Inverse proportion and the homographic function

There is a trip next week and 240 crates to move. One lorry takes them all, two take 120 each, four take 60 each. The product stays the same even though both factors change. That is inverse proportion.

Direct or inverse

Two quantities can be tied together in two opposite ways, and mixing them up is the commonest mistake in word problems.

Proportions and scale has a lesson of its own; here only the second column matters — and the function it describes.

The hyperbola y = a/x

The domain is every real number except zero, because dividing by zero is impossible. That single exclusion explains the entire shape of the graph.

The curve has two branches, because the excluded zero cuts the argument axis in two with no crossing between them. For $a > 0$ the branches sit in the first and third quadrants, for $a < 0$ in the second and fourth.

The constant from a single point

Since $y = \tfrac{a}{x}$, multiplying both sides by $x$ gives

The product of the coordinates of every point on the hyperbola is the same, and equal to $a$. One point is therefore enough to pin down the whole function — unlike a line, which needs two.

Asymptotes

Watch what happens to $y = \tfrac{6}{x}$ as the argument shrinks towards zero:

The values grow without bound. The line $x = 0$ is a vertical asymptote: the curve hugs it ever more tightly and never touches it.

At the other end, for large arguments:

The values shrink towards zero without being zero. The line $y = 0$ is a horizontal asymptote.

An asymptote is a line the curve approaches arbitrarily closely without ever reaching it. It does not belong to the graph — it is a reference line saying where the graph is headed.

The same idea returns later at the limit of a function, where "arbitrarily close" gets a precise definition, and at rational functions, where a third kind of asymptote — the oblique one — joins in.

The homographic function

A hyperbola can be translated like any other graph:

Such a function is called homographic. The translation by $[p,\, q]$ takes the asymptotes with it:

The domain is every real number except $p$, because at $x = p$ the denominator vanishes.

Vertex form

In problems, a homographic function usually arrives as a quotient of two linear expressions:

In that spelling the asymptotes are invisible. To read them off, split as much of the denominator out of the numerator as possible:

and then divide term by term:

Now both numbers are written out: the asymptotes are $x = 2$ and $y = 3$.

In general:

The horizontal asymptote sits at the height of the coefficient of $x$ in the numerator, and the vertical one at the zero of the denominator.

Note that cancelling a common factor in rational expressions is a different operation: there a fraction is simplified, here its spelling changes while the function does not.

Where you meet it

Everywhere something is split into parts, or the product of two quantities is fixed:

• $t = \dfrac{s}{v}$ — travel time at a given speed (see speed, distance and time); to convert the units quickly, the speed converter does it;
• $p \cdot V = \text{const}$ — Boyle's law for a gas at constant temperature;
• $I = \dfrac{U}{R}$ — current at constant voltage;
• $\text{cost per person} = \dfrac{\text{total cost}}{\text{number of people}}$.

In all four the graph falls ever more slowly — and that is the practical meaning of a horizontal asymptote: each additional unit brings a smaller gain than the one before.

Practice

The set checks the four steps of this lesson: the constant $k$ read off a single point, a value of a homographic function, the change to vertex form, and both asymptotes at once. For the asymptotes you type two numbers — the vertical one as $x$, the horizontal one as $y$. For the vertex form, copying the prompt back will not pass: the point is to change the spelling, not to repeat it.

Common mistakes

• Confusing inverse proportion with direct — check what is constant: the ratio or the product.
• Finding $a$ by dividing the coordinates — for a hyperbola $a = x \cdot y$, not $\tfrac{y}{x}$.
• Drawing the hyperbola as one stroke through zero — there are two branches and nothing joins them.
• Treating an asymptote as part of the graph — it is a reference line, not a set of points of the function.
• Reading the horizontal asymptote straight off $\tfrac{ax+b}{x-p}$ — convert to vertex form first; the asymptote is $y = a$, not $y = b$.
• Forgetting the domain — $x = p$ is excluded and must be stated in the answer.