Rectangular Hyperbolas
Understand rectangular hyperbolas for Queensland Year 11 Mathematical Methods (QCAA). A rectangular hyperbola is the graph of a reciprocal such as one over x, a hyperbolic shape with two branches.
You will learn to find the asymptotes and intercepts, describe how the curve behaves for large positive and negative values of x, see the effect of the parameters that shift the curve, and determine its rule from a graph.
Every question with a fully worked solution.
- Rectangular Hyperbolas - Video - The rectangular hyperbola Watch
Theory
In Year 11 Mathematical Methods (QCAA, Unit 1), a rectangular hyperbola is the graph of a reciprocal function such as \(y=\dfrac{1}{x}\) or \(y=\dfrac{a}{x-h}+k\). This page shows how to find its asymptotes, intercepts, domain and range, and how to determine the rule from a graph.
A rectangular hyperbola is the graph of \(y=\dfrac{1}{x}\). It has two branches that sit in opposite quadrants and get closer and closer to the axes without ever touching them.
A line the curve approaches but never meets is an asymptote. For \(y=\dfrac{a}{x-h}+k\) the vertical asymptote is \(x=h\) (the value that makes the denominator zero) and the horizontal asymptote is \(y=k\) (the value \(y\) approaches as \(x\to\pm\infty\)). Here \(a\) is a dilation; a negative \(a\) also reflects the curve.
The domain is every \(x\) except \(x=h\), and the range is every \(y\) except \(y=k\).
The general reciprocal function and its asymptotes:
Domain and range in set notation:
How to analyse \(y=\dfrac{a}{x-h}+k\)
- Asymptotes: write the vertical asymptote \(x=h\) and the horizontal asymptote \(y=k\) straight from the rule.
- Intercepts: substitute \(x=0\) for the \(y\)-intercept, then set \(y=0\) and solve for the \(x\)-intercept.
- Domain and range: exclude \(x=h\) and \(y=k\); note whether \(a<0\) reflects the branches.
Values — substitute each \(x\):
| \(y|_{x=2}\) | \(=\) | \(\dfrac{6}{2}=3\) |
| \(y|_{x=-3}\) | \(=\) | \(\dfrac{6}{-3}=-2\) |
So the curve passes through \((2,\,3)\) and \((-3,\,-2)\).
Asymptotes — the rule is \(\dfrac{6}{x-0}+0\):
| \(x\) | \(=\) | \(0\) |
| \(y\) | \(=\) | \(0\) |
The two branches sit in opposite quadrants.
Asymptotes \(x=0,\ y=0\); domain \(x\in\mathbb{R}\setminus\{0\}\); range \(y\in\mathbb{R}\setminus\{0\}\).
Asymptotes — read \(h\) and \(k\):
| \(x\) | \(=\) | \(2\) |
| \(y\) | \(=\) | \(1\) |
\(y\)-intercept — put \(x=0\):
| \(y\) | \(=\) | \(\dfrac{3}{0-2}+1\) |
| \(=\) | \(-\dfrac{3}{2}+1\) | |
| \(=\) | \(-\dfrac{1}{2}\) |
\(x\)-intercept — put \(y=0\):
| \(0\) | \(=\) | \(\dfrac{3}{x-2}+1\) |
| \(-1\) | \(=\) | \(\dfrac{3}{x-2}\) |
| \(-(x-2)\) | \(=\) | \(3\) |
| \(x-2\) | \(=\) | \(-3\) |
| \(x\) | \(=\) | \(-1\) |
Asymptotes \(x=2,\ y=1\); intercepts \(\left(0,\,-\tfrac{1}{2}\right)\) and \((-1,\,0)\).
Frame — the asymptotes give \(h\) and \(k\):
| \(h\) | \(=\) | \(-1\) |
| \(k\) | \(=\) | \(2\) |
So \(y=\dfrac{a}{x+1}+2\).
Find \(a\) — substitute the point \((0,\,5)\):
| \(5\) | \(=\) | \(\dfrac{a}{0+1}+2\) |
| \(5\) | \(=\) | \(a+2\) |
| \(a\) | \(=\) | \(3\) |
Rule: \(y=\dfrac{3}{x+1}+2\).
Asymptotes:
| \(x\) | \(=\) | \(-2\) |
| \(y\) | \(=\) | \(3\) |
\(y\)-intercept — put \(x=0\):
| \(y\) | \(=\) | \(-\dfrac{4}{0+2}+3\) |
| \(=\) | \(-2+3\) | |
| \(=\) | \(1\) |
\(x\)-intercept — put \(y=0\):
| \(0\) | \(=\) | \(-\dfrac{4}{x+2}+3\) |
| \(\dfrac{4}{x+2}\) | \(=\) | \(3\) |
| \(4\) | \(=\) | \(3(x+2)\) |
| \(4\) | \(=\) | \(3x+6\) |
| \(3x\) | \(=\) | \(-2\) |
| \(x\) | \(=\) | \(-\dfrac{2}{3}\) |
Asymptotes \(x=-2,\ y=3\); intercepts \((0,\,1)\) and \(\left(-\tfrac{2}{3},\,0\right)\); domain \(\mathbb{R}\setminus\{-2\}\), range \(\mathbb{R}\setminus\{3\}\).
Common pitfalls
Frequently asked questions
What is a rectangular hyperbola?
It is the graph of a reciprocal function such as \(y=\dfrac{1}{x}\): two smooth branches in opposite quadrants that approach the axes without touching them.
How do you find the asymptotes of a hyperbola?
For \(y=\dfrac{a}{x-h}+k\), the vertical asymptote is \(x=h\) (denominator zero) and the horizontal asymptote is \(y=k\).
What are the domain and range of a hyperbola?
The domain is all real \(x\) except \(x=h\), and the range is all real \(y\) except \(y=k\).
How do you find the intercepts of a rectangular hyperbola?
Substitute \(x=0\) for the \(y\)-intercept and set \(y=0\), then solve, for the \(x\)-intercept.
What is the difference between the graphs of one over x and one over x squared?
\(y=\dfrac{1}{x}\) has branches in opposite quadrants (one positive, one negative), while \(y=\dfrac{1}{x^2}\) has both branches above the \(x\)-axis.