Problem #9 Write Up
by Jeff Hall
graph of the first equation looks like this:
This is a standard parabola.
Now let's add the second equation:
The graph now looks like two parabolas
crossing over one another. The red parabola is a standard parabola,
but the blue parabola is tilted at
an unusual angle.
Are we getting the whole picture?
Since the blue parabola
looks unusual, let's zoom out to see if anything else is going
As it turns out, we were not getting
the whole picture before. When you zoom out,
you realize that the blue
parabola we saw earlier was just part of the equation.
What would happen if I changed the
coefficient in front of the xy variable? I predict that
the two blue parabolas will extend to meet each other.
In this next equation, I placed a 2 in front of
the xy variable. Here are the equation and graph:
The two parabolas are moving closer,
but they are starting to get wider, too.
I predict that the parabolas will continue
to grow wider and closer as the variable in front
of the xy (let's call it "n")
Let's see what happens when I change the n from
2 to 10.
As expected, the blue parabolas continued
to get wider and closer to each other.
However, I am starting to suspect that
the two parabolas will not actually connect with each other,
at least not with a positive n.
try one more positive n just to be sure. Let n=100.
Notice that I had to zoom in...a lot...in
order to see what's happening here.
Our suspicions have been confirmed.
The two parabolas will not connect when n is positive.
try some negative numbers now. Let xy be negative.
The negative xy caused the parabolas
to flip quadrants around a vertical axis.
Will the graphs of negative n-values
continue to be a mirror-image of the positive n-value graphs?
Let's try n as -2.
Interesting. The two parabolas have
shifted so that they form an hourglass shape.
This means that the two parabolas connected
somewhere between n=-1 and n=-2.
What will the graph look like
as n continues to get even more negative? Let's try n=-10.
The graph is again starting to look
like a mirror-image of the positive n equations.
Let's try one last n just to be sure. Let n=-100.
Again, a close zoom
was required to see the the two parabolas better.
As you can see, the
two parabolas are not going to connect again as n approaches negative
We have explored the equation as n cycled from 100
Let's see most of it
together in one movie.
I placed the variable
n, which cycles between -10 and +10, in front of the xy to get
this equation and graph:
Notice what happens
when n=0. The blue and red parabolas
Notice also the point
between n=-1 and n=-2 where the two parabolas finally connect
before diverging again.