Quick questions on Reciprocal models and inverse variation for HSC Maths Standard 2

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

<svg viewBox="0 0 480 360" role="img" aria-labelledby="rec-t rec-d" style="max-width: 100%; height: auto; display: block; margin: 1em auto;"> <title id="rec-t">Reciprocal function y equals k over x with two branches</title> <desc id="rec-d">A rectangular hyperbola with one branch in the first quadrant and a mirror branch in the third quadrant. Both axes are asymptotes; the curve approaches them but never touches.</desc> <g font-family="system-ui, sans-serif" font-size="13" fill="currentColor"> <line x1="40" y1="180" x2="460" y2="180" stroke="currentColor" stroke-width="1.5"/> <polygon points="460,180 452,176 452,184" fill="currentColor"/> <line x1="240" y1="30" x2="240" y2="335" stroke="currentColor" stroke-width="1.5"/> <polygon points="240,30 236,38 244,38" fill="currentColor"/> <text x="455" y="198">x</text> <text x="224" y="32">y</text> <text x="248" y="195" font-size="11">0</text> <path d="M 252 50 Q 270 80 290 110 Q 320 150 360 168 Q 400 175 450 178" fill="none" stroke="currentColor" stroke-width="2"/> <path d="M 228 310 Q 210 280 190 250 Q 160 210 120 192 Q 80 185 30 182" fill="none" stroke="currentColor" stroke-width="2"/> <text x="350" y="115" font-style="italic">y = k/x, k &gt; 0</text> <text x="55" y="245" font-size="11" opacity="0.7">third quadrant branch</text> <text x="380" y="200" font-size="11" opacity="0.7">first quadrant branch</text> </g> </svg>

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

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.

:::worked Worked example ### Sydney to Newcastle drive

A household budget for electricity is \1200$ per quarter. Cost per kWh used is

A pump empties a tank in $T$ hours when running $n$ pumps simultaneously, modelled by $T = \frac{12}{n}$ (so $1$ pump takes $12$ hours).

In a physical problem, $k$ is almost always positive. If you get a negative $k$, recheck your $(x, y)$ pair.

The hyperbola $y = k/x$ never passes through the origin; it has both axes as asymptotes.

Speed cannot be zero or negative; pressure cannot be zero. Only the first-quadrant branch matters.

In inverse variation, $k = x y$ is constant. Use that to check every answer. :::