The graph for the function **y = 2x^2 + 3x - 1** is hardly
mysterious: The parabola opens up (since the coefficient of a^2
is positive) and has vertex at (-3/4, -17/8), as you can see below.

What happens to the graph if we add in an **xy**
term? That is, what changes if we graph **y = 2x^2 + xy
+ 3x - 1** ?

the purple graph is |
At first glance the graph might appear to be
a rotated parabola (which is not an outlandish guess,
since the presence of an xy term in a conic can indicate a rotation
of the conic section.) |

But expanding the scales on the axes reveals
that in fact, the relation describes a hyperbola. |

So, as the coefficient of the xy term varies, will the relation
always describe a hyperbola (except when the xy term vanishes
and the graph becomes a parabola)? In other words, for **y =**
**2x^2 + nxy + 3x - 1**, what
types of relation(s) is/are described as **n**
varies?

Well, one of the best ways to get a feel for this question
is to **view a movie**. **When
you finish viewing the movie, press the Back button to return
to this page.**

In the movie, as **n** varies from -10 to 10, the graph appears to move
from a hyperbola to a parabola (at **n** = 0) and back to a hyperbola. Notice that for **n** close to 0, further
investigation is necessary to determine whether ** h**
is a parabola or a hyperbola, since for small values of

Let's see what happens when **n** = -0.2, -0.1,
0.1, and 0.2. We want to determine visually whether we have hyperbolas
or parabolas for these values of **n**. In the image below, the purple graph is **y
= 2x^2 + 0xy + 3x - 1** (that is, **n** = 0.) Then **n** = -0.2 (light
blue graph), -0.1 (green graph), 0.1 (red graph), and 0.2 (dark
blue graph).

While the scale has to be considerably expanded
to view the "second parts" of the hyperbolas (especially
for the red and green hyperbolas, i.e. for **n** *very* close to 0), this graph provides evidence
for the conjecture that **y =** **2x^2 + nxy
+ 3x - 1** is a parabola only when **n**
= 0, and is otherwise a hyperbola.

Also notice that for **n** < 0, the rotation of the hyperbolas appears
to proceed to the right, or clockwise (see the light blue and
green graphs above.) For **n** > 0, the rotation of the hyperbolas appears
to proceed to the left, or counterclockwise, as shown by the red
and dark blue graphs above. (This observation can also be confirmed
by viewing the movie, which, if you like, you can see again by
scrolling up.)

In the movie, an interesting switch seems
to occur at **n** = approximately 3.5. The hyperbola passes from
vertices with "branches" opening in the second and fourth
quadrants to vertices with branches that open mostly in the first
and third quadrants.

Therefore, let's zoom in: view the **short zoom movie**, and **when you
finish viewing the movie, press the Back button to return to this
page**.

I conjecture that there is a value of **n** between 3.48
and 3.62 that makes the hyperbola become degenerate (i.e., two
intersecting lines.) Before this moment of becoming degenerate,
the hyperbola opens in one orientation or direction, and after
it opens with a different orientation or direction.

Well, first an example, then a proof which relies on technology.

(1) To

verifythat the equation produces a hyperbola for a specific non-zero value ofn, view thisexample.(2) To

provethat the equation produces a hyperbola whennis not 0, view thisproof.

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