Direct-proportion tables and graphs | Year 8

01 / LEARN

The idea, step by step

One constant links every pair

Two quantities are directly proportional when multiplying one by a factor multiplies the other by the same factor. For nonzero $x$ , the ratio $y / x$ must be constant. We write $y = k x$ , where $k$ is the unit rate.

A suitable table includes zero: buying zero identical items costs zero when there is no fixed fee. A fixed starting charge changes the relationship even if its graph is still a straight line.

y = k x

Read the graph and the context

A direct-proportion graph is a straight line through the origin. Its steepness represents the rate. On the supplied graph, each extra hour adds 40 km, so $d = 40 t$ .

Do not extend a model without checking the situation. A steady-speed travel graph is appropriate only while that speed model remains sensible; an actual stop would create a different section.

A common trap

Every straight-line graph represents direct proportion.

Instead: The line must also pass through the origin. A taxi fare with a starting fee can be linear without being directly proportional.

02 / WORKED EXAMPLES

See the thinking

Example 1

Build a cost rule

Notebooks cost AUD 4 each, with no other charge. Write a rule for the cost C of n notebooks.

Notebook cost
Notebooks n	Cost C (AUD)
0	0
1	4
2	8
5	20

The cost per notebook is 4 dollars, so $C / n = 4$ for positive $n$ .
Therefore $C = 4 n$ .
For $n = 0$ , the cost is zero, so the graph passes through the origin.

C = 4 n

Example 2

Read a travel graph

Using the supplied steady-speed graph, find the distance after 2.5 hours.

The rate is 40 km for each hour.
Use $d = 40 t$ .
$d = 40 \times 2.5 = 100$ .

100 km.

Check your understanding

Does $y = 3 x + 5$ show direct proportion?

Show answer

No.

At $x = 0$ , $y = 5$ , so the graph does not pass through the origin and $y / x$ is not constant.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A table has x-values 1, 2, 4 and y-values 5, 10, 20. Find k in $y = k x$ .
2 marks
For $y = 7 x$ , find y when x is 0, 2 and 5.
2 marks
Six identical tickets cost AUD 45 with no fee. Find the cost of nine tickets.
2 marks
Explain whether $y = 6 x$ represents direct proportion.
2 marks
A table contains (1,4), (2,7), (3,10). Is y directly proportional to x? Explain.
2 marks
Using the supplied graph, how long does it take to travel 80 km?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A table has x-values 1, 2, 4 and y-values 5, 10, 20. Find k in $y = k x$ .
2 marks
$5 / 1 = 10 / 2 = 20 / 4 = 5$ .
$k = 5$ .
For $y = 7 x$ , find y when x is 0, 2 and 5.
2 marks
Multiply each x-value by 7.
0, 14 and 35.
Six identical tickets cost AUD 45 with no fee. Find the cost of nine tickets.
2 marks
Each ticket costs $45 \div 6 = 7.50$ .
Nine cost $9 \times 7.50 = 67.50$ .
AUD 67.50.
Explain whether $y = 6 x$ represents direct proportion.
2 marks
The ratio $y / x = 6$ whenever $x \neq 0$ .
The graph passes through $(0, 0)$ .
Yes: the constant of proportionality is 6.
A table contains (1,4), (2,7), (3,10). Is y directly proportional to x? Explain.
2 marks
The ratios are 4, 3.5 and $10 / 3$ , which are not equal.
The rule $y = 3 x + 1$ has a nonzero starting value.
No.
Using the supplied graph, how long does it take to travel 80 km?
2 marks
Read 80 km across to the line and down to the time axis.
Equivalently, $t = 80 \div 40 = 2$ .
2 hours.

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Ratio scaling and division review; unit rates; comparing rates and best buys; speed, distance and time; converting compatible units; direct-proportion tables and graphs; modelling practical rate problems and checking assumptions.

Official curriculum source ↗

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