Real Life Graphs

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Lesson notes

Conversion Graphs

A conversion graph is a straight-line graph used to convert between two quantities (e.g., °C to °F, currency, litres to gallons).
The gradient represents the rate of change (e.g., cost per mile on a taxi fare graph).
To convert a value, draw a vertical/horizontal line to the graph, then read off the other axis.
If the graph starts at the origin, you can use proportion to find values not on the axes.
If the graph does not start at the origin (e.g., fixed callout fee), the y-intercept represents the fixed cost.
Always double-check the scales on both axes before reading values.

Conversion graph: mass to cost

Distance-Time Graphs

Distance-time graphs show distance on the vertical axis and time on the horizontal axis.
The gradient of the graph equals the speed (speed = distance/time).
A steeper line means faster speed; a horizontal line means the object is stationary.
A line with positive gradient indicates moving away from the start; negative gradient indicates moving back towards the start.
Overall average speed = total distance travelled ÷ total time (including rests).
To complete a journey graph, add a horizontal line for rests and a straight line (with appropriate gradient) for return trips.

Mary's journey to her grandfather's house

Speed-Time Graphs

Speed-time graphs show speed on the vertical axis and time on the horizontal axis.
The gradient equals acceleration (acceleration = speed/time). Positive gradient = speeding up; negative gradient = deceleration.
A horizontal line indicates constant speed (if speed = 0, the object is at rest).
Distance travelled is the area under the graph – split into rectangles and triangles to calculate.
For a curved graph, draw a tangent at a point to find instantaneous acceleration.
Always check the vertical axis to distinguish speed-time from distance-time graphs.

Speed–time graph

Rates-of-Change Graphs

Rates-of-change graphs show how one variable changes with respect to another (often time).
The gradient of the graph gives the rate of change (e.g., litres per kilometre, cm³ per second).
A constant gradient means a constant rate of change; a horizontal line means zero rate of change.
For a curve, draw a tangent to estimate the rate of change at a specific point.
The units of the gradient are the y-axis units divided by the x-axis units.
In filling containers, the shape of the depth-time graph reflects the container's width: narrow parts give steeper gradients (faster depth increase).

Rate of change from a curve

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Practice questions

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1.On a conversion graph, the gradient represents:
Easy
- Athe rate of change between the two quantities
- Bthe total amount of the first quantity
- Cthe starting value of the second quantity
- Dthe difference between the two quantities
2.In a speed-time graph, a horizontal line represents:
Easy
- Aconstant speed
- Bacceleration
- Cdeceleration
- Drest
3.The distance travelled by an object is equal to the area under which type of graph?
Easy
- Aspeed-time graph
- Bdistance-time graph
- Cacceleration-time graph
- Dconversion graph
4.On a distance-time graph, a horizontal line indicates:
Easy
- Athe object is stationary
- Bthe object is moving at constant speed
- Cthe object is accelerating
- Dthe object has returned to the start
5.The gradient of a distance-time graph gives the:
Easy
- Aspeed
- Bacceleration
- Cdistance
- Dtime
6.A car travels at 20 m/s for 15 seconds, then decelerates at 2.5 m/s² until it stops. Find the total distance travelled.
Medium
- A380 m
- B300 m
- C450 m
- D500 m
7.The speed-time graph shows a car accelerating from rest to 20 m/s in 10 seconds, then travelling at constant speed for 20 seconds, then decelerating to rest in 10 seconds. What is the total distance travelled?
Medium
- A600 m
- B500 m
- C700 m
- D400 m
8.Using a conversion graph, if 10 kg costs $6, what is the cost of 35 kg?
Medium
- A$21
- B$18
- C$24
- D$30

