Proportion Graphs and Equations [H]

This is a Higher-tier only topic that builds on the direct and inverse proportion work covered earlier. You need to recognise and interpret graphs of proportional relationships, set up equations involving squares, cubes, and square roots, and use them to solve problems. This topic connects algebra and proportion and is frequently tested on AQA Higher papers.

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Direct Proportion: y = kx

When y is directly proportional to x:

• The equation is y = kx
• The graph is a straight line through the origin
• The gradient of the line equals the constant k

Worked Example 1: y is directly proportional to x. When x = 5, y = 35. Find the equation and use it to find y when x = 12.

1. y = kx
2. 35 = k x 5, so k = 7
3. Equation: y = 7x
4. When x = 12: y = 7 x 12 = 84

x	y = 7x
0	0
1	7
2	14
5	35
10	70
12	84

Exam Tip: If the graph passes through the origin and is a straight line, the relationship is y = kx (direct proportion). If it does not pass through the origin, it is NOT direct proportion — it may be a linear relationship of the form y = mx + c.

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Direct Proportion to Powers: y = kx squared, y = kx cubed

When y is directly proportional to x squared:

• The equation is y = kx squared
• The graph is a parabola passing through the origin
• As x doubles, y quadruples (because 2 squared = 4)

When y is directly proportional to x cubed:

• The equation is y = kx cubed
• The graph is a steeper curve through the origin
• As x doubles, y is multiplied by 8 (because 2 cubed = 8)

Worked Example 2: y is directly proportional to x squared. When x = 3, y = 45. Find y when x = 6.

1. y = kx squared
2. 45 = k x 9, so k = 5
3. y = 5x squared
4. When x = 6: y = 5 x 36 = 180

Notice: x doubled from 3 to 6, and y quadrupled from 45 to 180. This confirms y is proportional to x squared.

Worked Example 3: y is directly proportional to x cubed. When x = 2, y = 48. Find y when x = 5.

1. y = kx cubed
2. 48 = k x 8, so k = 6
3. y = 6x cubed
4. When x = 5: y = 6 x 125 = 750

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Direct Proportion to the Square Root

When y is directly proportional to the square root of x:

• The equation is y = k times the square root of x
• The graph curves upwards but flattens out (grows more slowly than y = kx)

Worked Example 4: y is directly proportional to the square root of x. When x = 16, y = 20. Find y when x = 49.

1. y = k times the square root of x
2. 20 = k x 4 (since the square root of 16 = 4), so k = 5
3. y = 5 times the square root of x
4. When x = 49: y = 5 x 7 = 35

graph TD
    A[Direct Proportion Types] --> B["y = kx (linear)"]
    A --> C["y = kx squared (quadratic)"]
    A --> D["y = kx cubed (cubic)"]
    A --> E["y = k times square root x"]
    B --> F[Straight line through origin]
    C --> G[Parabola through origin]
    D --> H[Steeper curve through origin]
    E --> I[Curve that flattens out]

Exam Tip: The key difference between y = kx and y = kx squared: for y = kx, doubling x doubles y. For y = kx squared, doubling x quadruples y. If a table shows y quadrupling when x doubles, the relationship is y = kx squared.

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Inverse Proportion: y = k/x

When y is inversely proportional to x:

• The equation is y = k/x
• The graph is a reciprocal curve (hyperbola)
• The curve approaches but never touches the axes (asymptotes)
• The product xy = k (constant)

Worked Example 5: y is inversely proportional to x. When x = 3, y = 8. Find y when x = 6.

1. y = k / x
2. 8 = k / 3, so k = 24
3. y = 24 / x
4. When x = 6: y = 24 / 6 = 4

Notice: x doubled from 3 to 6, and y halved from 8 to 4. This confirms inverse proportion.

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Inverse Proportion to Powers: y = k/x squared

When y is inversely proportional to x squared:

• The equation is y = k / (x squared)
• The graph is a steeper reciprocal curve
• As x doubles, y is divided by 4

Worked Example 6: y is inversely proportional to x squared. When x = 2, y = 50. Find y when x = 5.

1. y = k / (x squared)
2. 50 = k / 4, so k = 200
3. y = 200 / (x squared)
4. When x = 5: y = 200 / 25 = 8

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Recognising Proportional Relationships from Graphs

Relationship	Graph Shape	Passes Through Origin?	Key Feature
y = kx	Straight line	Yes	Constant gradient
y = kx squared	Parabola	Yes	y increases more steeply as x increases
y = kx cubed	Cubic curve	Yes	Even steeper than quadratic
y = k/x	Reciprocal curve	No	Never touches the axes
y = k/x squared	Steeper reciprocal	No	Approaches zero faster

How to Identify the Relationship from a Table

Worked Example 7: The table below shows corresponding values of x and y.

x	1	2	3	4
y	12	48	108	192

Test for y = kx: y/x gives 12, 24, 36, 48 — NOT constant, so not y = kx.

Test for y = kx squared: y/(x squared) gives 12/1 = 12, 48/4 = 12, 108/9 = 12, 192/16 = 12 — CONSTANT.

Therefore y = 12x squared.