continue an existing curve at similar slope?

Given a third degree polynomial curve, such as the GREEN curve below,

and a cutt-off point, indicated by the ORANGE line below, where the cutt-of point is placed on or near the cusp/inflection point of the GREEN line... like so:

How do I produce a left-most line segment that acts as a continuation of the green line?

I am looking for something like so, where the curve equation segment I am seeking is within the blue box:

And the x value for first inflection point is known.

The equation for the original GREEN line can be something like so (example only)

$$-3.5*10^-6 x^3 - 0.0066 x^2 + 1.3 x - 30$$

I am looking for methods, techniques, suggestions, actual steps to take for this class of problem... to replace the part of the curve with a new curve, of similar slope.. and I am seeking the equation for the segment of the new curve.

• you can use interpolation and for that you need to know a few data points in the blue box. Without those data points, any continuation of the line is just a wild guess. – Vasya Oct 10 '18 at 19:20
• Thanks ...... intuitively I will just "extend the curve towards the left". is there a way to do that mathematically? For example, I won't go straight up, and I won't go straight down. I will "extend the curve", using the similar slope value. i.e. ask someone to draw an extension on paper. Perhaps I could just do a linear approximation, using a few points from the curve to the right of the blue box, and then use those to fit a straight line that will be contained in the blue box. But that is crude, I wanted something ab it more advanced, such as a spline curve perhaps or another polynomial – Dennis Oct 10 '18 at 21:35