I have 5 points which I would like to fit onto a curve. Mostly this is just for fun, but trying to do it is hopefully teaching me and my daughter some mathematics. We're using shape-building to learn about the properties of equations.
We would like the curve to be concave to the left like a parabola and we would like it to hit each point.
We tried a linear regression of a quadratic function and the software did not hit all points. Is that a limitation of the software or are we simply trying to do something that a parabola cannot do?
The points are:
1, 2 1.5, 1.9 2, 1.5 1.5, 1.1 1, 1
Essentially a sideways, symmetrical parabola-like shape.
I tried higher order functions but as you might expect, the shape that the software produced was not concave to the left.
I'd like to know both whether there exists a perfect-fit equation which is concave forever, and also whether there exists a perfect-fit equation which is simply concave in the region 1,1
In general I'd like to understand whether every concave curve can be described as a single equation and if so, how.