More questions

9.A speed-time graph shows a constant acceleration of 2 m/s² for 5 seconds. What is the speed after 5 seconds?
Easy
- A10 m/s
- B2.5 m/s
- C5 m/s
- D20 m/s
10.In a distance-time graph, a straight line with negative gradient represents:
Easy
- Amoving back towards the start
- Bmoving away from the start
- Cstationary
- Daccelerating
11.A car accelerates from rest at 4 m/s² for 6 seconds, then travels at constant speed for 10 seconds, then decelerates at 2 m/s² until it stops. Find the total distance travelled.
Hard
- A456 m
- B360 m
- C480 m
- D540 m
12.The speed-time graph for a journey consists of a triangle (0-10 s, speed from 0 to 20 m/s), a rectangle (10-30 s, constant speed 20 m/s), and a triangle (30-40 s, speed from 20 to 0). What is the average speed for the whole journey?
Hard
- A15 m/s
- B12.5 m/s
- C17.5 m/s
- D20 m/s
13.A conversion graph for litres to gallons passes through (0,0) and (10, 2.2). How many gallons are in 25 litres?
Medium
- A5.5 gallons
- B5 gallons
- C6 gallons
- D4.5 gallons
14.A speed-time graph shows a curve. To find the acceleration at a specific time, you should:
Medium
- Adraw a tangent and find its gradient
- Bfind the gradient of the chord between two points
- Cfind the area under the graph up to that time
- Dread the speed from the graph directly
15.The graph below shows the speed-time graph for a car journey. Which statement is correct?
Medium
- AThe car accelerates, then travels at constant speed, then decelerates.
- BThe car moves at constant speed throughout.
- CThe car accelerates throughout.
- DThe car decelerates, then accelerates.
16.The units of acceleration on a speed-time graph are:
Easy
- Am/s²
- Bm/s
- Cm
- Ds
17.A conversion graph for kilograms to pounds passes through (0,0) and (5, 11). How many pounds are in 20 kilograms?
Medium
- A44 pounds
- B40 pounds
- C55 pounds
- D22 pounds
18.The speed-time graph for a car journey shows a horizontal line at 30 m/s from 10 s to 20 s. What does this represent?
Easy
- AConstant acceleration
- BConstant speed
- CDeceleration
- DStationary
19.A distance-time graph shows a straight line from (0,0) to (4, 100). What is the speed?
Easy
- A25 km/h
- B100 km/h
- C40 km/h
- D20 km/h
20.The speed-time graph for a car consists of a triangle from 0 to 10 s (speed 0 to 20 m/s) and a rectangle from 10 to 30 s (constant speed 20 m/s). What is the total distance travelled?
Medium
- A500 m
- B400 m
- C600 m
- D300 m
21.On a speed-time graph, a line with negative gradient represents:
Easy
- AAcceleration
- BDeceleration
- CConstant speed
- DStationary
22.A conversion graph for euros to dollars passes through (0,0) and (50, 60). How many euros can be bought for 90 dollars?
Medium
- A75 euros
- B54 euros
- C108 euros
- D45 euros
23.A car accelerates from rest at 4 m/s² for 5 s, then travels at constant speed for 10 s, then decelerates at 2 m/s² until it stops. Find the total distance travelled.
Hard
- A300 m
- B250 m
- C350 m
- D400 m
24.The speed-time graph for a car shows a straight line from (0,0) to (6, 12). What is the acceleration?
Easy
- A2 m/s²
- B0.5 m/s²
- C12 m/s²
- D6 m/s²
25.A car journey lasts 8 seconds: it accelerates at a constant rate from rest to v m/s during the first 6 seconds, then travels at a constant speed of v m/s for the final 2 seconds. The total distance travelled is 150 m. Find v.
Medium
- A30 m/s
- B25 m/s
- C20 m/s
- D15 m/s
26.A conversion graph for litres to gallons passes through (0,0) and (10, 2.2). How many litres are in 5.5 gallons?
Easy
- A25 litres
- B20 litres
- C30 litres
- D15 litres
27.The speed-time graph for a car shows a straight line from (0,10) to (5,0). What is the deceleration?
Easy
- A2 m/s²
- B0.5 m/s²
- C10 m/s²
- D5 m/s²
28.The gradient of a distance-time graph gives the acceleration.
Easy
True or false?
29.Match each graph type to what it represents.
Easy
- Distance-time graph
- Speed-time graph
- Conversion graph
- Gradient gives speed
- Gradient gives acceleration
- Gradient gives rate of change
30.Order the following steps to find the distance from a speed-time graph:
Easy
- Find the area under the graph
- Split the area into simple shapes
- Add the areas together
- Identify the time interval
31.Which of the following statements about a speed-time graph are true? (Select all that apply)
Medium
- AThe gradient gives acceleration
- BThe area under the graph gives distance
- CA horizontal line indicates constant speed
- DA negative gradient indicates deceleration
- EThe graph always starts at the origin
32.Match each graph type to the quantity it represents.
Medium
- Gradient of a distance-time graph
- Gradient of a speed-time graph
- Area under a speed-time graph
- Acceleration
- Distance travelled
- Speed
33.Order the following steps to find the distance travelled from a speed-time graph.
Easy
- Identify the shapes under the graph (triangles, rectangles, etc.)
- Calculate the area of each shape
- Add the areas together
- Read the time and speed values from the axes
34.On a distance-time graph, a steeper line indicates a faster speed.
Easy
True or false?
35.The gradient of a conversion graph for dollars ($) to pounds (£) is 0.8. What does this represent?
Medium
- A$0.80 per £1
- B£0.80 per $1
- C$1 per £0.80
- D£1 per $0.80
36.A speed-time graph shows a constant acceleration of 2 m/s² for 5 seconds. What is the speed after 5 seconds in m/s?
Easy
37.A distance-time graph shows a straight line from (0,0) to (4, 100). What is the speed in km/h?
Easy
38.In a distance-time graph, what does a horizontal line represent?
Easy
39.A car's speed-time graph over 100 seconds is a trapezium: it accelerates from 0 to V for the first 20 s, travels at constant speed V for 60 s, then decelerates to 0 over the last 20 s. The total distance travelled is 1200 m. Find V in m/s.
Hard
40.A speed-time graph for a car journey consists of a triangle from 0 to 10 seconds (speed from 0 to 20 m/s), a rectangle from 10 to 30 seconds (constant speed 20 m/s), and a triangle from 30 to 40 seconds (speed from 20 m/s to 0). Calculate the total distance travelled in metres.
Medium
41.Complete the sentence: In a distance-time graph, a horizontal line indicates that the object is _____.
Easy
In a distance-time graph, a horizontal line indicates that the object is ____.
42.For how long was the car travelling at constant speed? (seconds)
Medium
43.Calculate the acceleration during the first 6 seconds in m/s².
Medium
44.Work out the distance covered by the car in metres.
Medium
45.A speed-time graph for a car journey shows a curve. Describe how you would estimate the acceleration at a specific time.
Medium
46.A car's speed-time graph over 100 seconds is a trapezium: it accelerates from 0 to V for the first 20 s, travels at constant speed V for 60 s, then decelerates to 0 over the last 20 s. The total distance travelled is 1200 m. Find V in m/s.
Hard
- A16 m/s
- B15 m/s
- C14 m/s
- D17 m/s
47.A speed-time graph for a car journey consists of a triangle from 0 to 10 seconds (speed from 0 to 20 m/s), a rectangle from 10 to 30 seconds (constant speed 20 m/s), and a triangle from 30 to 40 seconds (speed from 20 m/s to 0). Calculate the total distance travelled in metres.
Medium
- A500 m
- B600 m
- C550 m
- D450 m
48.For how long was the car travelling at constant speed? (seconds)
Medium
- A8 s
- B9 s
- C11 s
- D10 s
49.Calculate the acceleration during the first 6 seconds in m/s².
Medium
- A0.5 m/s²
- B1.5 m/s²
- C2.5 m/s²
- D3.5 m/s²
50.Work out the distance covered by the car in metres.
Medium
- A117.4 m
- B143.6 m
- C156.7 m
- D130.5 m

