Reciprocal Models & Inverse Variation
Master reciprocal models and inverse variation for NSW Year 12 Mathematics Standard 2. In this topic two quantities have a constant product, \(y=\dfrac{k}{x}\): you find the constant of variation \(k=xy\) from one known pair, write the model, and predict any other value.
You will learn to construct a reciprocal model, solve inverse-variation problems algebraically or from the hyperbola graph, tell inverse variation apart from direct variation, and explain the limitations of the model — a core Standard 2 skill for speed and time, workers and time, and pressure and volume problems.
Theory
Inverse variation (a reciprocal model) has two quantities with a constant product, \(y=\dfrac{k}{x}\). This Year 12 Standard 2 (NSW) guide shows how to find the constant of variation \(k=xy\), construct the model, solve inverse-variation problems algebraically or from the hyperbola graph, and explain the limitations of the model.
Two quantities are in inverse variation (a reciprocal model) when their product is constant. The equation is \(y=\dfrac{k}{x}\), where \(k\) is the constant of variation. As \(x\) increases, \(y\) decreases, but the product \(xy=k\) never changes.
To construct the model you find \(k\) from one known pair by multiplying: \(k=xy\). Writing \(y=\dfrac{k}{x}\) then lets you predict any missing value — divide to find \(y\) from \(x\), or rearrange to \(x=\dfrac{k}{y}\) to find \(x\) from \(y\).
The graph of \(y=\dfrac{k}{x}\) (with \(k>0\) and \(x>0\)) is a hyperbola: it falls steeply, hugging both axes without ever touching them. This is the opposite of direct variation \(y=kx\), whose graph is a straight line through the origin where both quantities rise together. A reciprocal model also has limitations: \(x\) can never be \(0\), and in real problems \(x\) is often a whole number.
Inverse variation has the two quantities in a constant product, written as a reciprocal model:
The constant of variation is the product of any matching pair of values:
Rearranging the model lets you find \(x\) when you know \(y\):
How to build and use a reciprocal model
- Recognise inverse variation: \(y=\dfrac{k}{x}\) — as \(x\) increases \(y\) decreases, and the product \(xy\) is constant.
- Find the constant from one known pair by multiplying: \(k=xy\).
- Write the model \(y=\dfrac{k}{x}\).
- Predict: substitute to find \(y=\dfrac{k}{x}\), or rearrange to \(x=\dfrac{k}{y}\) to find \(x\).
- Check the limitations: \(x\) cannot be \(0\), values are often whole numbers, and extreme predictions may be unrealistic.
For inverse variation \(k\) is the product \(xy\).
| \(y\) | \(=\) | \(\dfrac{k}{x}\) |
| \(k\) | \(=\) | \(xy = 4\times 15 = 60\) |
| \(y\) | \(=\) | \(\dfrac{60}{x}\) |
| \(\text{At } x=10:\ y\) | \(=\) | \(\dfrac{60}{10} = 6\) |
So \(k=60\), the model is \(y=\dfrac{60}{x}\), and \(y=6\) when \(x=10\).
Find \(k=st\) from the first ride, then use the model at the new speed.
| \(t\) | \(=\) | \(\dfrac{k}{s}\) |
| \(k\) | \(=\) | \(st = 6\times 8 = 48\) |
| \(t\) | \(=\) | \(\dfrac{48}{s}\) |
| \(\text{At } s=8:\ t\) | \(=\) | \(\dfrac{48}{8} = 6\) |
At \(8\) km/h the ride takes \(6\) hours.
Test a few values to see the trend, then name the shape.
| \(R=3:\ I\) | \(=\) | \(\dfrac{36}{3} = 12\) |
| \(R=6:\ I\) | \(=\) | \(\dfrac{36}{6} = 6\) |
| \(\text{As } R\uparrow,\ I\downarrow\) | \(\Rightarrow\) | \(\text{hyperbola}\) |
| \(\text{At } R=9:\ I\) | \(=\) | \(\dfrac{36}{9} = 4\) |
The graph is a hyperbola falling toward (but never reaching) \(0\); \(I=4\) amps when \(R=9\).
Substitute to find the cost, then consider where the model breaks down.
| \(c\) | \(=\) | \(\dfrac{1800}{n}\) |
| \(c\) | \(=\) | \(\dfrac{1800}{30} = 60\) |
The cost is \(\$60\) per student. Limitation: \(n\) must be a whole number and cannot be \(0\); for very small \(n\) the model predicts an unrealistically large cost, and \(n\) is capped by the bus capacity.
Common pitfalls
Frequently asked questions
What is inverse variation?
Inverse variation is a relationship where two quantities have a constant product. It is written as the reciprocal model y equals k over x, where k is the constant of variation. As x increases, y decreases, but the product xy always equals k.
How do you find the constant of variation k?
Multiply a known pair of values, because k equals x times y. For example, if y is 15 when x is 4, then k is 4 times 15, which is 60, so the model is y equals 60 over x.
What does the graph of y = k/x look like?
For k greater than 0 and x greater than 0 it is a hyperbola in the first quadrant: a curve that falls steeply and then levels off, getting closer and closer to both axes but never touching them. The x-axis and y-axis are asymptotes.
How is inverse variation different from direct variation?
In direct variation y equals kx, so both quantities rise together and the graph is a straight line through the origin. In inverse variation y equals k over x, so one quantity falls as the other rises and the graph is a hyperbola.
How do you find x when you know y?
Use the model y equals k over x and rearrange it to x equals k over y. Substitute the known value of y and divide k by it. For example, with y equals 60 over x, if y is 5 then x is 60 divided by 5, which is 12.
What are the limitations of a reciprocal model?
The value of x can never be 0 because you cannot divide by zero, and for very small x the model predicts values that grow without limit. In real contexts the quantities are often whole numbers (such as people or workers), so only whole-number answers make sense, and very large or very small predictions may be unrealistic.