A proportional relationship graphs as a straight line through the origin.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Why the origin matters
Zero of one must give zero of the other. No weight, no cost.
A fixed charge breaks it
A taxi charging 3 pounds plus 2 a mile is still a straight line, but it misses the origin.
Three ways to check
Table: every output over input agrees. Graph: straight, through the origin. Equation: y = kx with nothing added.
Two tests, both required
A relationship is proportional if every ratio y/x is the same, and if it passes through the origin. A taxi charging £2 plus £3 per mile fails the second test — at zero miles the cost is £2, not £0 — so it is not proportional.
Why the origin matters
Proportional means "twice as much input gives twice as much output". A fixed starting charge breaks that: doubling the miles does not double the fare. The graph passing through the origin is the visible sign that no fixed amount is present.
Proportional and linear are not the same
Every proportional relationship is linear, but not every linear one is proportional. y = 3x + 2 is a straight line that misses the origin. Grade 8 will treat the general case; Grade 7 concentrates on the special one where the constant does all the work.
Where each kind appears
Price per kilogram is proportional. A phone plan with a monthly fee plus a per-minute charge is not. Recognising which kind you are dealing with decides whether you can simply scale — and scaling a non-proportional relationship is a common and costly mistake.
Step 2: Try It Yourself
Set the starting amount above zero and the line stops passing through the origin.
The slope is 2: for every 1 across, the line goes 2 up.
Step 3: Watch an Example
One step at a time.
Watch Leo Reject a Table
A table pairs 1 with 5, 2 with 8, and 3 with 11.
- Step 1
He divides the first pair: 5 ÷ 1 = 5.
Step 4: Your Turn
Practice makes it stick.
The Taxi
Problem 1 of 2
A taxi charges 3 pounds plus 2 per mile. What is the cost of 5 miles?
pounds
The Test
Problem 2 of 2
Is that taxi relationship proportional?
Proportional or Not?
1 of 4
Is y = 4x proportional?
2 of 4
Is y = 4x + 1 proportional?
3 of 4
2 pairs with 6 and 4 pairs with 12. Is it proportional?
4 of 4
y = 5x + 2. What is y when x = 4?
Step 5: Quick Check
Show what you know.
Question 1 of 2
y = 3x + 7. What is y when x = 6?
Question 2 of 2
What must a proportional graph pass through?
What You Learned
- Proportional means a straight line through the origin.
- A fixed charge makes it linear but not proportional.
- Test with a table, a graph, or the equation.