Model practical problems involving reciprocal functions and inverse variation of the form $y = \frac{k}{x}$

What this dot point is asking

NESA wants you to recognise inverse variation in worded problems (speed-time at fixed distance, pressure-volume at fixed temperature, workers-hours for a fixed job), set up the reciprocal model $y = \frac{k}{x}$, find the constant of proportionality, and use the model to predict.

The answer

Direct vs inverse variation

• Direct. $y$ doubles when $x$ doubles. Equation: $y = k x$.
• Inverse. $y$ halves when $x$ doubles. Equation: $y = \frac{k}{x}$.

In inverse variation, the product $x y = k$ is constant.

The reciprocal function

• For $k > 0$: branches in the first and third quadrants.
• For $k < 0$: branches in the second and fourth quadrants.
• Two asymptotes: the $x$-axis ($y = 0$) and the $y$-axis ($x = 0$).
• The graph is a rectangular hyperbola.

Finding IMATH_19

Given one $(x, y)$ pair from the problem, compute

Then write the full model and use it for other values.

Domain considerations

In a real-world problem, the variables are usually positive (speed cannot be negative, pressure cannot be zero), so you only use the first-quadrant branch. Always state the practical domain when the question asks for a graph.

Practical applications

• Speed and time at fixed distance. $t = \frac{d}{s}$. The constant is the trip distance.
• Pressure and volume of a gas at fixed temperature (Boyle's Law). $P V = k$.
• Workers and time for a fixed job. $T = \frac{W}{n}$ where $W$ is total worker-hours and $n$ is number of workers.
• Per-unit cost and number of units when total cost is fixed. Cost per pizza when splitting a \50$order$n$ways is$\frac{50}{n}$.