Graphing proportional relationships: what the graph looks like (a straight line through the origin), how to read the constant of proportionality from the graph (the y-value at x = 1, which is also the unit rate and the slope), how to tell a proportional graph from a non-proportional one, comparing two proportional relationships given in different forms, worked examples, the mistakes, and a printable sheet with graphs and practice

"Proportional relationship graph" is searched about 3,000 times a month in the United States, by Grade 7 and 8 students whose standards (7.RP.2 and 8.EE.5) ask them to recognise a proportional relationship on a graph, read its constant from it and compare two of them given in different forms. The whole topic is one line through one point. This page says what the graph looks like, how to draw it from a table, an equation or a rate, how to read the constant (which is also the unit rate and the slope), how to tell proportional from merely linear, how to compare two relationships, the mistakes, and gives a printable sheet with worked graphs and practice. Finding the constant from tables, equations and word problems is on our constant of proportionality page; this page is about the graph.

Download the proportional graphs sheet with practice (PDF) (A4, the two tests, the worked graphs and the comparison method on page 1, blank grids and the practice set with answers on page 2, free).

What the graph looks like

A proportional relationship is one where y is always the same multiple of x: y = kx, with k the constant of proportionality. Its graph is

1. a straight line, and
2. through the origin (0, 0).

Both are required. Doubling x doubles y, so zero x gives zero y (the origin), and equal steps in x give equal steps in y (a straight line). The constant k sets the steepness: y = 3x climbs faster than y = ½x.

Reading the constant from the graph

Three equivalent readings, all giving k:

• The point (1, k). Find x = 1 on the horizontal axis, go up to the line, read y. That y is k. On a line through (1, 2.5), k = 2.5.
• Any point, divided. Take a point where the line crosses a grid intersection cleanly, (x, y), and compute k = y ÷ x. The point (4, 10) gives 10 ÷ 4 = 2.5. Every point on the line gives the same k, which is what "proportional" means.
• The slope. Rise over run between two points; for a line through the origin this equals y ÷ x at any point. Slope 2.5.

So on a proportional graph, constant of proportionality = unit rate = slope, three names for one number.

What the points mean in context

Suppose the graph shows cost (y, dollars) against tickets (x).

• (0, 0): zero tickets cost zero dollars. Every proportional graph starts here.
• (1, k): one ticket costs k dollars, the unit rate. If (1, 12) is on the line, tickets are $12 each.
• (5, 60): five tickets cost $60. Check: 60 ÷ 5 = 12.
• The equation: y = 12x. The slope: 12 dollars per ticket.

Distance against time: (1, 60) means 60 miles per hour; (0, 0) means no time, no distance. Flour against cakes: (1, 200) means 200 g per cake.

How to graph a proportional relationship

From an equation, y = 2.5x: plot (0, 0) and (1, 2.5), or for accuracy (0, 0) and a further point such as (4, 10). Join with a ruler through the origin and beyond.

From a table:

Check it is proportional (y ÷ x = 2.5 each time). Plot the three points, add (0, 0), join.

From a rate, "a car travels 60 miles per hour": k = 60, so plot (0, 0) and (1, 60), label the axes time (h) and distance (mi), join. Scale the axes to fit: 0 to 5 hours, 0 to 300 miles.

Proportional against non-proportional: the two tests

To decide from a graph: is it a straight line? Does it pass through (0, 0)? Both yes means proportional.

To decide from a table: is y ÷ x the same for every row? (And would (0, 0) fit?) A table with (2, 7), (4, 11), (6, 15) has differences of 4 but ratios 3.5, 2.75, 2.5: linear, not proportional. Its graph would miss the origin (it is y = 2x + 3).

To decide from an equation: is it of the form y = kx with nothing added? y = 4x yes; y = 4x − 1 no; y = x ÷ 3 yes (k = ⅓); y = 3 ÷ x no.

The difference matters in context: a taxi fare with a $3 flag fall plus $2 a mile is linear, not proportional, because two miles do not cost twice one mile. A pay rate of $15 an hour with nothing else is proportional.

Comparing two proportional relationships in different forms

The Grade 8 standard (8.EE.5) gives one relationship as a graph and another as a table or equation and asks which is greater. Find each k and compare.

Example. Runner A's graph passes through (4, 32) (metres against seconds). Runner B's table: 3 s, 21 m; 5 s, 35 m. Who is faster? A: k = 32 ÷ 4 = 8 m/s. B: k = 21 ÷ 3 = 7 m/s. A is faster; A's line is steeper.

Example. Shop P charges by the equation y = 1.8x (dollars per kg of apples). Shop Q's graph shows (5, 8). Q: k = 8 ÷ 5 = 1.6. Q is cheaper per kilogram.

On one grid, the line with the larger k is steeper; two proportional lines can only cross at the origin.

The mistakes that cost marks

• Calling any straight line proportional. It must pass through the origin.
• Reading k at x = 0 (which is always 0) instead of x = 1.
• Dividing x by y instead of y by x.
• Forgetting to label the axes with the quantities and units, which is where the "what does (1, k) mean" mark lives.
• Joining the points but not extending the line to the origin, or drawing a line that starts at the first data point.
• Comparing slopes by eye across two graphs with different scales. Compute k for each.

Where it sits in each curriculum

Practice (answers in the PDF)

1. A line passes through (0, 0) and (3, 12). Is it proportional? What is k, and what is the slope?
2. On a graph of cost against number of pens, the point (1, 0.75) is on the line. What does it mean? What does (8, 6) mean?
3. Which of these graphs show a proportional relationship: (a) a straight line through (0, 2) and (4, 10); (b) a straight line through (0, 0) and (5, 20); (c) a curve through (0, 0), (1, 1) and (2, 4)?
4. Graph y = 1.5x from x = 0 to 6. Read the value of y when x = 4.
5. Machine A's graph passes through (6, 90) (parts against minutes). Machine B is described by y = 14x. Which machine is faster, and by how much per minute?
6. A table shows (2, 9), (4, 17), (6, 25). Is the relationship proportional? Explain using the ratios and the graph.

Download the sheet with the grids and answers (PDF)

Where to get help

A 1-on-1 online maths class with a tutor who holds a master's in mathematics and teaches Grade 7 and 8, KS3 and GCSE, the Australian Years 7 to 9 and CBSE Class 8 and 9 every week: proportion taught as one idea across tables, graphs, equations and words, so the child sees the same k in all four and can move between them on a test. The Grade 7 maths page lists the units; the free 30-minute first class is a real lesson on this week's homework. Related pages: the constant of proportionality from tables, equations and word problems and graphing linear equations.

Questions parents ask

See how we teach this, 1-on-1 online →