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.