Real Life Graphs

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Notes

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.

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 $\div time)$ .
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.

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.

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.

Calculating Speed from Graphs

Speed = gradient $= (change$ in $distance) \div (change$ 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 $(8-2)/(1) = 6 km/h$ .

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 $\times d +$ fixed charge.
Example: A plumber charges £45 callout plus £60 per hour → $c = 60t + 45$ .

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.

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.

Conversion Graph Example

Distance-Time Graph with Stop

Speed-Time Graph Constant Speed

Comparing Two Taxi Services

Practice questions

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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?
Medium
- AFixed charge
- BCost per mile
- CTotal cost for 1 mile
- DDistance travelled
5.In a distance-time graph, a steeper line means:
Medium
- 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?
Medium
- 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?
Hard
- 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:
Hard
- A200 m
- B20 m
- C2 m
- D0.5 m

Summary

## 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).

Key terms

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).

Slides

Real-Life Graphs: Introduction

Real-life graphs show relationships between two quantities, like distance and time or cost and weight.
Common types include conversion graphs, distance-time graphs, and speed-time graphs.
The gradient (slope) of a line tells you the rate of change, e.g., speed or cost per unit.

What does the gradient of a distance-time graph represent?

Distance
Speed
Time
Acceleration

Conversion Graphs

A conversion graph is a straight-line graph used to convert between two units, e.g., £ to $ or °C to °F.
To convert from one unit to the other, read across from the known value to the line, then down/up to the other axis.
If the graph starts at the origin, you can also use proportion to find values.

To convert 20 kg to pounds using a conversion graph, start at 20 kg on $the ____$ axis, draw a vertical line to the graph, then a horizontal line to the y-axis.

Distance-Time Graphs

Distance is on the vertical axis, time on the horizontal axis.
A horizontal line means the object is stationary (not moving).
The steeper the line, the faster the speed. A line sloping downwards means returning to the start.

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

Constant speed
Accelerating
Stationary
Returning to start

Speed-Time Graphs

Speed is on the vertical axis, time on the horizontal axis.
A horizontal line means constant speed (if speed $> 0)$ or at rest (if speed $= 0)$ .
A line sloping upwards means acceleration, sloping downwards means deceleration.

Which of these are true about speed-time graphs?

A horizontal line at speed 10 m/s means constant speed.
A line sloping upwards means the object is slowing down.
The gradient represents acceleration.

Reading Values from a Conversion Graph

To find the cost of 20 kg: start at 20 on x-axis, go up to the line, then left to y-axis to read cost.
To find how many kg for £30: start at 30 on y-axis, go right to the line, then down to x-axis.
Always check the scales on both axes carefully.

Using a conversion graph, to find the number of kilograms you can buy for £30, you start on the:

x-axis
y-axis
line
origin

Average Speed from Distance-Time Graphs

Overall average speed = total distance ÷ total time (including rests).
For a journey with multiple parts, add all distances and all times.
Example: 8 km in 1.75 hours gives average speed $= 8 \div 1.75$ ≈ 4.57 km/h.

A car travels 120 km in 2 hours. Its average speed $is ____ km/h$ .

Interpreting Rest and Return on Distance-Time Graphs

A horizontal section means the object is at rest (stopped).
A line with negative gradient means the object is returning towards the start.
The total time includes all rests.

On a distance-time graph, a line with negative gradient shows:

Moving away from start
Returning to start
Stationary
Constant speed away

Speed-Time Graphs: Constant Speed and Acceleration

Constant speed is shown by a horizontal line (speed does not change).
Acceleration is shown by a line sloping upwards (speed increases).
Deceleration is shown by a line sloping downwards (speed decreases).

Which statements about speed-time graphs are correct?

A horizontal line at speed 0 means the object is at rest.
A line sloping upwards means the object is decelerating.
The area under the graph represents distance travelled.

Using Graphs with Fixed Charges

Some conversion graphs do not start at the origin, e.g., a taxi fare with a fixed callout fee.
The y-intercept represents the fixed charge (cost at zero usage).
To find the cost for a given amount, read from the graph as usual.

A taxi fare graph shows a y-intercept of £5. This represents:

Cost per km
Fixed callout fee
Total cost for 1 km
Distance travelled

Flashcards

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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