Real-life graphs

A container takes 8 minutes to be filled with water.

The graph shows the rate of flow, in litres per minute, of the water into the container during the 8 minutes.

Show that 126 litres of water flows into the container between 0.5 minutes and 7.5 minutes.

Work out the average rate of flow, in litres per minute, of water into the container during the 8 minutes.

(b)........ litres per minute

Zayn drives from home to a shopping centre. The graph shows information about the journey and the time spent at the shopping centre.

07 00 08 00 09 00 10 00 11 00 12 00 13 00 14 00

State an assumption that has been made when the graph was drawn.

Write down the distance from Zayn’s home to the shopping centre.

Calculate Zayn’s average speed, in km/h, from home to the shopping centre.

Zayn leaves the shopping centre at 11 00. Zayn drives home without stopping. The journey home takes 1 hours.

Complete the graph to show this information.

The graph shows the speed of an object during the first 20 seconds of its motion.

Calculate the distance travelled by the object during the 20 seconds.

The graph shows the distance travelled by an object during the first 4 seconds of its motion.

Work out the average speed of this object between 2 and 4 seconds.

Use the graph to estimate the speed of this object at 3 seconds. You must show working to support your estimate.

What happens to the speed of this object during these 4 seconds of motion. Explain how you know.

I know this because........

Below are six graphs, numbered 1 to 6, that show different relationships between x and y.

Graph 1
Graph 2
Graph 3

Graph 4
Graph 5
Graph 6

y y y

For each description below, write down the number of the graph that best represents the relationship between x and y.

x: temperature in °C y: temperature in °F, where y = 1.8 x + 32. Graph........

x: average speed when running 200 m y: time taken to run 200 m. Graph........

x: mass of a solid object made from clay y: volume of the same solid object. Graph........

The graph shows Taylor’s journey from home to an airport. During the journey Taylor stops for petrol.

For how long did Taylor stop for petrol?

Taylor drives the same route back home from the airport at an average speed of 45 km / h. Taylor leaves the airport at 22:00.

Work out the time when Taylor arrives home.

The graph shows Taylor’s journey from home to an airport. During the journey Taylor stops for petrol.

For how long did Taylor stop for petrol?

Taylor drives the same route back home from the airport at an average speed of 45 km / h. Taylor leaves the airport at 22:00.

Work out the time when Taylor arrives home.

Here are sketches of six graphs, labelled A to F.

Write the letter of the graph that represents the following relationships.

y is directly proportional to x.

y is inversely proportional to x.

A sales representative and a manager discuss this graph of sales over the last year.

The sales representative says

I can tell from the graph that, over the last year, sales have risen every month.

Is the sales representative correct? Give a reason for your answer.

The manager says

I can tell from the graph that sales are now more than double what they were at the start of the year.

Is the manager correct? Give a reason for your answer.

Hiro and Taylor are both electricians. Hiro does not charge to visit a house but charges a fixed rate per hour for the work needed.

This graph shows the relationship between the hours worked and the total charge made by Hiro.

Explain how this graph shows that Hiro’s total charge is directly proportional to the hours worked.

Taylor does charge to visit a house and charges the same fixed rate per hour as Hiro for the work needed.

On the axes above, draw a graph to show the relationship between the hours worked and

the total charge made by Taylor.

The graph shows the velocity of a particle over the first 20 minutes of its motion.

Between 7 minutes and 15 minutes the velocity of the particle is v metres per minute. The average velocity of the particle over the 20 minutes is 11.55 metres per minute.

Find the value of v. You must show your working.

The time taken to complete a journey halves as the speed doubles.

On the axes below, sketch a graph to show this relationship.

It takes 40 minutes to fill a garden pond using water from 5 identical hose pipes.

Assuming the rate of flow of water from each hose pipe is the same, work out how many minutes it would take to fill the same garden pond using 2 of these hose pipes.

The graph shows information about the population of a village.

The population of the village in 2021 was 4740.

Plot this point on the graph.

Work out the increase in the population of the village between 2016 and 2018.

Rowan says that there was a huge increase in the population of the village between 2015 and 2020.

Describe how Rowan may have been misled by the graph.

Blake says that the population of the village will be greater than 4800 in 2022.

Write down an assumption Blake has made.

The graph shows information about the population of a village.

The population of the village in 2021 was 4740.

Plot this point on the graph.

