Hyperbolas
Ellipses are different from circles because of numerical coefficients for the x2 and y2 terms. Hyperbolas are different from ellipses and circles because one of the coefficients of these x2 and y2 terms is negative. This makes the analysis somewhat more complicated. Hyperbolas are written in one of two forms, both of which are sometimes needed in the graphing.
The ± and upside down ± are to emphasize that the signs of the two terms must be opposite.
It is a bit difficult to describe the graphing technique for hyperbolas. Hyperbolas are unique in that when one variable is set equal to zero the other variable has positive and negative values while when the other variable is set equal to zero the values appear to be the square roots of a negative number. This is a strange result, but it is valuable in that it tells you where the function does not exist. Plotted on an x-y graph the hyperbola is two separate curves with an area between where the curve cannot exist. These two separate curves are symmetrical in either the horizontal or vertical directions. Rather than continue to describe the analysis it is best to actually graph a parabola explaining the logic of the steps in the process.
The form of the equation tells us this is a hyperbola. Now proceed as if this were a circle or ellipse and set x = 0, so that y = ±2. This produces two points that can be placed on the graph.
Now if we set y = 0, there are no real values of x!
If the curve goes through the points (0,2) and (0,-2)
and does not exist along the line y = 0, then the curve must have two
separate parts! In order to sketch the rest of
the curve first rearrange the equation to 4x2 = 25y2
This rearrangement where
the variables are both positive, but on opposite sides of the equation allows
further analysis. Note immediately that for real values of x, y
has to be greater than 2 or less than -2. This little step shows that the
curve does not exist in the region bounded by the lines y = 2 and y
= -2. At this point in the analysis we
have two points and a region where the curve does not exist. Further analysis
requires a departure from the usual techniques applied to conics. Rewrite the equation again, but
this time in the form y = …
How this helps in
graphing is that for large values of x, the function begins to
look like a straight line, y »
±(2/5)x (for large x
the +4 is small compared to 4x2/25). Use these
two straight lines, one of slope (2/5) and the other of slope
-(2/5), as guides in drawing
the curve. With the points (0,2) and (0,-2)
and these lines as guides, the curve can be sketched as shown.
In the language of mathematics
these straight lines are asymptotes or asymptote lines.
Asymptotes are
lines the curve approaches but does not touch. Now that you know the general shape of
hyperbolas, we can look at some hyperbolas that are not symmetric about the
origin. The next problem is somewhat artificial, but it is instructive and
illustrates a situation that comes up in the graphing of hyperbolas. The
remaining problems are increasingly difficult and one requires the completing
the square approach used for circles and ellipses. HYP20
Hyperbola requiring completing the square
