Proportion and graphs Revision Notes for Edexcel GCSE Maths

Proportion and graphs

Understanding proportion symbols and graphs

Proportion shows how quantities relate to each other. You can use the symbol ∝ to show direct proportion, and you can display these relationships clearly using graphs.

Direct proportion

Definition and key features

When two quantities are in direct proportion, as one quantity increases, the other increases by the same factor.

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Key characteristics of direct proportion:

You can write it using the symbol: $y ∝ x$
You can write it as an equation: $y = kx$ (where $k$ is a constant number)
The graph of $x$ against $y$ creates a straight line passing through the origin
If one quantity doubles, the other quantity also doubles

Graph appearance

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A direct proportion graph always:

Forms a straight line
Passes through the point $(0,0)$ - the origin
Has a constant gradient (slope)

Inverse proportion

Definition and key features

When two quantities are in inverse proportion, as one quantity increases, the other decreases. The quantities are inversely proportional to the reciprocal of each other.

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Key characteristics of inverse proportion:

You can write it using the symbol: $y ∝ \frac{1}{x}$
You can write it as an equation: $y = \frac{k}{x}$ (where $k$ is a constant number)
The graph of $x$ against $y$ looks like a reciprocal graph (curved shape)
If one quantity doubles, the other quantity halves

Graph appearance

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An inverse proportion graph always:

Forms a curved line (hyperbola shape)
Never touches or crosses the axes
Gets closer to the axes but never reaches them
Is symmetrical about the line $y = x$

Identifying equations in exam questions

For inverse proportionality questions:

Look for equations in the form $y = \frac{k}{x}$
The constant k appears in the numerator
The variable appears in the denominator

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Worked Example: Identifying Inverse Proportion

If $p$ and $q$ are inversely proportional, which equation describes their relationship?

$p = 2q$ ❌ (this is direct proportion)
$p = q + 5$ ❌ (this is linear but not proportional)
$p = \frac{q}{10}$ ❌ (this is direct proportion with a fraction)
$p = \frac{5}{q}$ ✅ (this is inverse proportion)

Real-world applications

Electrical power relationships

Power and current in electrical appliances follow inverse proportion:

Power (watts) ÷ Current (amps) = constant
If you know the power and current for one appliance, you can find the current for another appliance with different power

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Worked Example: Electrical Power Calculation

Given information:

Electric drill: 2.2A current, 528W power
Television: 132W power, current = ?

Step 1: Find the constant using the drill $528 ÷ 2.2 = 240$ (constant)

Step 2: Apply the relationship to find the television current $132 ÷ 240 = 0.55A$ for the television

Using graphs for conversions

Proportion graphs can help convert between different units:

The relationship between inches and centimetres is directly proportional
A straight line through the origin shows this relationship
You can read values directly from the graph
The straight line confirms the direct proportional relationship

Graph reading techniques

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When using conversion graphs:

Find your starting value on one axis
Draw a line to meet the graph
Read the corresponding value on the other axis
Check your answer makes sense

Checking your work

For direct proportion: When one quantity decreases, the other should decrease proportionally.

For inverse proportion: When one quantity decreases, the other should increase proportionally.

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Graph check:

Direct proportion = straight line through origin
Inverse proportion = curved line (reciprocal shape)

Summary

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Key Points to Remember:

Direct proportion gives straight line graphs through the origin with equation $y = kx$
Inverse proportion gives curved reciprocal graphs with equation $y = \frac{k}{x}$
Real-world examples include electrical power relationships and unit conversions
Graph shapes immediately tell you the type of proportion relationship
Always check your answers make sense in the context of the problem

Proportion and graphs (Edexcel GCSE Maths): Revision Notes