Real Life Graphs

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Apuntes de la lección

Conversion Graphs

A conversion graph is a straight-line graph relating two quantities (e.g., currency, temperature).
To convert from one quantity to the other, read across from the given value to the graph line, then read down/up to the other axis.
The gradient of a conversion graph represents the rate of change (e.g., cost per kilometre).
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, the y-intercept often represents a fixed cost (e.g., callout fee).
Always check the scales on both axes carefully before reading values.

Conversion graph: mass to cost

Distance-Time Graphs

A distance-time graph shows distance travelled (vertical axis) against time (horizontal axis).
The gradient of the graph equals the speed (speed = distance ÷ time).
A steeper line means a faster speed; a horizontal line means the object is stationary (rest).
A line with positive gradient indicates moving away from the start; negative gradient indicates moving back towards the start.
The overall average speed for a journey = total distance travelled ÷ total time (including rests).
To find the distance at a given time, draw a vertical line to the graph then horizontal to the distance axis.

Mary's journey to her grandfather's house

Speed-Time Graphs

A speed-time graph shows speed (vertical axis) against time (horizontal axis).
A horizontal line means constant speed (if speed = 0, the object is at rest).
A line with positive gradient means acceleration (speeding up); negative gradient means deceleration (slowing down).
The area under a speed-time graph represents the distance travelled (not required at Core level but useful).
Always check the vertical axis label to distinguish speed-time from distance-time graphs.

Speed–time graph

Interpreting Travel Graphs

A travel graph is a distance-time graph showing a journey with possible stops.
A horizontal section indicates a stop (rest) – read the time duration from the time axis.
To find the distance of a stop from home, read the distance at the horizontal section.
To find arrival time, read the time at the final distance point.
To complete a travel graph, add horizontal lines for stops and straight lines for travel at constant speed.

Mary's complete return journey

Calculating Speed from Graphs

Speed = gradient = (change in distance) ÷ (change in time).
Use a straight section of the graph to calculate speed (rise over run).
If the graph has multiple sections, calculate speed separately for each.
For a return journey, the gradient is negative but the speed is the absolute value of the gradient.
Example: A line from (1:45, 2 km) to (2:45, 8 km) has gradient (8-2)/(1) = 6 km/h.

Mary's journey to her grandfather's house

Using Conversion Graphs with Fixed Charges

Some conversion graphs do not pass through the origin – the y-intercept is a fixed charge (e.g., callout fee).
To find the fixed charge, read the value on the y-axis when x = 0.
The gradient then represents the cost per unit (e.g., per hour or per km).
To find the equation of the line: c = gradient × d + fixed charge.
Example: A plumber charges £45 callout plus £60 per hour → c = 60t + 45.

Plumber's charge graph

Comparing Two Linear Graphs

When comparing two services (e.g., taxis), plot both lines on the same axes.
The point of intersection shows where the costs are equal.
For distances less than the intersection, the cheaper service is the one with the lower y-intercept.
For distances greater than the intersection, the cheaper service is the one with the lower gradient.
Always state which is cheaper for a given distance using the graph.

Comparing two taxi services

Completing Travel Graphs

To complete a travel graph, first add any stops as horizontal lines for the correct duration.
For travel at constant speed, draw a straight line from the end of the stop to the destination at the correct time.
The gradient of the return line should match the speed given (use rise/run to check).
Ensure the graph is continuous and labelled with time and distance as required.

