Quadratic functions in two variables

Symmetric Matrices > Definitions > Example

Two examples of quadratic function are $p,q : mathbf{R}^2 rightarrow mathbf{R}$ , with values

$begin{array}{rcl} p (x) &=& 4x_1^2 + 2x_2^2 + 3 x_1x_2 + 4 x_1 + 5 x_2 + 2 times 10^5, q(x) &=& 4x_1^2 - 2x_2^2 + 3 x_1x_2 + 4 x_1 + 5 x_2 + 2 times 10^5. end{array}$

The function

is a form, since it has no linear or constant terms in it.

Level sets and graph of the quadratic function . The epigraph is anything that extends above the graph in the -axis direction. This function is ‘‘bowl-shaped’’, or convex.

Level sets and graph of the quadratic function . This quadratic function is not convex.