**The graph of our original ellipse**

**is**

**Here we see that the asymptotes are y =
-2 and x = -2 in rectangular form.**

**Let's begin by changing the sign of the
2 from positive to negative.**

**Speaking in terms of rectangular coordinates,
we see that this moved the vertical asymptote to x = 2 and the
horizontal asymptote to y = 2. This movement can be viewed as
a translation along the line y = x. **

**Now, let's change the sign of the 1 in denominator.**

**There is no difference between the original
graph and this one.**

**Now, let's see what happens when we make
the equation**

**Here we see the original graph with the graph of the altered equation.**

**Notice that the new graph is a reflection
of the original across the line y = 0. Also, the horizontal asymptote
has shifted, as we may have expected remembering that ,
so by changing the coefficient of sine, we have altered the y
values. The asymtotes (in rectangular form) are now x = -2 and
y = 2. **

**Finally, let us investigate how the graph
of the original equation alters when we change the sign of the
cosine term in the denominator. If the graph acts in a manner
similar to the previous example, we would expect the y asymptote
to remain unchanged while the x asymptote does change.**

**The original
equation and the new equation.**

**Here, we readily see that
the new graph is the reflection image of the original across the
y axis ( x = 0). So, indeed the horizontal asymptote y = -2 did
not change while the vertical asymptote moved to x = 2.**