A student claims: 'If y decreases when x increases, the relationship is inverse variation.' Which data table below disproves this claim?
Ax=1, y=12; x=2, y=6; x=4, y=3 — product xy = 12 in each row
Bx=1, y=−2; x=2, y=−4; x=3, y=−6 — ratio y/x = −2 in each row
Cx=2, y=10; x=4, y=5; x=5, y=4 — product xy = 20 in each row
Dx=1, y=8; x=2, y=4; x=4, y=2 — product xy = 8 in each row
Table B shows y decreasing as x increases, yet y/x is constant (= −2) — making this direct variation (y = −2x), not inverse variation. The student's claim confuses 'y decreases as x increases' with inverse variation, but a line with negative slope also has this property. Inverse variation requires the product xy to be constant, which is not the case in table B. Tables A, C, and D all have constant products and are genuine inverse variation.
Question 2 Multiple Choice
A car travels a fixed distance of 300 km. As average speed increases, travel time decreases. What type of relationship describes speed and time?
ADirect variation — time = k × speed, where k = 300
BInverse variation — speed × time = 300 (a constant product)
CNeither — the relationship is exponential because time decreases rapidly at first
DDirect variation, because distance acts as the constant of proportionality
Speed × time = distance = 300, a constant. This is the definition of inverse variation: xy = k. As speed doubles, time halves; the product is always 300. Option A has the equation backwards — if time = k × speed, time would increase as speed increases, which is wrong. Options C and D misidentify the structure of the relationship.
Question 3 True / False
In direct variation, the graph always passes through the origin (0, 0).
TTrue
FFalse
Answer: True
The defining equation y = kx produces y = 0 when x = 0, regardless of k. This is built into the equation — the origin is always on the line. This distinguishes direct variation from other linear functions like y = kx + b (b ≠ 0), which are proportional relationships but not direct variations. If a linear graph does not pass through the origin, it cannot be a direct variation.
Question 4 True / False
Inverse variation means y = −kx — the two quantities vary inversely because one is the negative of the other.
TTrue
FFalse
Answer: False
Inverse variation means y = k/x (equivalently, xy = k), which produces a hyperbola — not a line. 'Inversely' here means the variables change in opposite directions with their product held constant. A line y = −kx with negative slope is actually direct variation with a negative constant of proportionality, because the ratio y/x = −k is constant. The graph of inverse variation never produces a straight line.
Question 5 Short Answer
How do you determine from a table of values whether two quantities have direct variation, inverse variation, or neither?
Think about your answer, then reveal below.
Model answer: Compute two quantities for each row: the ratio y/x and the product xy. If y/x is constant across all rows, the relationship is direct variation and k = y/x. If xy is constant across all rows, the relationship is inverse variation and k = xy. If neither is constant, the relationship is neither type. Only one should be constant for any given non-trivial relationship.
This diagnostic procedure is the practical core of the topic. It works because direct variation requires y = kx (so y/x = k), and inverse variation requires xy = k. Running both checks on a table quickly identifies the variation type and extracts the constant k, which then gives the full equation for predicting any additional values.