A proportional relationship always graphs as a straight line passing through (0, 0).
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Why the origin
Nought hours of work earns nought pounds. If nothing gives nothing, the line starts at the origin.
Reading k off the graph
The height of the line at x = 1 is the constant of proportionality.
Straight is not enough
A straight line that misses the origin — a taxi with a £3 pick-up fee — is not proportional.
A straight line through the origin
The graph of y = kx is a straight line passing through (0, 0). Both features are diagnostic: curved means not proportional, and straight but missing the origin means linear but not proportional.
The constant is the steepness
A bigger k gives a steeper line, because each step across produces a bigger step up. The constant of proportionality is exactly what Grade 8 will call the gradient. Seeing it as steepness makes comparing two relationships a visual task.
The point (1, k)
Every proportional graph passes through (1, k), because when x is 1, y is k. Reading the height of the line above x = 1 gives the constant directly off the graph, with no calculation at all.
What a point on the line means
Any point (a, b) on the line is a valid pair for the relationship: a units of input give b of output. That is why a graph is more than a picture — it answers questions about values you never tabulated.
Step 2: Try It Yourself
Change the slope and watch the height at x = 1.
The slope is 3: for every 1 across, the line goes 3 up.
Step 3: Watch an Example
One step at a time.
Watch Ana Read k Off a Graph
A straight line passes through (0, 0) and (4, 10).
- Step 1
She checks it goes through the origin, so it is proportional.
Step 4: Your Turn
Practice makes it stick.
The Line
Problem 1 of 2
A proportional graph passes through (3, 12). What is k?
The Taxi
Problem 2 of 2
A taxi charges £3 to start plus £2 a mile. Is it proportional? Answer 1 for yes, 0 for no.
Read the Graph
1 of 4
Proportional graph through (2, 10). What is k?
2 of 4
Proportional graph through (5, 15). What is k?
3 of 4
Must a proportional graph pass through the origin? Answer 1 for yes, 0 for no.
4 of 4
If k = 6, what is y when x is 4?
Step 5: Quick Check
Show what you know.
Question 1 of 1
Proportional graph through (4, 24). What is k?
What You Learned
- A proportional relationship is a straight line through the origin.
- The height of the line at x = 1 is the constant k.
- Straight alone is not enough — it must also pass through (0, 0).