Quartic Polynomials and the Golden Ratio

Totland investigates graphs of quartic polynomials with inflection points by means of certain naturally defined points and length ratios. He sees that the graph of every quartic polynomial with inflection points can be obtained as the image of the graph of the symmetric quartic subject to an appropriate affine transformation. An affine transformation consists of an invertible linear transformation followed by translation along a constant vector. An affine transformation of the plane has the following properties: straight lines are mapped to straight lines, parallel lines to parallel lines, and tangents to tangents, while length ratios between parallel line segments are preserved.