Cambridge Lower Secondary · Mathematics 0862 · Stage 8 · Algebra

Reading and Interpreting Real-life Graphs

Read and interpret real-life graphs with several sections or two lines — journeys, costs, filling containers — explain their shape and say what an intersection means.

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Read and interpret real-life graphs that have more than one part — for example a journey made of several stages, or two lines drawn on the same axes. Explain why a graph has its shape (steep, flat, straight or curved) and what it means when two graphs cross (intersect). [8As.07]

Graphs are used everywhere to show how one quantity changes with another: distance with time, cost with distance, the depth of water with time. In Stage 8 the graphs have more than one component — several straight sections joined together, or two lines on the same grid. You must be able to read values accurately, explain what each part of the graph shows in the real situation, and say what is happening where two lines cross.

1. Key words

Word / phrase	Meaning / use
axis (plural axes)	محور — the horizontal and vertical number lines of a graph
scale	مقياس الرسم — what each square on an axis is worth
component / section	جزء — one part (one straight piece) of a graph
distance–time graph	منحنى المسافة والزمن — shows how far something is from a place as time passes
steep / steeper	شديد الانحدار — rising or falling quickly
horizontal	أفقي — flat: the quantity is not changing
rate	معدل — how fast something changes, e.g. km per hour, litres per minute
speed	السرعة — distance ÷ time
intersection	نقطة التقاطع — the point where two graphs cross
fixed charge	رسوم ثابتة — an amount you pay before anything else

2. Read the graph before you answer

Read the title and the labels on both axes, including the units (minutes or hours? km or m? $?).
Work out the scale: what one small square is worth on each axis. For example, if 5 small squares cover 10 km, one small square is 2 km.
Look at each section of the graph separately: is it going up, going down, or flat? Is it steep or gentle? Straight or curved?
Look for intersections — places where two graphs cross.
When you read a value, use a ruler: go across or up from the axis to the line, then to the other axis. Give the units.

Shape of a section	What it means
Straight line sloping up	The quantity increases at a steady (constant) rate.
Steeper straight line	A faster rate (e.g. a higher speed or a higher price per unit).
Horizontal (flat) line	No change — on a distance–time graph, the person has stopped.
Straight line sloping down	The quantity decreases at a steady rate — e.g. coming back towards the start.
Curve getting steeper	The rate of change is increasing (speeding up / filling faster).
Curve getting less steep	The rate of change is decreasing (slowing down / filling more slowly).
Two lines cross	At that moment both quantities are equal: same place at the same time, or same cost.

3. Distance–time graphs with several sections

On a distance–time graph, time goes across and distance from a fixed place goes up. The steepness shows the speed: speed = distance ÷ time for that section.

Yasmin's cycle ride from her home to a park and back. The sections are labelled A to E.

Section	What the graph shows	What is happening
A (0 to 40 min)	Up from 0 to 8 km	Cycles away from home: 8 km in 40 minutes.
B (40 to 60 min)	Flat at 8 km	Stopped for 20 minutes (distance not changing).
C (60 to 90 min)	Up from 8 to 12 km, less steep than A	Cycles another 4 km in 30 minutes — slower than in A.
D (90 to 110 min)	Flat at 12 km	Stopped at the park for 20 minutes.
E (110 to 150 min)	Down from 12 km to 0, the steepest section	Cycles home: 12 km in 40 minutes — her fastest section.

Speeds. A: 8 km in 40 min. 40 min is 2/3 of an hour, so speed = 8 ÷ 2/3 = 12 km/h. C: 4 km in 30 min = 8 km/h. E: 12 km in 40 min = 18 km/h. The steepest section has the highest speed.

4. Two graphs on the same axes — what an intersection means

When two people or two plans are shown on one grid, compare them section by section, and pay special attention to where the lines cross.

Ahmed walks from Town A to Town B, 10 km away. Karim cycles from Town B to Town A. The dot is where the lines cross.

Ahmed starts at 09:00 from Town A (distance 0) and reaches Town B (10 km) at 11:30.
Karim starts at 09:30 from Town B (10 km from A), so his line starts at 10 and goes down — he is getting closer to Town A. He arrives at 10:20.
The lines cross at (10:00, 4 km): at 10:00 they are both 4 km from Town A — they meet.
Karim's line is steeper, so Karim is faster: 10 km in 50 minutes = 12 km/h. Ahmed: 10 km in 2 1/2 hours = 4 km/h.

Comparing costs. Two taxi companies charge differently. Company P has a fixed charge of $12 plus $3 per km. Company Q charges $5 per km with no fixed charge.

Cost against distance for two taxi companies. The dot shows where the lines cross.

Why straight lines? Each extra kilometre always costs the same amount, so the cost rises at a constant rate.
Why does P start at $12? That is the fixed charge you pay even for 0 km. Q starts at $0 (the origin).
Why is Q steeper? Q charges more for each km ($5 instead of $3).
The lines cross at (6 km, $30): for a 6 km journey both companies cost $30.
Shorter than 6 km → Q's line is lower, so Q is cheaper. Longer than 6 km → P is cheaper.

5. Curved graphs: filling containers

Water is poured into each container at a steady rate. The graph shows the depth of the water against time. The shape of the container decides the shape of the graph.

Three containers filled at the same steady rate, and the shape of each depth–time graph.

X (straight sides): every centimetre of height needs the same amount of water, so the depth rises steadily — a straight line.
Y (wide at the bottom): at first the water spreads over a wide area, so the depth rises slowly. As the container gets narrower, the same amount of water makes the depth rise faster — the graph gets steeper.
Z (narrow at the bottom): the depth rises quickly at first and then more slowly as the container gets wider — the graph gets less steep.
A container with straight sides that suddenly changes width (e.g. a wide box with a narrow tube on top) gives two straight sections, the second steeper.

