Variation · 1.0.1
Direct variation
This standard asks you to understand what it means when one quantity varies directly as another: as x increases, y increases in the same proportion, and as x decreases, y decreases too, so the ratio y/x stays constant. You should recognise direct variation from a table of values, a graph, a real situation, or the equation y = kx.
The official learning standard (1.0.1)
“Explain the meaning of direct variation.”
What it means
This standard asks you to understand what it means when one quantity varies directly as another: as x increases, y increases in the same proportion, and as x decreases, y decreases too, so the ratio y/x stays constant. You should recognise direct variation from a table of values, a graph, a real situation, or the equation y = kx.
How it is examined
In Paper 1, expect short objective items asking you to identify which table, graph or statement describes direct variation, or to choose the correct equation y = kx for a given situation. In Paper 2, this idea usually forms the opening part of a longer structured question on variation, setting up later parts that find the constant k or an unknown value.
Worked example
The cost, C (RM), of printing photographs varies directly as the number of photographs printed, n. It costs RM12 to print 8 photographs.
Explain what this means, and state whether tripling n will also triple C.
- Direct variation means C = kn for a constant k, so C is always a fixed multiple of n.
- Substitute C = 12 and n = 8: k = C ÷ n = 12 ÷ 8 = 1.5.
- So the relation is C = 1.5n, meaning the cost per photograph is a constant RM1.50.
- If n is tripled to 3n, then C = 1.5(3n) = 3(1.5n) = 3C, so C is also tripled.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How is direct variation different from just any relationship between two quantities?
In direct variation, the ratio y/x is always the same constant k, and the graph of y against x is a straight line through the origin. Not every relationship behaves this way, for example, y = x + 3 is not direct variation, because y/x is not constant.
Does 'y varies directly as x' always mean y increases whenever x increases?
Yes, provided the constant k is positive, which is the usual case in SPM questions. As x increases, y = kx increases in the same proportion, and as x decreases, y decreases too.
For example, if x doubles, y also doubles.
Can direct variation apply to real-life quantities like cost or distance?
Yes, many everyday situations follow direct variation, such as cost against the quantity bought at a fixed price, or distance travelled against time at constant speed. Once you know the constant k, you can predict one quantity from the other.