Mary's complete return journey

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Preguntas de práctica

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1.What does the gradient of a distance-time graph represent?
Easy
- ASpeed
- BDistance
- CTime
- DAcceleration
2.On a distance-time graph, a horizontal line indicates that the object is:
Easy
- AStationary
- BMoving at constant speed
- CAccelerating
- DDecelerating
3.What does the gradient of a conversion graph represent?
Easy
- ARate of change
- BFixed cost
- CTotal cost
- DInitial value
4.A conversion graph shows the cost of a taxi journey. The y-intercept is £5. What does this represent?
Easy
- AFixed charge
- BCost per mile
- CTotal cost for 1 mile
- DDistance travelled
5.In a distance-time graph, a steeper line means:
Easy
- AFaster speed
- BSlower speed
- CLonger time
- DShorter distance
6.A car travels 120 km in 2 hours. What is its average speed in km/h?
Easy
- A60 km/h
- B240 km/h
- C30 km/h
- D120 km/h
7.A journey consists of 3 parts: 2 hours at 50 km/h, then 1 hour rest, then 1 hour at 30 km/h. What is the total distance travelled?
Medium
- A130 km
- B160 km
- C100 km
- D80 km
8.On a speed-time graph, a horizontal line at 20 m/s for 10 seconds means the object travels:
Easy
- A200 m
- B20 m
- C2 m
- D0.5 m

