Real Life Graphs

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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.

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.

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.

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, $cm^{3}$ 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).

Conversion Graph Example

Distance-Time Graph

Speed-Time Graph

Rates of Change: Filling Containers

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 $2.5 \frac{m}{s}^{2}$ 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

Summary

## 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

Key terms

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.

Slides

Real-Life Graphs Overview

Real-life graphs show relationships between two quantities, like distance and time, or speed and time.
Common types: conversion graphs, distance-time graphs, speed-time graphs, and rates-of-change graphs.
The gradient of a graph represents the rate of change between the two variables.

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

Acceleration
Speed
Distance
Time

Conversion Graphs

Conversion graphs are straight-line graphs used to convert between two units (e.g., °C to °F, £ to $).
To read a conversion: start at the known value on one axis, draw a line to the graph, then read the other axis.
If the graph starts at the origin, you can use proportion to find values not on the axes.

To convert from pounds to kilograms using a conversion graph, start at the value on $the ____$ axis, draw $a ____$ line to the graph, then $a ____$ line to the other axis.

Distance-Time Graphs

Distance is on the vertical axis, time on the horizontal axis.
Gradient = speed. Steeper line = faster speed.
Horizontal lines mean the object is stationary (rest).
Overall average speed = total distance / total time (including rests).

Which of these are true about distance-time graphs?

A horizontal line means the object is moving at constant speed.
The gradient represents speed.
A line with negative gradient means the object is returning to the start.
The area under the graph gives distance travelled.

Speed-Time Graphs

Speed is on the vertical axis, time on the horizontal axis.
Gradient = acceleration. Positive gradient means speeding up, negative means deceleration.
Horizontal lines mean constant speed (zero acceleration).
Distance travelled = area under the graph (split into rectangles and triangles).

On a speed-time graph, what does a horizontal line at 10 m/s represent?

Accelerating at $10 \frac{m}{s}^{2}$
Decelerating
Moving at constant speed 10 m/s
Stationary

Calculating Acceleration from Speed-Time Graph

Acceleration = change in speed / change in time (rise/run).
For a straight line, use two points on the line.
Units: $\frac{m}{s}^{2} (metres$ per second squared).
Deceleration is negative acceleration.

A car's speed increases from 0 to 20 m/s in 10 seconds. Its acceleration $is ____ \frac{m}{s}^{2}$ .

Calculating Distance from Speed-Time Graph

Distance travelled = area under the speed-time graph.
Split the area into simple shapes: rectangles, triangles, trapeziums.
Area of triangle = ½ × base × height.
Area of trapezium = ½ $\times (sum$ of parallel $sides) \times$ height.

A speed-time graph shows a triangle from 0 to 10 s with speed from 0 to 20 m/s, then a rectangle from 10 to 20 s at 20 m/s. Total distance travelled is:

200 m
300 m
400 m
100 m

Rates-of-Change Graphs

Rate-of-change graphs show how one variable changes with another (often time).
Gradient = rate of change (e.g., litres per km, depth per second).
A steeper gradient means a faster rate of change.
For curves, draw a tangent to estimate the rate at a point.

Which statements about rates-of-change graphs are true?

The gradient gives the rate of change.
A horizontal line means a constant positive rate.
Tangents are used to find rates on curved graphs.
The units of gradient are the y-axis units divided by x-axis units.

Interpreting Gradients in Context

The gradient's units help interpret its meaning (e.g., £/hour = cost per hour).
A negative gradient indicates a decrease (e.g., fuel consumption).
Example: a graph of fuel volume vs distance; gradient $= -0.09 litres/km$ means 0.09 L used per km.

A graph shows depth of water (cm) against time (s). The gradient is 2 cm/s. What does this mean?

Depth increases by 2 cm every second
Depth decreases by 2 cm every second
Time increases by 2 s per cm
The container is empty

Exam Tips

Always check the axes: is it distance-time or speed-time?
Double-check scales on axes before reading values.
For multi-part journeys, total time includes rests.
Use simple shapes to find area under speed-time graphs.

On a speed-time graph, the area under the graph gives $the ____$ travelled.

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