Rational Functions: Reciprocal Proportionality and the Hyperbola y=a/x
A rational function is a function that is the quotient of two polynomials, and the simplest case is the reciprocal function y = a/x, whose graph is a hyperbola with the axes as asymptotes. In brief: in y = a/x the product xy = a is constant (inverse proportionality). For a > 0 the two branches lie in quadrants I and III, for a < 0 in quadrants II and IV. For y = 12/x, the points (1, 12), (2, 6), (3, 4), (4, 3), (6, 2) and (12, 1) lie on the graph.
Reciprocal proportionality describes situations where one quantity increases as the other decreases so that their product stays constant: speed and travel time over a fixed distance, the number of workers and the time to finish a job, pressure and volume of a gas. This guide explains the function y = a/x, its graph and properties, how to find a from data, the general form y = a/(x − h) + k and a table of values with a figure.
What is inverse proportionality?
Two quantities x and y are inversely proportional if their product is a nonzero constant: xy = a, so y = a/x. If x is multiplied by k, then y is divided by k: doubling x halves y. The constant a is the constant of proportionality.
Example: a car travelling 120 km at speed v takes t = 120/v hours. At 40 km/h the time is 3 h, at 60 km/h it is 2 h and at 120 km/h it is 1 h.
What does the graph of y = a/x look like?
The graph of y = a/x is a hyperbola with two branches. The function is undefined at x = 0, so the graph has a vertical asymptote x = 0 (the y-axis). As |x| grows, y approaches 0, so y = 0 (the x-axis) is a horizontal asymptote.
For a > 0 the branches are in quadrants I and III. For a < 0 they are in quadrants II and IV, and the graph of y = −a/x is the reflection of y = a/x in the x-axis (or in the y-axis). The larger |a|, the farther the branches are from the origin.
What are the properties of the reciprocal function?
- Domain: x ≠ 0. Range: y ≠ 0.
- Odd function: f(−x) = −f(x), so the graph is symmetric about the origin. It is also symmetric about the lines y = x and y = −x (for a > 0, y = x is the axis through the vertices (√a, √a) and (−√a, −√a)).
- For a > 0, decreasing on (−∞, 0) and on (0, ∞), but not on the whole domain, since f(−1) < f(1).
- No zeros and no y-intercept.
- The inverse of y = a/x is itself: x = a/y gives y = a/x.
The reciprocal function is a basic example of a function with asymptotes.
What is the form y = a/(x − h) + k?
Shifting the hyperbola h units to the right and k units up gives y = a/(x − h) + k, with the vertical asymptote x = h and the horizontal asymptote y = k. The domain is x ≠ h and the range is y ≠ k.
Value table for y = 12/x
| x | −12 | −6 | −4 | −2 | −1 |
|---|---|---|---|---|---|
| y | −1 | −2 | −3 | −6 | −12 |
| x | 1 | 2 | 4 | 6 | 12 |
|---|---|---|---|---|---|
| y | 12 | 6 | 3 | 2 | 1 |
Every pair has the product 12. Values of x between −1 and 1 give large values: 12/0.5 = 24 and 12/0.1 = 120.
Worked examples
Example 1: finding a
y is inversely proportional to x, and y = 8 when x = 3. Then a = 24 and y = 24/x. For x = 6: y = 4. For y = 2: x = 12.
Example 2: work
6 workers finish a job in 10 days. Assuming the same pace, the time for n workers is t = 60/n. With 15 workers: 4 days. With 4 workers: 15 days.
Example 3: a rectangle of fixed area
A rectangle has area 36 cm2. The width w and length l satisfy lw = 36, so l = 36/w. For w = 4: l = 9. For w = 6 the rectangle is a square. The perimeter 2(w + 36/w) is smallest at w = 6.
Example 4: the gas law
At constant temperature, pressure and volume are inversely proportional (Boyle’s law): pV = constant. If a gas at 100 kPa occupies 6 litres, compression to 2 litres raises the pressure to 100 · 6/2 = 300 kPa.
Example 5: finding the intersection with a line
Where does y = 12/x meet y = x + 4? 12/x = x + 4 gives x2 + 4x − 12 = 0, so (x + 6)(x − 2) = 0, x = 2 or x = −6. The points are (2, 6) and (−6, −2).
Where does the hyperbola come from?
The hyperbola is one of the conic sections studied by Apollonius of Perga, and the name hyperbola was introduced by him. The equation xy = a, in which the hyperbola has the axes as asymptotes, was studied after the development of coordinate geometry in the seventeenth century. Robert Boyle’s law, published in 1662, is the first well-known application of inverse proportionality in physics.
Where is it used today?
Inverse proportion appears in unit pricing (the number of items you can buy for a fixed budget), in the relation between frequency and wavelength (λ = c/f), in Ohm’s law (I = V/R for a fixed voltage) and in gear ratios. The average speed and the time over a fixed distance are inversely proportional, and fuel consumption in litres per 100 km is the reciprocal of distance per litre.
Which mistakes are common?
Mixing up direct and inverse proportion
In direct proportion the ratio y/x is constant. In inverse proportion the product xy is constant.
Connecting the branches
The two branches of the hyperbola are separate, because the function is undefined at x = 0.
Thinking that the graph touches the axes
The curve approaches the axes but never reaches them: there is no x with 12/x = 0.
Saying the function is decreasing everywhere
For a > 0 it decreases on each branch separately.