Reciprocal Graphs (Hyperbolas)
Master reciprocal graphs and the hyperbola for NSW Year 12 Mathematics Standard 2. In this topic you recognise inverse variation \(y=\dfrac{k}{x}\) (constant product \(xy=k\)), find the constant \(k\) from a point or a table, and sketch the two branches of the curve.
You will learn how the sign of \(k\) places the branches in quadrants 1 & 3 or 2 & 4, how the \(x\)-axis and \(y\)-axis act as asymptotes the curve approaches but never touches, and why a reciprocal model has real limitations — a core Standard 2 skill for modelling quantities whose product stays fixed.
Theory
A reciprocal relationship is inverse variation \(y=\dfrac{k}{x}\), where the product \(xy=k\) is constant. This Year 12 Standard 2 (NSW) guide shows the hyperbola shape and its two branches, how the sign of \(k\) sets the quadrants, how to find \(k\) from a point, the \(x\)-axis and \(y\)-axis asymptotes, and the limitations of a reciprocal model.
A reciprocal relationship is inverse variation: as one quantity grows the other shrinks so that their product stays constant, \(xy=k\). Rearranged, this is \(y=\dfrac{k}{x}\) with \(k\neq 0\), where \(k\) is the constant of variation.
The graph of \(y=\dfrac{k}{x}\) is a hyperbola — two separate smooth branches, not a single line. When \(k>0\) the branches sit in the first and third quadrants; when \(k<0\) they sit in the second and fourth quadrants. A larger \(|k|\) pushes the curve further from the origin.
Both the \(x\)-axis \((y=0)\) and the \(y\)-axis \((x=0)\) are asymptotes: the curve gets ever closer but never touches them, so it has no intercepts. Because \(x=0\) is undefined and \(y\) is never \(0\), a reciprocal model has real limitations — it cannot describe a quantity that actually reaches zero.
Inverse variation has a constant product, giving the reciprocal equation:
The constant of variation is the product of the coordinates of any point on the curve, so one known point fixes the whole graph:
Translating the curve moves its asymptotes:
How to work with a reciprocal graph
- Check it is inverse variation: the product \(xy\) is constant, so the equation is \(y=\dfrac{k}{x}\).
- Find \(k\): multiply the coordinates of a known point, \(k=xy\) (or read a complete column of a table).
- Evaluate: substitute into \(y=\dfrac{k}{x}\) to find \(y\) from \(x\), or rearrange to find \(x\) from \(y\).
- Sketch the shape: two branches — quadrants 1 & 3 if \(k>0\), quadrants 2 & 4 if \(k<0\).
- Mark the asymptotes: the axes for \(y=\dfrac{k}{x}\); shift to \(y=c\) or \(x=r\) for a translated curve. The curve nears them but never touches.
Use the sign of \(k\) to place the two branches.
| \(k\) | \(=\) | \(-8 < 0\) |
| \(x>0\) | \(\Rightarrow\) | \(y<0\ \text{(quadrant 4)}\) |
| \(x<0\) | \(\Rightarrow\) | \(y>0\ \text{(quadrant 2)}\) |
A hyperbola with two branches in the second and fourth quadrants, approaching but never touching the \(x\)- and \(y\)-axes.
The constant \(k\) is the product of the coordinates.
| \(k\) | \(=\) | \(xy = 3\times 5 = 15\) |
| \(y\) | \(=\) | \(\dfrac{15}{x}\) |
| \(\text{At } x=5:\ y\) | \(=\) | \(\dfrac{15}{5} = 3\) |
\(k=15\), so \(y=\dfrac{15}{x}\), and \(y=3\) when \(x=5\).
| \(x\) | 1 | 2 | 4 | 5 |
|---|---|---|---|---|
| \(y\) | 20 | 10 | 5 | ? |
For inverse variation the product \(xy\) is constant.
| \(k\) | \(=\) | \(xy = 1\times 20 = 20\) |
| \(\text{At } x=5:\ y\) | \(=\) | \(\dfrac{20}{5} = 4\) |
| \((10,2):\ 10\times 2\) | \(=\) | \(20 = k\) |
\(k=20\); the missing value is 4; and \((10,2)\) does lie on the curve since \(10\times 2=20\).
Let \(x\) grow large, then let \(x\) approach 0.
| \(x\to\pm\infty:\ \dfrac{4}{x}\) | \(\to\) | \(0,\ \text{so } y\to 1\) |
| \(x\to 0:\ \dfrac{4}{x}\) | \(\to\) | \(\pm\infty\) |
Horizontal asymptote \(y=1\), vertical asymptote \(x=0\). The curve gets ever closer, but \(\dfrac{4}{x}\) is never exactly \(0\) and \(x=0\) is undefined.
Common pitfalls
Frequently asked questions
What is a reciprocal relationship?
It is inverse variation: two quantities whose product is constant, xy equals k, which rearranges to y equals k over x. As one quantity increases the other decreases so that the product stays the same.
What shape is the graph of y = k/x?
A hyperbola. It has two separate smooth branches, not a single straight line. Each branch curves toward the axes without ever touching them.
How does the sign of k change the graph?
When k is positive the two branches lie in the first and third quadrants. When k is negative they lie in the second and fourth quadrants. A larger size of k pushes the curve further from the origin.
What are the asymptotes of y = k/x?
The x-axis (y = 0) and the y-axis (x = 0). The curve approaches both lines as x or y become very large or very small, but it never actually touches them, so there are no intercepts.
How do you find k from a point on the curve?
Multiply the coordinates of the point. If y equals k over x passes through the point (a, b), then k equals a times b, and the equation is y equals that value over x.
Why does a reciprocal model have limitations?
Because y equals k over x is undefined at x = 0 and can never give y = 0, the model cannot describe a situation where a quantity reaches exactly zero, and it breaks down near x = 0 where the value grows without bound.