Work out the increase in the population of the village between 2016 and 2018.

Rowan says that there was a huge increase in the population of the village between 2015 and 2020.

Describe how Rowan may have been misled by the graph.

Blake says that the population of the village will be greater than 4800 in 2022.

Write down an assumption Blake has made.

The graph shows the speed of a car during the first 40 seconds of a journey.

Write down the acceleration of the car between 10 seconds and 25 seconds.

Work out the average speed of the car, in m/s, during the 40 seconds. You must show your working.

1600 fish are released into a new lake which has no fish. The number of fish is expected to increase by 5% each year.

The table shows the expected number of fish in the lake at the end of 1 year and at the end of 2 years.

Complete the table. Round your answers to the nearest integer.

Use the table to draw a suitable graph to show the expected number of fish in the lake.

A maximum of 2000 fish can live in the lake.

What effect would you expect this to have on the shape of your graph after 4 years?

TURN OVER FOR QUESTIONS 25 AND 26

An athlete goes for a training run. The graph shows their speed as they run.

Write down the athlete’s acceleration between 400 seconds and 1000 seconds.

Work out the athlete’s average speed, in m/s, during the 1200 seconds. You must show your working.

1600 fish are released into a new lake which has no fish. The number of fish is expected to increase by 5% each year.

The table shows the expected number of fish in the lake at the end of 1 year and at the end of 2 years.

Complete the table. Round your answers to the nearest integer.

Use the table to draw a suitable graph to show the expected number of fish in the lake.

A maximum of 2000 fish can live in the lake.

What effect would you expect this to have on the shape of your graph after 4 years?

A conversion graph between pounds (£) and dollars ($) is shown below.

Explain fully how the graph shows that the number of dollars is directly proportional to the number of pounds.

Use the conversion graph to change £20 into dollars.

Some trainers cost £170 in the UK. The same trainers cost $195 in the USA.

Show that the trainers cost less in the USA.

If the trainers are brought from the USA there is an extra charge for tax and delivery.

Alex wants to pay the lowest total amount for the trainers.

Write down the maximum extra charge for tax and delivery that Alex should be willing to pay. Give your answer in dollars.

Here are sketches of five graphs.

Write the letter of the graph that represents the following relationships.

y is directly proportional to x.

y is inversely proportional to x.

Here are sketches of five graphs.

Write the letter of the graph that represents the following relationships.

y is directly proportional to x.

y is inversely proportional to x.

The mass of flour used in a recipe doubles as the number of biscuits made doubles.

On the axes below, sketch a graph to show this relationship.

Here are some of the ingredients for a recipe to make 10 biscuits.

Jane followed the recipe and used 432 g of butter. All of the sugar used came from a new 2 kg bag.

Find the number of biscuits she made.

Find the mass of the sugar, in grams, that Jane has left in the bag.

The graph shows the speed of a vehicle during the first 40 seconds of motion.

Calculate the distance travelled by the vehicle during the 40 seconds.

The graph shows the distance travelled by a particle over 6 seconds.

Work out the average speed of the particle between 2 and 4 seconds.

Estimate the speed of the particle at 4 seconds.

The graph shows the cost of electricity with Company A.

Use the information in the graph to estimate the cost of electricity for a customer who uses 450 kwH of electricity.

Company B charges 14.3 pence per kwH of electricity used.

If Company B’s cost of electricity was plotted on the same axes as Company A’s cost of electricity, which line would be steeper? Explain how you know.

Company........ would have the steeper line because........

The graph shows Sarah’s journey from her home to a shopping centre.

State an assumption that has been made when the graph was drawn.

What is the distance from Sarah’s home to the shopping centre?

Between which two times did Sarah stop? Explain how the graph shows this.

From........ to........ shown on the graph by........

Sarah stays at the shopping centre until 13:00. She then travels home without stopping. Her journey home takes 40 minutes.

Complete the graph to show this information.

Work out Sarah’s average speed for her journey home. Give your answer in kilometres per hour.

50 sweets weigh 200 g.

If each sweet weighs the same, work out the weight of 7 sweets.

b is directly proportional to a. b is 10 when a is 8.

Work out b when a is 9.

A graph is drawn below.

Explain how you know that y is not directly proportional to x.

The graph shows the speed of a train during the first 60 seconds of motion.

What is the speed of the train after 9 seconds?