Revision notes

## Real-Life Graphs - Quick Revision ### Conversion Graphs - Straight-line graph to convert between two quantities (e.g., °C ↔ °F, currency, litres ↔ gallons). - **Gradient** = rate of change (e.g., cost per mile). - To read: from x-axis → vertical to graph → horizontal to y-axis (and vice versa). - If graph starts at origin, use proportion for values off axes. - If not at origin, the y-intercept is a fixed cost (e.g., callout fee). ### Distance-Time Graphs - Distance on y-axis, time on x-axis. - **Gradient = speed** = distance/time. - Steeper line = faster. - Positive gradient = moving away; negative = moving back; horizontal = stationary. - **Average speed** = total distance / total time (including rests). ### Speed-Time Graphs - Speed on y-axis, time on x-axis. - **Gradient = acceleration** = change in speed / time. Positive = speeding up; negative = deceleration. - Horizontal line = constant speed (zero = at rest). - **Distance travelled = area under graph** (split into rectangles, triangles, trapeziums). - For a trapezium: Area = (a+b)/2 × h. ### Rates-of-Change Graphs - Gradient = rate of change of y with respect to x. - Units: y-axis units per x-axis unit (e.g., litres per km). - For curves, draw a tangent to estimate gradient at a point. - Constant gradient → constant rate; steeper → higher rate. ### Key Formulas - Speed = distance / time - Acceleration = (change in speed) / time - Distance (from speed-time) = area under graph - Average speed = total distance / total time

Conversion graph: A straight-line graph used to convert between two quantities, e.g., temperature or currency.
Gradient: The slope of a line, calculated as rise/run; on a distance-time graph it represents speed; on a speed-time graph it represents acceleration.
Speed: The rate of change of distance with time, found from the gradient of a distance-time graph.
Acceleration: The rate of change of speed with time, found from the gradient of a speed-time graph. Positive = speeding up, negative = deceleration.
Deceleration: Negative acceleration; slowing down.
Distance-time graph: A graph showing distance travelled against time; gradient = speed.
Speed-time graph: A graph showing speed against time; gradient = acceleration, area under graph = distance travelled.
Area under graph: On a speed-time graph, the area between the graph and the time axis equals the distance travelled.
Average speed: Total distance travelled divided by total time taken (including any stops).
Rate of change: How one quantity changes with respect to another; represented by the gradient of a graph.
Tangent: A straight line that touches a curve at a point; its gradient gives an estimate of the rate of change at that point.

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