**An Exploration of Two Conic
Sections**

**By**

**Ken Montgomery**

**Equation 1**:

This is the graph of *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)

**Figure 6**: *a* = 0 (Purple), *a* = 1 (Red), *a* =2 (Blue), *a* = -2
(green), *a* = -0.2 (light blue)

**Figure 7**: *a* = 0 (Purple), *a* = 1 (Red), *a* =2 (Blue), *a* = -2
(green), *a* = -0.2 (light blue)

Download Assign2KM.gcf to further explore these two examples of conic sections in Graphing Calculator.

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