I understand that, because the slope is negative we assume the height is negative. But geometrically this does not makes sense to me. How should I interpret it?

I do not think this is correct. What exactly do you mean by â€śheightâ€ť?

In the lecture, the instructor said that if we take the tangent of a point in J(w).

We ends up with a positive or a negative slop. When the slope is negative, the value for the slope height is negative, but that is confusing me, perhaps height is not the most appropriate term?

Right, slope has nothing to do with â€śheightâ€ť as an absolute quantity. You can take that curve as shown and shift it up or down and the slopes stay the same, right? Slope just describes the direction of the tangent line to the curve at a given point on the curve.

Yes, the â€śslopeâ€ť is the quotient of the â€śriseâ€ť over the â€śrunâ€ť meaning the change in height divided by the change in the w coordinate. So if the slope is negative, that means the â€śdeltaâ€ť or change of the height is negative as you move w to the right.

Yes, I would just ignore the word â€śheightâ€ť in that context. â€śThe slope is negativeâ€ť is the key concept.