Más preguntas

9.Which of these is a common use of a conversion graph?
Easy
- AConverting Celsius to Fahrenheit
- BDrawing a distance-time graph
- CCalculating acceleration
- DFinding area under a curve
10.A distance-time graph shows a line from (0,0) to (2,100) then a horizontal line to (3,100). The speed during the first part is:
Medium
- A50 km/h
- B100 km/h
- C25 km/h
- D33.3 km/h
11.A taxi charges a fixed fee of £3 plus £2 per km. Which equation represents the cost c for d km?
Easy
- Ac = 2d + 3
- Bc = 3d + 2
- Cc = 2d - 3
- Dc = 3d - 2
12.On a speed-time graph, a line with negative gradient represents:
Easy
- ADeceleration
- BAcceleration
- CConstant speed
- DStationary
13.A train travels from A to B (100 km) at 80 km/h, stops for 15 minutes, then travels B to C (60 km) at 60 km/h. Total journey time in hours is:
Medium
- A2.75 h
- B2.5 h
- C3 h
- D2.25 h
14.What is the unit of speed on a distance-time graph if distance is in km and time in hours?
Easy
- Akm/h
- Bkm
- Ch
- Dkmh
15.A conversion graph line passes through (0,10) and (5,30). The gradient is:
Medium
- A4
- B2
- C10
- D20
16.A person walks 3 km in 40 minutes, then runs 2 km in 10 minutes. Average speed for whole journey in km/h is:
Medium
- A6 km/h
- B5 km/h
- C7.5 km/h
- D10 km/h
17.A conversion graph shows the exchange rate between pounds (£) and dollars ($). The line passes through (0,0) and (5,8). How many dollars are equivalent to £20?
Medium
- A$32
- B$40
- C$25
- D$16
18.On a distance-time graph, a line segment has a gradient of 40. What does this represent if distance is in km and time in hours?
Easy
- ASpeed of 40 km/h
- BDistance of 40 km
- CTime of 40 hours
- DAcceleration of 40 km/h²
19.A taxi company charges a fixed fee of £3 plus £2.50 per km. Which equation represents the cost c (£) for a journey of d km?
Easy
- Ac = 2.5d + 3
- Bc = 3d + 2.5
- Cc = 2.5d − 3
- Dc = 3d − 2.5
20.A distance-time graph shows a journey: from 0 to 2 hours, distance increases from 0 to 120 km; then a horizontal line from 2 to 3 hours; then from 3 to 5 hours, distance decreases to 0. What is the total distance travelled?
Medium
- A240 km
- B120 km
- C60 km
- D180 km
21.A speed-time graph has a horizontal line at 30 m/s from 0 to 10 seconds, then a straight line down to 0 m/s at 20 seconds. What is the total distance travelled?
Hard
- A450 m
- B300 m
- C600 m
- D150 m
22.A travel graph shows a journey: from 0900 to 1000, distance increases from 0 to 40 km; then a horizontal line to 1030; then from 1030 to 1130, distance increases to 80 km. What is the average speed for the whole journey in km/h?
Hard
- A32 km/h
- B40 km/h
- C48 km/h
- D24 km/h
23.A car travels at 60 km/h for 1.5 hours, then at 40 km/h for 1 hour. What is the total distance travelled?
Medium
- A130 km
- B100 km
- C150 km
- D120 km
24.A speed-time graph shows a straight line from (0,0) to (10,20). What is the acceleration?
Medium
- A2 m/s²
- B0.5 m/s²
- C20 m/s²
- D200 m/s²
25.On a distance-time graph, a line with negative gradient represents:
Easy
- AMoving back towards the start
- BMoving away from the start
- CStationary
- DAccelerating
26.A conversion graph line passes through (0,20) and (10,50). The gradient is:
Easy
- A3
- B2
- C5
- D0.3
27.A train travels 120 km at 80 km/h, then 60 km at 60 km/h. What is the average speed for the whole journey in km/h?
Medium
- A72 km/h
- B70 km/h
- C75 km/h
- D68 km/h
28.A taxi charges £2 per km plus a fixed fee of £4. What is the cost of a 10 km journey?
Easy
- A£24
- B£20
- C£40
- D£14
29.A speed-time graph has a straight line from (0,10) to (5,30). What is the acceleration?
Medium
- A4 m/s²
- B2 m/s²
- C5 m/s²
- D20 m/s²
30.A conversion graph shows the exchange rate between pounds (£) and euros (€). The line passes through (0,0) and (5,6). How many euros are equivalent to £30?
Medium
- A€36
- B€25
- C€30
- D€40
31.A taxi company charges a fixed fee of £4 plus £3 per kilometre. Which equation represents the cost c (£) for a journey of d km?
Medium
- Ac = 3d + 4
- Bc = 4d + 3
- Cc = 3d - 4
- Dc = 4d - 3
32.On a distance-time graph, a line segment has a gradient of 50. What does this represent if distance is in km and time in hours?
Medium
- AA speed of 50 km/h
- BA distance of 50 km
- CA time of 50 hours
- DAn acceleration of 50 km/h²
