Quadratic functions in two variables

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
 r(x) = 4x_1^2 + 2x_2^2 + 3 x_1x_2
is a form, since it has no linear or constant terms in it.

Level curve of $p$ 

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

Level curve of $p$ 

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