Quadratic functions

In this assignment we will study quadratic functions of 2 variables, i.e. any function of the form: c1x2 + c2xy + c3y2 + c4x + c5y + c6 where the ci are all constants. Recall that the graph of a quadratic function in one variable is a parabola with a vertex that is either a minimum or a maximum of the function. We will try to generalize the idea of a vertex for two variables as well as get at the idea of a minimum or maximum. One problem in two variables is that a quadratic function may have a single vertex or it may instead have a line of vertices. There are 5 categories for a quadratic function:

 I.          Single vertex, saddle. Example: x2 – y2

II.         Single vertex, minimum, Example: x2 + y2

III.       Single vertex, maximum, Example: -x2 – y2

IV.       Line of vertices, all maxima, Example: -x2

V.        Line of vertices, all minima, Example: x2


Note that in the examples for I, II, and III (0, 0, 0) is the single vertex, and that in the examples IV and V (0, 0, 0) is on the line of vertices. You should look at the graphs of all five examples to see what they are trying to demonstrate.

1. Every quadratic function of the form c1x2 + c2xy + c3y2 has a vertex at the point (0, 0, 0). Look at a graph of each of the following functions and decide to which of the 5 categories it belongs.

a) 3xy – x2 – 5y2
b) 4x2 – y2
d) 6xy – 2x2 – y2
e) 4xy – 2x2 – 2y2
f) 2x2 – xy + y2

2. For the following quadratic functions, find a vertex. Then by looking at the graph, decide to which of the 5 categories the function belongs.

a) 3x2 – y2 – 12x – 2y + 13
b) 3xy – 5x2 – y2 + 13x – 5y – 7
c) x2 – xy + y2 – 3x + 3y + 1
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