Direct Proportion Graphs (y = kx): KS3 Maths

What makes a graph show direct proportion?

Two features together confirm direct proportion:

1. The graph is a straight line.
2. The line passes through the origin (0, 0).

A straight line that does NOT pass through the origin (e.g., y = 2x + 3) represents a linear relationship but NOT direct proportion — the y-intercept of 3 means y is not zero when x is zero.

What does the equation y = kx mean?

The equation y = kx says: "y is always k times x." The constant k is the gradient — how steep the line is.

• If k = 2, then every time x increases by 1, y increases by 2.
• If k = 0.5, then y is always half of x.

To find k from two coordinates: use k = y ÷ x (since y = kx means k = y/x).

Example: A graph passes through (0, 0) and (5, 20). Find k.

k = 20 ÷ 5 = 4. The equation is y = 4x.

How do you draw a direct proportion graph from a table?

Worked example: Oranges cost 35p each. Draw a graph showing cost (y pence) against number of oranges (x).

1. The equation is y = 35x.
2. Make a table of values:

3. Plot the points and draw a straight line through them and the origin.
4. Label the axes and the line y = 35x.

The gradient is 35 — it represents the price per orange.

How do you read values from a direct proportion graph?

Once drawn, a direct proportion graph lets you find y for any x (and x for any y) by reading off the graph.

Example using the orange graph (y = 35x):

• "How much do 5 oranges cost?" → Go to x = 5, read up to the line, read across to y = 175p = £1.75.
• "How many oranges can I buy for £2.10?" → Go to y = 210, read across to the line, read down to x = 6.

This reading-off method is especially useful when the numbers are awkward (e.g., fractions of oranges would be hard to calculate mentally, but the graph gives an instant visual answer).

How does the gradient link to the constant of proportionality?

The gradient of y = kx equals k exactly. You can measure the gradient from the graph by choosing two points on the line:

gradient = (change in y) ÷ (change in x)

Example: A line passes through (0, 0) and (8, 12).

gradient = k = 12 ÷ 8 = 1.5

The equation is y = 1.5x.

Practical meaning: If this were a conversion graph for pounds to dollars, k = 1.5 means £1 = $1.50.

What are real-life examples of direct proportion graphs?

• Currency conversion: euros against pounds (both quantities are 0 when the other is 0; the gradient is the exchange rate).
• Distance vs time at constant speed: distance = speed × time, so d = kt where k = speed.
• Recipe scaling: 250 g flour per batch → total flour = 250 × batches, a line with gradient 250.
• Petrol cost: total cost = price per litre × litres bought.

In every case, if one quantity is zero, the other must also be zero — a key feature to check.

Frequently asked questions

How do I check from a table of values whether a relationship is direct proportion?

Divide each y value by its corresponding x value. If you always get the same number (constant of proportionality k), the relationship is direct proportion. For example: (x = 2, y = 6) → 6/2 = 3; (x = 5, y = 15) → 15/5 = 3. Since both ratios equal 3, the relationship is direct proportion with k = 3.

Why must the line pass through the origin for direct proportion?

Because direct proportion means y = kx. When x = 0, y = k × 0 = 0. If the line has a y-intercept other than 0, it cannot be modelled by y = kx and the two quantities are not in direct proportion. An example where the line does NOT pass through the origin is cost = £3 delivery charge + £2 per kg — the £3 shows the non-proportional starting point.

What if the graph is curved but still passes through the origin?

A curve through the origin does not show direct proportion. Direct proportion requires y to change by a CONSTANT amount for each unit of x (a straight line). A curve through the origin might represent y = kx² (y proportional to x squared) or another power, not y = kx.

Can direct proportion graphs have negative gradients?

No, not in the usual physical or financial contexts at KS3. If x and y are both positive quantities (cost, distance, weight), k is positive and the line slopes upwards. A negative k would mean y decreases as x increases, which is the opposite behaviour — that would be a different type of relationship, not typical direct proportion at KS3.

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Professor Pi can walk you through plotting and reading direct proportion graphs, one step at a time — visit aitutors.me.