Two examples of quadratic functions are [latex]p,q: \mathbb{R}^2 \rightarrow \mathbb{R}[/latex], with values

[latex]\begin{align*} p(x) &= 4x_1^2 + 2x_2^2 + 3x_1 x_2 +4x_1 + 5x_2 + 2\times 10^5 \end{align*}[/latex]

[latex]\begin{align*} q(x) &= 4x_1^2 - 2x_2^2 + 3x_1 x_2 +4x_1 + 5x_2 + 2\times 10^5 \end{align*}[/latex]

The function

[latex]\begin{align*} r(x) &= 4x_1^2 + 2x_2^2 + 3x_1 x_2 \end{align*}[/latex]

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

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

	Level sets and graph of the quadratic function [latex]q[/latex]. This quadratic function is not convex.