Variation · Form 5
Direct variation, explained
Direct variation means two quantities grow (or shrink) together at a fixed rate: when one is multiplied by a number, so is the other. We write y ∝ x, which becomes y = kx for a constant k.
What direct variation says
If y varies directly as x, then doubling x doubles y and tripling x triples y, the ratio y/x always stays the same number, k. That constant k is called the constant of variation, and it is simply the fixed 'exchange rate' between the two quantities.
For example, if 3 litres of petrol cost RM9, then y/x = 3 stays constant, so 5 litres cost RM15.
How to recognise it in a question
Look for wording like 'varies directly as', 'is proportional to', or a table where every pair gives the same y/x value. On a graph, direct variation is a straight line that passes through the origin (0, 0), because when x is 0, y must be 0 too.
The relationship can also involve a power, such as y ∝ x² (then y/x² is constant) or y ∝ √x, and this is still called direct variation.
The usual misconception
Not every relationship where 'y goes up as x goes up' is direct variation. The line y = 2x + 5 rises as x rises, but it does not pass through the origin and y/x is not constant, so it is not direct variation.
The real test is a constant ratio (or a line through the origin), not merely 'increasing together'.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How can I tell from a table of values that y varies directly as x?
Check whether y/x stays the same for every pair of values in the table, that constant is k. If the ratio is fixed and the graph would pass through the origin as a straight line, it's direct variation, y = kx.
If the ratio changes, it isn't direct variation.
How do I find k and use it to solve for unknown values?
Substitute one known pair (x, y) into y = kx to solve for k, then rewrite the equation with k's value. Use this equation to find any missing x or y by substituting the given value and solving.
Exam questions usually give one pair to find k, then ask for another value.
What's the biggest mistake students make with direct variation?
Many students write y = kx + c, adding a constant that doesn't belong, direct variation always passes through the origin, so there's no '+c'. Another slip is assuming any straight-line graph is direct variation; only lines through (0, 0) qualify.
Always check the ratio y/x is truly constant.