33.A car travels at 60 km/h for 2 hours, then at 40 km/h for 1 hour. What is the average speed for the whole journey in km/h?
Medium
- A53.3 km/h
- B50 km/h
- C55 km/h
- D48 km/h
34.A speed-time graph has a straight line from (0,0) to (10,30). What is the acceleration?
Medium
- A3 m/s²
- B0.3 m/s²
- C30 m/s²
- D10 m/s²
35.A train travels 150 km at 75 km/h, then 60 km at 60 km/h. What is the total time for the journey in hours?
Medium
- A3 hours
- B2.5 hours
- C3.5 hours
- D2 hours
36.A distance-time graph shows a journey: from 0 to 1.5 hours, distance increases from 0 to 90 km; then a horizontal line from 1.5 to 2.5 hours; then from 2.5 to 4 hours, distance increases to 150 km. What is the speed during the last part?
Medium
- A40 km/h
- B60 km/h
- C50 km/h
- D30 km/h
37.A taxi charges a fixed fee of £5 plus £2.50 per km. What is the cost of a 12 km journey?
Easy
- A£35
- B£30
- C£32.50
- D£37.50
38.On a speed-time graph, a horizontal line at 15 m/s for 8 seconds means the object travels:
Easy
- A120 m
- B15 m
- C8 m
- D23 m
39.On a distance-time graph, what does a horizontal line represent?
Easy
- AThe object is stationary
- BThe object is moving at constant speed
- CThe object is accelerating
- DThe object is returning to the start
40.On a speed-time graph, a line with positive gradient represents an object that is accelerating.
Easy
True or false?
41.A conversion graph shows the cost (£) of a taxi journey against distance (km). The line passes through (0,4) and (10,29). What does the y-intercept represent?
Medium
- AThe fixed charge of £4
- BThe cost per km of £2.50
- CThe total cost for 10 km
- DThe distance travelled for £0
42.A distance-time graph shows a journey: from 0 to 1 hour, distance increases from 0 to 60 km; then a horizontal line from 1 to 1.5 hours; then from 1.5 to 2 hours, distance increases to 100 km. What is the average speed for the whole journey in km/h?
Hard
- A50 km/h
- B60 km/h
- C40 km/h
- D66.7 km/h
43.Match each type of graph to its correct description.
Medium
- Conversion graph
- Distance-time graph
- Speed-time graph
- Shows relationship between two units or quantities
- Shows distance travelled over time; gradient = speed
- Shows speed over time; gradient = acceleration
44.Arrange the following steps in the correct order to find the cost of 20 kg using a conversion graph that starts at the origin.
Medium
- Read the value on the y-axis
- Draw a horizontal line from the graph to the y-axis
- Draw a vertical line from the x-axis to the graph
- Start at 20 kg on the x-axis
45.Which of the following statements about a distance-time graph are true? (Select all that apply.)
Hard
- AA horizontal line means the object is stationary.
- BA steeper line means a slower speed.
- CThe gradient of the line represents the speed.
- DA line with negative gradient means the object is moving back towards the start.
- EThe area under the graph represents the distance travelled.
46.On a speed-time graph, a straight line with a positive gradient represents an object that is accelerating.
Easy
True or false?
47.A conversion graph for litres and gallons passes through (0,0) and (10,2.2). How many litres are in 1 gallon? Give your answer correct to 1 decimal place.
Medium
48.Complete the sentence: On a distance-time graph, the steeper the line, the ______ the speed.
Easy
On a distance-time graph, the steeper the line, the ____ the speed.
49.A car travels 150 km in 3 hours. What is its average speed in km/h?
Easy
50.Explain what the gradient of a speed-time graph represents.
Easy
51.A speed-time graph shows a straight line from (0,0) to (8,24). What is the acceleration in m/s²?
Medium
52.A car travels at a constant speed of 20 m/s for 15 seconds. How far does it travel in metres?
Easy
53.A conversion graph for litres and gallons passes through (0,0) and (10,2.2). How many litres are in 1 gallon? Give your answer correct to 1 decimal place.
Medium
- A6.5
- B5.5
- C3.5
- D4.5
54.A car travels 150 km in 3 hours. What is its average speed in km/h?
Easy
- A60 km/h
- B55 km/h
- C45 km/h
- D50 km/h
55.A speed-time graph shows a straight line from (0,0) to (8,24). What is the acceleration in m/s²?
Medium
- A3 m/s²
- B4 m/s²
- C2 m/s²
- D5 m/s²
56.A car travels at a constant speed of 20 m/s for 15 seconds. How far does it travel in metres?
Easy
- A360 m
- B300 m
- C330 m
- D270 m

