Quick questions on Direct variation for HSC Maths Standard 2

The whole method turns on one move: you are given a single pair of matching values, you substitute them into $y = kx$, and you solve for $k$. After that the equation is fully known and you can do anything with it. NESA examines this as a clean three-step routine.

Once $k$ is known and the equation is written, using it is the same two-way skill as any model. To predict, substitute the input and evaluate. With $p = 24.5h$, the pay for a $30$-hour week is

A question often gives a table and asks whether the quantities are in direct variation, without using the words. The test is the constant-ratio test: work out $\dfrac{y}{x}$ for every column. If the ratio is the same number every time, the quantities are in direct variation and that number is $k$; if the ratios differ, they are not.

The graph of a direct-variation relationship $y = kx$ is a straight line that passes through the origin, and its gradient is the constant of variation $k$. Both features must hold: a straight line that misses the origin is a linear model with a non-zero intercept, not direct variation, and a curve is not direct variation at all. To sketch one, you can use the methods in Graphing linear functions: plot the origin and one other point from the equation, then rule the line.

For contrast only, NESA names the opposite case, inverse variation (or inverse proportion), written $y = \dfrac{k}{x}$. It is the reverse of direct variation: as one quantity goes up the other goes down (for example, more workers means fewer days to finish a job), so its graph is a falling curve rather than a rising straight line through the origin. It is met here just as this brief point of comparison; the focus of this dot point stays on direct variation $y = kx$.