Graphing simple relationships | Year 7

01 / LEARN

Understand the idea.

Make a table before drawing

Choose axes that show what the variables mean. Put the input horizontally and the output vertically, label both quantities, and include units. Read the numerical scale: equal distances along one axis must represent equal changes on that axis.

For notebooks costing $4$ dollars each, $C = 4 n$ . Make a table before plotting, including $n = 0$ . Whole notebooks allow only whole-number counts, so the model consists of separate cost points at those inputs.

Notebook prices
Notebooks n	Cost C (dollars)
0	0
1	4
2	8
3	12
4	16

Connect the graph to the situation

Each plotted pair links two quantities. In a time-depth graph, $(4, 14)$ means a time of $4$ minutes and a depth of $14$ centimetres. Read the axis labels before interpreting either number.

Join points only when intermediate inputs are meaningful and the model supports those values. The steady filling model $d = 3 t + 2$ allows times between whole minutes, with initial depth $2$ centimetres and increase $3$ centimetres per minute.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Graph a counted quantity

Plot notebook cost for 0 to 4 notebooks.

Separate points (0,0), (1,4), (2,8), (3,12), (4,16) give notebook count horizontally and cost in dollars vertically. The unchanged plotting window is 0 to 5 notebooks and 0 to 20 dollars.

Worked solution

1
Label the quantities and choose suitable scales.
$C = 4 n$
Use notebook count n horizontally and cost C in dollars vertically; each notebook adds 4 dollars.
2
Calculate the allowed input-output pairs.
$\begin{aligned} 0 & \mapsto 0 \\ 1 & \mapsto 4 \\ 2 & \mapsto 8 \\ 3 & \mapsto 12 \\ 4 & \mapsto 16 \end{aligned}$
Substitute each whole-number count from 0 to 4 into the cost rule.
3
Plot the five separate points.
$(0, 0), (1, 4), (2, 8), (3, 12), (4, 16)$
Whole notebooks are sold, so fractional counts between these points are not allowed.

Answer

Each extra notebook adds 4 dollars.

EXAMPLE 02

Read a continuous model

A depth model is $d = 3 t + 2$ , where t is minutes and d is centimetres. Explain (4,14).

Worked solution

1
Read the input from the ordered pair.
$t = 4 min$
The first coordinate measures time on the horizontal axis.
2
Read the output from the ordered pair.
$d = 14 cm$
The second coordinate measures water depth on the vertical axis.
3
Check the pair against the model.
$\begin{aligned} d & = 3 \times 4 + 2 \\ = 14 cm \end{aligned}$
The rule confirms that after 4 minutes the depth is 14 cm.

Answer

After 4 minutes the water is 14 cm deep.

03 / YOUR TURN

Try it yourself.

Check your understanding

For $C = 2 d + 5$ , where d is kilometres and C is dollars, explain (3,11).

Show answer & reasoning

A 3 km trip costs 11 dollars.

The pair $(3, 11)$ records a distance input of $3$ kilometres and a cost output of $11$ dollars. The rule confirms $2 \times 3 + 5 = 11$ , so a $3$ km trip costs $11$ dollars.

Keep practising

6 worksheet questions · 15 marks

Open worksheet Worked solutions

Read the questions online

For $C = 3 n$ , find costs for 0,1,2,3 tickets.
2 marks
Plot the pairs for $C = 3 n$ at $n = 0, 1, 2, 3$ .
3 marks
For $h = 2 t + 4$ , where $t$ is weeks and $h$ is cm, find the height at $t = 3$ .
2 marks
For the same plant model, explain $(2, 8)$ .
2 marks
A horizontal axis uses the same physical gap for 0 to 1 and for 1 to 5. Explain the error.
3 marks
A steady filling model gives depths 1,4,7,10 cm at times 0,1,2,3 min. Plot and join the points.
3 marks

Read the worked solutions

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

For $C = 3 n$ , find costs for 0,1,2,3 tickets.
2 marks
Multiply each count by 3.
0,3,6,9 dollars.
Plot the pairs for $C = 3 n$ at $n = 0, 1, 2, 3$ .
3 marks
Label tickets on the horizontal axis and dollars vertically.
Only whole-ticket inputs are allowed.
$(0, 0)$ , $(1, 3)$ , $(2, 6)$ , $(3, 9)$ , as separate points.
For $h = 2 t + 4$ , where $t$ is weeks and $h$ is cm, find the height at $t = 3$ .
2 marks
$2 \times 3 + 4 = 10$ .
10 cm.
For the same plant model, explain $(2, 8)$ .
2 marks
Read time first and height second.
At 2 weeks the model predicts 8 cm.
A horizontal axis uses the same physical gap for 0 to 1 and for 1 to 5. Explain the error.
3 marks
The first gap represents 1 unit; the next represents 4 units.
An ordinary uniform scale needs equal numerical changes in equal physical gaps.
A steady filling model gives depths 1,4,7,10 cm at times 0,1,2,3 min. Plot and join the points.
3 marks
The depth increases by 3 cm per minute.
Steady filling justifies intermediate depths.
$(0, 1)$ , $(1, 4)$ , $(2, 7)$ , $(3, 10)$ , joined by a straight segment.

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Curriculum reference & lesson scope

Australian Curriculum v9.0 / South Australian Curriculum

Year 7 coordinate, pattern and authentic-graph interpretation scope. Includes prerequisite four-quadrant plotting; formal gradient/intercept techniques are not required here.

Official curriculum source ↗

In this lesson

Understand the idea.

Make a table before drawing

Connect the graph to the situation

See the method in action.

Graph a counted quantity

Label the quantities and choose suitable scales.

Calculate the allowed input-output pairs.

Plot the five separate points.

Read a continuous model

Read the input from the ordered pair.

Read the output from the ordered pair.

Check the pair against the model.

Try it yourself.

Check your understanding

Keep practising