Apuntes de repaso

## Real-Life Graphs - Quick Revision ### Conversion Graphs - Straight-line graph to convert between two quantities (e.g., currency, temperature). - **Reading values**: From x-axis → vertical line to graph → horizontal to y-axis (and vice versa). - **Gradient** = rate of change (e.g., cost per km). - If graph starts at origin, use proportion. If not, the y-intercept often represents a fixed charge. ### Distance-Time Graphs - Distance on y-axis, time on x-axis. - **Gradient = speed** (speed = distance / time). - Steeper line = faster. - Positive gradient = moving away; negative gradient = returning; horizontal = stationary. - **Overall average speed** = total distance / total time (including rests). ### Speed-Time Graphs - Speed on y-axis, time on x-axis. - Positive gradient = accelerating; negative gradient = decelerating; horizontal = constant speed. - Constant speed zero means at rest. ### Key Formulas - Speed = distance / time - Average speed = total distance / total time ### Tips - Always check axis scales. - For distance-time: gradient = speed. - For conversion graphs: use lines to read values; proportion works if graph starts at (0,0).

Conversion graph: A straight-line graph used to convert between two quantities, e.g., currency or temperature.
Distance-time graph: A graph with distance on the vertical axis and time on the horizontal axis; the gradient represents speed.
Speed-time graph: A graph with speed on the vertical axis and time on the horizontal axis; the gradient represents acceleration.
Gradient: The steepness of a line, calculated as rise/run; on a distance-time graph it equals speed.
Average speed: Total distance travelled divided by total time taken (including any stops).

Tarjetas

What is a conversion graph?

A conversion graph is a straight-line graph relating two quantities, used to convert between them by reading values off the graph.

What does the gradient of a conversion graph represent?

The gradient represents the rate of change. For example, in a cost-distance graph, the gradient is the cost per kilometre.

How do you use a conversion graph to convert from the x-axis to the y-axis?

Start at the value on the x-axis, draw a vertical line to the graph, then a horizontal line to the y-axis, and read off the value.

How do you use a conversion graph to convert from the y-axis to the x-axis?

Start at the value on the y-axis, draw a horizontal line to the graph, then a vertical line to the x-axis, and read off the value.

What does the y-intercept represent in a conversion graph that does not start at the origin?

The y-intercept often represents a fixed cost or starting value, such as a callout fee for a plumber or the freezing point in temperature conversion.

On a distance-time graph, what does the gradient represent?

The gradient represents speed (distance/time).

On a distance-time graph, what does a horizontal line indicate?

A horizontal line indicates that the object is stationary (not moving).

On a distance-time graph, what does a line with negative gradient represent?

A line with negative gradient represents movement towards the start point (returning).

How do you calculate overall average speed from a distance-time graph?

Overall average speed = total distance travelled / total time (including rests).

On a speed-time graph, what does a horizontal line indicate?

A horizontal line indicates constant speed. If the speed is zero, the object is at rest.

On a speed-time graph, what does a line with positive gradient represent?

A line with positive gradient represents acceleration (speeding up).

On a speed-time graph, what does a line with negative gradient represent?

A line with negative gradient represents deceleration (slowing down).

How do you find the distance travelled from a speed-time graph?

The distance travelled is the area under the graph (between the graph and the time axis).

Sara walks from home to school. She stops at a shop. The travel graph shows her journey. How do you find the distance of the shop from her home?

Read the distance from the graph at the time when she stops (horizontal line). The distance at that point is the distance from home.

Michael's travel graph shows a rest. How do you find how many minutes he stopped?

Find the horizontal section of the graph. The length of that section on the time axis gives the duration of the stop in minutes.

How does a distance-time graph show that an object travelled faster before a stop than after?

The line before the stop is steeper (greater gradient) than the line after the stop, indicating higher speed.

Shohan cycles to the library, stops at the post office. How do you complete the travel graph after he stays at the library for 25 minutes and cycles home at 18 km/h?

Draw a horizontal line for 25 minutes at the library distance, then a straight line back to home (distance zero) with slope corresponding to 18 km/h (distance/time).

Use a conversion graph to change £5 to dollars.

Draw a vertical line from £5 on the x-axis to the graph, then a horizontal line to the y-axis. Read the dollar amount.

Hua cycles to the library, stops for a rest. How do you find her distance from home when she stops?

Read the distance on the y-axis at the time where the graph becomes horizontal.

Juan travels to a shop. The graph shows his journey. What happens at 1400?

At 1400, the graph becomes horizontal, indicating he stopped (arrived at the shop).

How do you use a conversion graph to change 5000 shillings to dollars?

Start at 5000 on the shillings axis (x-axis), draw a vertical line to the graph, then a horizontal line to the dollars axis, and read the value.

A train travels from station A to B (distance 60 km) in 30 minutes. Calculate its speed in km/h.

Speed = distance/time = 60 km / 0.5 h = 120 km/h.

A train stops at station B from 14:30 to 14:40. How many minutes did it stop?

10 minutes.

Saanvi's Taxis cost graph: find the cost of a 4 km journey.

Read the cost from the graph at 4 km on the x-axis.

Krishna's Taxis cost $5 plus $2 per km. Write an equation for the cost c for d km.

c = 5 + 2d.

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