An Exploration of Two Conic
Sections
By
Ken Montgomery
In this exploration,
we first plot the graph of Equation 1, in purple (Figure 1). We see that this
is a parabola.
Equation 1:
This is the graph of
,
with a = 0.
Figure 1:
So letting a = 1, we obtain
an xy term and the graph of Equation 2, in red (Figure 2).
Equation 2:
This looks, at first to be another
parabola, but with a slightly different major axis.
Figure 2:
,
a = 0 (Purple), a = 1 (Red)
However, redrawing these functions,
with range on the y-axis changed to –25 to 25, we obtain the graphs in Figure
3.
Figure 3:
,
a = 0 (Purple), a = 1 (Red)
What appeared to be a parabola is
actually the top branch of a hyperbola.
Now, we will systematically change
the coefficient of the xy term and observe how this affects the graph.
Letting a = 2, we have the graph of Equation 3, in blue (Figure 4).
Equation 3:
Figure 4:
,
a = 0 (Purple), a = 1 (Red), a =2 (Blue)
The vertices of the hyperbola have
moved closer together and toward the second quadrant, and the general shape has
become sharply defined, as the negative slope of the graph’s horizontal
asymptote appears to have decreased and the slope of the vertical asymptote
appears to have increased. However, letting a = -2, we obtain the graph of
Equation 4 in green (Figure 5).
Equation 4:
Figure 5:
,
a = 0 (Purple), a = 1 (Red), a =2 (Blue), a = -2
(green)
In Equation 4, we
observe a more rounded hyperbola, less sharply defined, with the vertices of
the hyperbola more closely approaching the second quadrant and with the slope
of the horizontal asymptote now having become positive. The change in sign of
the coefficient seems to have changed the sign of the horizontal asymptote and
the change in magnitude of the coefficient seems to have changed the magnitude
of the slope of the horizontal asymptote. Otherwise the graphs still represent
a family of hyperbolas. Lastly, we choose values very close to either side of
zero, to observe this affect. First, letting a = 0.1, we obtain the graph of
Equation 5, in light blue (Figure 6).
Figure 6:
,
a = 0 (Purple), a = 1 (Red), a =2 (Blue), a = -2
(green), a = -0.2 (light blue)
Changing the vertical
range to
and
the horizontal range to
we
obtain the graph of Equation 5 in light blue (Figure 7).
Figure 7:
,
a = 0 (Purple), a = 1 (Red), a =2 (Blue), a = -2
(green), a = -0.2 (light blue)
This verifies that the
graph is still a hyperbola, although its vertices have moved apart by a great
distance and the graph’s shape seems to be changing from hyperbolic in nature
to parabolic, as we would expect.
Download Assign2KM.gcf
to further explore these two examples of conic sections in Graphing Calculator.
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