We have the function:
Find the approximating quadratic polynomial with the development points:
I've found it to be:
Now find the type, vertex and line of symmetry for the graph the approximating quadratic polynomial produces.
I had to translate it to English, but hopefully it is understandable and if not I'll try to explain the question differently.
The quadric is . The associated quadratic form is
The eigenvalues are . An orthonormal basis of eigenvectors is:
The equation of the basis change is:
and the quadric has the expression:
Using the translation:
we can cancel the linear terms choosing . The quadric on the axis has the equation:
that is, a hyperbolic paraboloid. You can easily identify its elemets of simmetry. After, use to find them on the axis.