Direct proportion means that when one quantity doubles or triples, the other does the same, and we write it as y = ax.
Walking at a steady speed is a good example: more time means proportionally more distance.
Inverse proportion, y = a/x, works the opposite way, since a faster speed over a fixed distance takes less time, so the product xy stays constant.
Here you slide the constant a and watch the line through the origin and the hyperbola change.
Direct Proportion: y = ax
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Direct-proportion graph — line through the origin
2
👀 What is Direct Proportion?
①Doubling x doubles y; tripling triples it
②Form y = ax (a ≠ 0)
③Graph: line through the origin
④a > 0 → upper right; a < 0 → lower right
Inverse Proportion: y = a/x
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Inverse-proportion graph — hyperbola
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🔍 What is Inverse Proportion?
①Doubling x halves y
②Form y = a/x (a ≠ 0)
③Graph: hyperbola (approaches axes without touching)
④xy = a (product is constant)
Meaning of the Constant
Direct Constant
a = yx
y/x is constant and equals a
Inverse Constant
a = xy
xy is constant and equals a
💡 Finding the Constant
①Direct: one point (x, y) gives a = y/x
②Inverse: one point gives a = xy
③The sign of the constant determines the graph's position
Work It Out
Example 1
y is directly proportional to x, and y = 6 when x = 2. Write y as an expression in x.
1
Direct proportion is y = ax. Substitute x = 2, y = 6 to find a.
6 = a × 2 ⇒ a = 3
2
Put a = 3 into the expression.
y = 3x
▸ y = 3x
For y = ax, find a from a single (x, y) pair.
Example 2
y is inversely proportional to x, and y = 4 when x = 3. Find y when x = 6.
1
Inverse proportion is y = a/x. So a = xy = 3 × 4 = 12.
y = a/x, a = 3 × 4 = 12
2
Substitute x = 6.
y = 12/6 = 2
▸ y = 2
In inverse proportion the product xy is constant (= a).
Exam Key Points
Direct Proportion
y = ax → line through origin
Larger |a| → steeper line (closer to y-axis)
Inverse Proportion
y = ax
a > 0: Q1 & Q3 / a < 0: Q2 & Q4
Grade-7 school exam type
The graph of y = 12/x passes through the point (a, 3). What is a?