Direct-proportion tables and graphs | Year 8

01 / LEARN

Understand the idea.

One constant links every pair

Two quantities are in direct proportion when the same multiplier scales them both. Their relationship is $y = k x$ , where $k$ is constant. For each nonzero $x$ , dividing the paired value $y$ by $x$ must give this same $k$ .

The rule also gives $y = 0$ when $x = 0$ . For identical notebooks with no other charge, doubling the number doubles the cost. A fixed fee breaks this relationship: doubling the number would not double the fixed part of the bill.

y = k x

Read the graph and the context

A direct-proportion graph is a straight line through the origin. Read both axis scales to find its unit rate. On the supplied travel graph, $40$ km is covered in one hour, giving the rule $d = 40 t$ .

Keep the context in mind when using the rule. Time can take fractional values, but a notebook count uses whole numbers. A straight travel line assumes steady speed over the modelled interval; a stop or speed change needs another section.

Distance in kilometres is plotted against time in hours. The sage line passes through the origin and the marked points 1 hour and 40 kilometres, 2 hours and 80 kilometres, and 3 hours and 120 kilometres.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

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

Worked solution

1
Identify the cost for one notebook.
$k = 4 dollars per notebook$
Every notebook has the same price and there is no additional charge.
2
Multiply by the number bought.
$C = 4 n$
A purchase of n notebooks contains n copies of the four-dollar cost.
3
Check zero and an entry in the table.
$\begin{aligned} n = 0 & : C = 4 (0) = 0 \\ n = 5 & : C = 4 (5) = 20 \end{aligned}$
The rule matches the table; zero notebooks cost zero, with n restricted to nonnegative whole numbers.

Answer

$C = 4 n$ .

EXAMPLE 02

Read a travel graph

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

Worked solution

1
Read a convenient point from the graph.
$t = 1 h, d = 40 km$
The labelled point gives the distance for one hour.
2
Write the steady-speed rule.
$d = 40 t$
The straight line through the origin represents a constant rate of 40 km/h.
3
Substitute the requested time.
$d = 40 \times 2.5 = 100 km$
The time 2.5 hours lies on the supplied steady-speed section.

Answer

100 km.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

No.

No. Substituting $x = 0$ into $y = 3 x + 5$ gives $y = 5$ , so the graph does not pass through the origin and the rule cannot have the form $y = k x$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet 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
$\frac{5}{1} = \frac{10}{2} = \frac{20}{4} = 5$ .
$k = 5$ .
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 $\frac{y}{x} = 6$ 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 $\frac{10}{3}$ , 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 ↗

In this lesson

Understand the idea.

One constant links every pair

Read the graph and the context

See the method in action.

Build a cost rule

Identify the cost for one notebook.

Multiply by the number bought.

Check zero and an entry in the table.

Read a travel graph

Read a convenient point from the graph.

Write the steady-speed rule.

Substitute the requested time.

Try it yourself.

Check your understanding

Keep practising