Mathematics Year 7: Direct proportionality as a function
Mathematics, Year 7 (Portugal): A direct-proportionality function is y = k × x with a domain: its table, its graph (points on a line through the origin) and its expression tell the same story.
An expression plus a domain
You already know the constant k and the expression y = k × x from proportional relations. Now see it as a function: the expression and the domain (the values allowed for x) define it completely. The same expression with different domains gives different functions: y = 2 × x with x in {1, 2, 3} has only 3 pairs, (1, 2), (2, 4) and (3, 6); with x taking any value from 0 upwards it has infinitely many. Two marks identify this function: the object 0 always has image 0, and doubling x doubles y.
Why the graph is a line through the origin
Take y = 3 × x. When x grows by 1, y always grows by 3, so the points (1, 3), (2, 6), (3, 9) are evenly spaced along a straight line. And (0, 0) belongs to it, because 3 × 0 = 0. If the domain is separate values, the graph is points on that line; if it is an interval, the graph is a piece of the line. So among several graphs, the ones for direct proportionality lie on a straight line through (0, 0). One function also solves every missing-value problem: from x you get y = k × x, and from y you get x = y ÷ k.
Map scale: expression, table and graph
On a map at scale 1 : 50 000, 3 cm on the map is 3 × 50 000 = 150 000 cm = 1.5 km on the ground. The constant is 1.5 ÷ 3 = 0.5 km per cm, so y = 0.5 × x, with x from 0 to 8 cm (the domain) and y in km. Table: x = 2, 4, 6, 8 gives y = 1, 2, 3, 4. Graph: the segment from (0, 0) to (8, 4). Use it: 5.5 cm are 0.5 × 5.5 = 2.75 km, and 3.5 km on the ground are 3.5 ÷ 0.5 = 7 cm. Backwards: from x = 3, 5, 7 and y = 1.5, 2.5, 3.5, y ÷ x is always 0.5, so y = 0.5 × x.
A straight line is not enough
The points (1, 5), (2, 7), (3, 9), (4, 11) lie on a straight line, and many people conclude that y is directly proportional to x. But the ratios y ÷ x are 5, 3.5, 3 and 2.75: they are not constant. Doubling x from 1 to 2 does not double y (5 → 7, not 10). Extended to x = 0, the line would give y = 3, so it does not pass through the origin. For direct proportionality you need a straight line through (0, 0), or all the ratios equal to the same k.
Density, map scales and enlarged photos
In Physics and Chemistry, the mass of a piece of a homogeneous material is directly proportional to its volume: for a density of 2.7 g/cm³, about that of aluminium, y = 2.7 × x, so 10 cm³ weigh 27 g and 25 cm³ weigh 67.5 g. In Geography the map scale links lengths on the map to real lengths. In geometry, if a 6 cm × 4 cm photo is enlarged 3 times, corresponding lengths become 18 cm and 12 cm, each 3 times the original. In all of them you can test a table or a graph: constant ratio, line through the origin.
Test what you learned
Which of these expressions defines a function of direct proportionality?
- a) y = 6 × x
- b) y = x + 6
- c) y = 6
Correct answer: a) y = 6 × x — It has the form y = k × x with k = 6: doubling x doubles y, and x = 0 gives y = 0.
In an experiment, the mass y (g) of a sample was measured for several volumes x (cm³): (2, 9), (4, 18), (6, 27), (8, 36). Which expression matches the table?
- a) y = x + 7
- b) y = 4.5 × x
- c) y = 9 × x
Correct answer: b) y = 4.5 × x — 9 ÷ 2, 18 ÷ 4, 27 ÷ 6 and 36 ÷ 8 are all 4.5, so y = 4.5 × x.
The points (1, 5), (2, 7), (3, 9), (4, 11) lie on a straight line. Is y directly proportional to x?
- a) Yes, because the points are on a straight line
- b) Yes, because y grows by 2 every time x grows by 1
- c) No: the ratios y ÷ x (5, 3.5, 3, 2.75) are not constant
Correct answer: c) No: the ratios y ÷ x (5, 3.5, 3, 2.75) are not constant — In direct proportionality y ÷ x is always the same k. Here it changes, and the line does not pass through the origin.
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