Hyperbola & reciprocal function

A pizza is shared among $x$ people. With 2 people everyone gets half, with 4 people a quarter, with 10 people only a tenth. The share one person receives is exactly $y = \frac{1}{x}$. This function is called the reciprocal function, and its graph has a shape that no line and no parabola can produce.

The larger x, the smaller y

Work out a few values: $\frac{1}{1} = 1$, $\frac{1}{2} = 0.5$, $\frac{1}{4} = 0.25$ and $\frac{1}{10} = 0.1$. It works the same way in the other direction: $\frac{1}{0.5} = 2$, because 0.5 fits into 1 exactly twice, and $\frac{1}{0.1} = 10$. When $x$ grows, $y$ shrinks, and when $x$ shrinks, $y$ grows. A small test tells you whether a point belongs to the curve: multiply its two coordinates, and for the reciprocal function you always get 1.

The forbidden zero

One spot stays empty: $x = 0$. Dividing by zero is not allowed, because no number times zero gives 1. In the pizza picture it makes no sense either to share a pizza among zero people. So the domain consists of every number except 0, written briefly as $x \neq 0$. At this spot the graph has a gap, and that splits it into two separate pieces, which are called branches.

Right next to the gap things get exciting. Put in $x = 0.1$ and you get $y = 10$, at $x = 0.01$ it is already 100, and at $x = 0.001$ a full 1000. As you approach the gap from the right, that is from the positive numbers, the curve shoots steeply upwards. Coming from the left it drops just as steeply, because $\frac{1}{-0.001} = -1000$. Far out the opposite happens: $x = 100$ gives 0.01 and $x = 1000$ gives 0.001. The value becomes arbitrarily small, yet it never reaches 0, because 1 divided by a number is never exactly zero.

Two branches, two asymptotes

Together this gives the typical shape: two branches that never touch each other. For positive $x$ the branch sits in the upper right, for negative $x$ in the lower left, because 1 divided by a negative number stays negative. This curve is called a hyperbola, and it is point-symmetric about the origin: for every point $(x \mid y)$ the point $(-x \mid -y)$ belongs to it as well. The two axes are its asymptotes, lines that the curve approaches ever more closely without ever reaching them: the $y$-axis at the gap and the $x$-axis far out.