Any equation of the form

a x^2 + b x + c y^2 + d y + e z^2 + f z + h = 0

has a set of solution points (x, y, z) that is either empty, consists of a single point, or defines a surface in 3-dimensional space.

When the solution set defines a surface, that surface is called a quadric surface. 

If this surface intersects a plane which is parallel to a coordinate plane (i.e., parallel to the xy plane, the xz plane or the yz plane) at more than one point, the intersection is a conic section, i.e., either an ellipse, a parabola or a hyperbola.

In class the group randomly selected numbers of a, b, c, d, e, f, and h, resulting in the example function

-3 x^2 - 4 x + 2 y^2 + 5 y + 4 z^2 - 3 z + 2 = 0.

The surface defined by this equation will intersect the x-y plane at an point for which z = 0.

Substituting z = 0 we get the equation

- 3 x^2 - 4 x + 2 y^2 + 5 y + 2 = 0.

By completing the square and performing the necessary algebraic maniulations we put this equation into the form

(x-h)^2 / a^2 - (y - k)^2 / b^2 = 1,

the equation of a hyperbola centered at (h, k) = (-2/3, -5/4), with a = sqrt(5/72) and b = sqrt(5/48).

This hyperbola is easily graphed by constructing a rectangle of dimensions (2 a)  by (2 b), centered at (-2/3, -5/4).

If x = h our equation results in a negative quantity being equal to 1, so the x = h points of this rectangle are not on the hyperbola. 

If y = k then we do get two possible values of x, indicating that the points (-2/3 + sqrt(5/72), -5/4) and (-2/3 - sqrt(5/72), -5/4) do lie on the hyperbola.  This allows us to sketch the hyperbola, which opens to the right and left.  The asymptotes of the hyperbola are the extended lines formed by the diagonals of our rectangle.

See the assigned review sections for Precalculus II for the details of graphing conic sections.

If we let z = 1 we get the intersection of the surface with the plane z = 1, which is the plane parallel to the x-y plane, lying 1 unit above that plane.  Working through the details we find that this gives us a hyperbola again centered above the point (-2/3, -5/4).  The dimensions of the rectangle defining the hyperbola are proportional to but somewhat greater than those of the preceding.  The asymptotes are the same as before.

If we substitute greater positive values for z, we get the intersection of the surface with 'higher' and 'higher' planes, i.e., with planes that lie further and further above the x-y plane.  You should convince yourself that the hyperbolas are all centered above (-2/3, -5/4) in the xy plane, and that the greater the value of z, the larger the rectangle that defines the hyperbola.  You should then visualize what the surface might look like (not an easy visualization but worth some effort).

We will go into more detail on quadric surfaces next week.  By then you should be comfortable with the details of completing the square, as well as the details of graphing conic sections (i.e., parabolas, ellipses and hyperbolas).

You might want to consider the following questions for the quadric surface of this example:

• What is the equation of the intersection of the surface with the x-z plane, and what type of conic section do we get (hint: to get the intersection with the x-z plane we substitute y = 0; be sure you understand why).
• What is the equation of the intersection of the surface with the y-z plane, and what type of conic section do we get (hint: to get the intersection with the xyz plane we substitute x = 0; be sure you understand why).
• One of the intersections is a hyperbola, the other an ellipse.  Which is which? Try to visualize the surface using all this information.