ازاي تقرا الرسم البياني

الأول اقرا عنوان كل محور والوحدات، واحسب كل مربع صغير يساوي كام.
الخط الأفقي (المستوي) يعني مفيش تغيير — في رسم المسافة والزمن يعني الشخص واقف.
كل ما الخط يبقى أشد انحدارًا، يبقى المعدل أسرع (سرعة أعلى أو سعر أعلى لكل وحدة).
نقطة تقاطع الخطين معناها إن القيمتين متساويتين في اللحظة دي: نفس المكان في نفس الوقت، أو نفس التكلفة.

6. Worked examples

Worked example 1 — Use the graph of Yasmin's ride in section 3. (a) How far from home was Yasmin after 30 minutes? [1] (b) How long did she stop for altogether? [1] (c) In which section was she cycling fastest? Explain how you know. [1] (d) Work out her speed on the way home in km/h. [2]

Part	Working and answer
(a)	Go up from 30 on the time axis to the line, then across: 6 km. (Section A rises 8 km in 40 min, i.e. 2 km every 10 min.)
(b)	Flat sections: 40–60 min and 90–110 min. 20 + 20 = 40 minutes.
(c)	Section E — it is the steepest line (12 km in 40 minutes).
(d)	12 km in 40 minutes. 40 min = 40/60 h = 2/3 h. Speed = 12 ÷ 2/3 = 12 × 3/2 = 18 km/h. (Or: 12 km in 40 min → 3 km in 10 min → 18 km in 60 min.)

Common mistake: reading the speed as 12 ÷ 40 = 0.3 and calling it km/h. That is km per minute; change the time to hours first.

Worked example 2 — Use the graph of Ahmed and Karim in section 4. (a) At what time do Ahmed and Karim meet, and how far are they from Town A? [2] (b) How far apart are they at 09:30? [1] (c) Explain why Karim's graph slopes downwards. [1]

Part	Working and answer
(a)	The lines cross at 10:00, 4 km from Town A.
(b)	At 09:30 Ahmed is 2 km from A and Karim is 10 km from A: 10 − 2 = 8 km apart.
(c)	The graph shows distance from Town A. Karim is travelling towards Town A, so his distance from A is getting smaller.

Common mistake: thinking that a line going down means "going downhill". On a distance–time graph it means moving back towards the starting place.

Worked example 3 — Use the taxi graph in section 4. (a) Find the cost of an 8 km journey with each company. [2] (b) Mona travels 4 km. Which company is cheaper, and by how much? [2] (c) What does the point where the lines cross tell you? [1]

Part	Working and answer
(a)	Up from 8 km: P = $36, Q = $40. (Check: 12 + 3 × 8 = 36; 5 × 8 = 40.)
(b)	P: 12 + 12 = $24; Q: 5 × 4 = $20. Q is cheaper by $4.
(c)	At 6 km both companies charge the same amount, $30. Below 6 km Q is cheaper; above 6 km P is cheaper.

Worked example 4 — The graph shows the amount of water in a tank. Describe what happens, giving the rates. [3]

Worked example 4: water in a tank.

Section	Description
0 to 4 min	The tank fills from 0 to 40 litres: 40 ÷ 4 = 10 litres per minute.
4 to 6 min	Flat: the amount stays at 40 litres for 2 minutes (the tap is off and the tank is not emptying).
6 to 14 min	The tank empties from 40 litres to 0 in 8 minutes: 40 ÷ 8 = 5 litres per minute. This line is less steep, so it empties more slowly than it filled.

Worked example 5 — Water is poured at a steady rate into container Z (narrow at the bottom, wide at the top) from section 5. Sara says: "The graph of depth against time is a straight line because the water is poured at a steady rate." Is she correct? Explain. [1]

Answer: No. The water flows in at a steady rate, but the container gets wider, so each litre raises the depth by less and less. The depth rises quickly at first and then more slowly, so the graph is a curve that gets less steep. A straight line would only happen if the sides were straight up (the same width all the way).

Common mistakes

Not checking the scale: one small square may be 2 km, 5 litres or 20 metres — not 1.
Mixing minutes and hours when working out speed in km/h.
Thinking a flat section means "moving at a steady speed". On a distance–time graph, flat means stopped.
Thinking a line going down means going downhill. It means the quantity (e.g. distance from home) is decreasing.
Reading the wrong line when two graphs are drawn — use the key and follow the colour or label.
Describing an intersection as "where the lines meet" without saying what it means in the situation (same place and time, same cost…).
Forgetting units in the answer (km, minutes, $, litres).

How Checkpoint asks this

"Use the graph to find …" — show your reading lines on the graph and write the value with units.
"Write down the time when …" or "How long did … stop?" — look for flat sections and read both ends.
"Explain why the graph is steeper / curved / flat in this part" — link the shape to the situation (faster, slower, stopped, container wider).
"What is the significance of the point where the graphs intersect?" — say both quantities are equal there, in context, with the values.
"Which company is cheaper for … ? Give a reason." — compare the heights of the lines at that value.
"Match each container to its graph" or "Sketch the graph" — straight sides give straight lines; wider gives less steep; narrower gives steeper.

Practice

Remember

Check the axis labels, units and scale before reading any value.
Steeper = faster rate; flat = no change (stopped); sloping down = decreasing (e.g. returning).
Speed = distance ÷ time — change minutes to hours for km/h.
Where two graphs intersect, both quantities are equal: say what that means in the story, with the values.
Containers filled at a steady rate: straight sides → straight line; getting wider → less steep; getting narrower → steeper.

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