What does the straight line suggest about the speed of the train over the first 15 seconds?

Work out the average speed of the train, in m / s, during the 60 seconds.

This graph shows the cost of buying potatoes from a farm.

How much does it cost to buy 70 kg of potatoes?

What weight of potatoes can be bought for £38?

The cost per kilogram of potatoes is the same for any weight of potatoes.

How much will it cost to buy 180 kg of potatoes from the farm?

This graph shows the annual profits, in thousands of pounds, of the farm between 2013 and 2017.

Describe one misleading feature of the graph.

These graphs show different relationships between the variables x and y.

Identify the graph which shows the following.

y is directly proportional to x.

y is inversely proportional to x.

The graph shows the speed of a cyclist during 20 seconds of a journey.

Find the acceleration of the cyclist

for the first 4 seconds

between 4 seconds and 14 seconds.

Work out the distance travelled by the cyclist during the 20 seconds.

These graphs show different relationships between the variables x and y.

Identify the graph which shows the following.

y is directly proportional to x.

y is inversely proportional to x.

Sketch a graph which shows that y is directly proportional to x².

Viraj cycled from his home to visit his aunt. He drew this graph to show his journey. He stopped at a shop 7 km from his home.

State one assumption that Viraj made when he drew his graph.

For how long did Viraj stop at the shop?

Work out Viraj’s average speed between his home and the shop. Give your answer in metres per minute.

(c)........ metres per minute

How can you tell, without doing any calculations, that Viraj’s average speed between his home and the shop is greater than his average speed between the shop and his aunt?

The graph shows the speed of a tram as it travels from the library to the town hall.

Calculate the deceleration of the tram as it approaches the town hall.

Calculate the distance travelled by the tram between the library and the town hall.

What was the maximum speed of the tram as it travelled between the library and the town hall? Give your answer in kilometres per hour.

This graph shows part of Lucy’s car journey from London to Sheffield. The car made one stop at a service station.

Use the graph to answer these questions.

For how long did the car stop at the service station?

Work out the average speed of the car, in miles per hour, between London and the service station.

Sheffield is 180 miles from London. Lucy arrived in Sheffield at 14 20.

Complete the graph.

A graph is drawn below.

Explain how you know that y is not directly proportional to x.

q is directly proportional to r. q is 68 when r is 20.

Work out q when r is 25.

A company tests a new battery for an electric car. The distance the car travels, d km, and the charge left in the battery, C %, are measured.

Some measurements are shown in the table.

Plot these values on the grid and use them to draw a straight line.

Use your line to estimate the greatest distance the car will travel.

What assumption is made when estimating the greatest distance?

For your line in part (a), find

the gradient,

the C-axis intercept.

Use your answers to part (c) to write down the equation of your graph.

Give your equation in the form C = ad + b.

Use your equation to find the value of C when d = 210.

Comment on your answer.

The graph shows information about the speed of a vehicle during the final 50 seconds of a journey. At the start of the 50 seconds the speed is k metres per second. The distance travelled during the 50 seconds is 1.35 kilometres.

Work out the average speed of the vehicle during the 50 seconds. Give your answer in metres per second.

Work out the value of k.

Calculate the gradient of the graph in the final 10 seconds of the journey.

Describe what this gradient represents.

This graph shows the world population, in billions, between 1951 and 2015.

Use the graph to estimate the average rate of growth of the world population between 1951 and 2015. Give suitable units for your answer.

The graph shows Mia’s journey from her home to university.

Calculate Mia’s average speed for the whole journey.

The graph shows the speed, v metres per second (m/s), of a car at time t seconds.

Find the speed of the car at t = 7.

It is claimed that the car has accelerated from 0 to 60 miles per hour in the first 10 seconds.

Does the graph support this claim? Show your reasoning.

Use 1 mile = 1.6 kilometres.

Use the graph to estimate the acceleration at t = 7.

The speed of this car is directly proportional to the square of the time.

Find a formula linking v and t.

Georgina says that the graph shows that the speed of the car will continue to increase after 10 seconds.

Make one comment to show that this statement is incorrect.

The graph shows the speed, in metres per second, of a particle over the first four seconds of

Use the graph to estimate the distance travelled by the particle in the four seconds.

The graph shows Mia’s journey from her home to university.

Calculate Mia’s average speed for the whole journey.