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Graphs
Graphs are used to visualise changes in quantities. The axes provide information about the name, value and unit of the measurement.
Example 1
A meteorological station recorded the changes in temperature over time on a winter day.
- At what time was it the warmest?
- What was the temperature at \latex{ 10 } o'clock?
- When did the temperature begin to turn colder?
- What was the total change in temperature over the day?
Temperature in Vienna (\latex{ 03 }. \latex{ 02 }. \latex{ 2010 }.)
\latex{temperature}
\latex{time}
(hour)
\latex{ 6 }
\latex{ 4 }
\latex{ 2 }
\latex{ 0 }
\latex{-2 }
\latex{-4 }
\latex{-6 }
\latex{-8 }
\latex{-10 }
\latex{ 2 }
\latex{ 4 }
\latex{ 6 }
\latex{ 8 }
\latex{ 10 }
\latex{ 12 }
\latex{ 14 }
\latex{ 16 }
\latex{ 18 }
\latex{ 20 }
\latex{ 22 }
\latex{ 24 }
\latex{ (°C) }
Solution
Read the recordings from the graph.
- It was at its warmest at \latex{ 14:00 }, \latex{ +2 }°C.
- It was \latex{ -4 }°C at \latex{ 10 } o'clock.
- The temperature was decreasing from \latex{ 0 } to \latex{ 6 } and from \latex{ 14 } to \latex{ 24 }.
- The daily change in temperature is the difference between the warmest and coldest temperature, \latex{ (+2) - (-8) = 10 } \latex{ (°C). }
Example 2
Rob and Sullivan were appointed as classroom helpers for a \latex{ week. } They agreed to toss a coin to decide who would clean the blackboard before lessons. There were six occasions when the blackboard needed cleaning, which meant six tosses in total. Make a graph and a table to illustrate all the possible outcomes.
Solution
The table of possible outcomes:
Rob
Jack
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 6 }
\latex{ 5 }
\latex{ 4 }
\latex{ 3 }
\latex{ 2 }
\latex{ 1 }
\latex{ 0 }
Illustrate the results on a coordinate system. \latex{ (4; 2) } represents the situation in which Rob cleans the board \latex{ 4 } times, and Jack cleans it \latex{ 2 } times. The points should not be connected as there are no possible outcomes where the number of cleanings is not an integer.
\latex{ x :} Number of cleanings:
Rob
Rob
\latex{ y: } Number of cleanings:
Sullivan
Sullivan
\latex{ (0; 6) }
\latex{ (1; 5) }
\latex{ (2; 4) }
\latex{ (3; 3) }
\latex{ (4; 2) }
\latex{ (5; 1) }
\latex{ (6; 0) }
\latex{ 6 }
\latex{ 5 }
\latex{ 4 }
\latex{ 3 }
\latex{ 2 }
\latex{ 1 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ x }
\latex{ y }
Example 3
Pete plays football every \latex{ day. } He walks \latex{ 420 } \latex{ m } from his house to the football pitch. The following graphs show Pete's journeys over three consecutive \latex{ days: }
Day \latex{ 1 }
\latex{distance\; from\; home\; (m)}
\latex{time\;(min)}
\latex{ 420 }
\latex{ 360 }
\latex{ 300 }
\latex{ 240 }
\latex{ 180 }
\latex{ 120 }
\latex{ 60 }
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
Day \latex{ 2 }
\latex{time\;(min)}
\latex{distance\;from\;home\;(m)}
\latex{ 420 }
\latex{ 360 }
\latex{ 300 }
\latex{ 240 }
\latex{ 180 }
\latex{ 120 }
\latex{ 60 }
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
Day \latex{ 3 }
\latex{time\;(min)}
\latex{distance\;from\;home\;(m)}
\latex{ 420 }
\latex{ 360 }
\latex{ 300 }
\latex{ 240 }
\latex{ 180 }
\latex{ 120 }
\latex{ 60 }
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
- How many \latex{ metres } per minute did Peter walk on the first \latex{ day? }
- On which \latex{ day } did he stop to have a chat, and for how long?
- On which \latex{ day } did he run back home to get his trainers? How did he make it to football practice on time?
Solution
One axis of the graph shows the time, the other the distance from Pete's house.
- Pete's speed was constant, \latex{ 60 } \latex{ m } per minute. It took him \latex{ 7 } \latex{ minutes } to walk \latex{ 420 } \latex{ m }.
- On day \latex{ 2 }, the distance from home remained the same from \latex{ minute } \latex{ 3 } to \latex{ minute } \latex{ 5 }. This means that he stopped to chat for \latex{ 2 } \latex{ minutes }.
- When Pete ran back home, his distance from his home decreased to \latex{ 0 }. This can be seen in Graph \latex{ 3 }. Then he had \latex{ 3 } \latex{ minutes } to walk the whole distance of \latex{ 420 } \latex{ m }, so he had to increase his speed a lot.
Exercises
{{exercise_number}}. Use the graphs from Example 3.
a) Complete the table. How far from home was Pete at the times given?
Day \latex{ 1 }
\latex{ time (min) }
\latex{ distance (m) }
\latex{ 1 }
\latex{ 2 }
\latex{ 4 }
\latex{ 6 }
\latex{ 60 }
\latex{ 180 }
\latex{ 300 }
\latex{ 420 }
b) When was Pete \latex{ 180 } \latex{ m } away from his house on each of the three \latex{ days? }
c) How many \latex{ metres } was he away from the football pitch after walking for \latex{ 4 } \latex{ minutes } on each \latex{ day? }
d) How many \latex{ metres } did he walk on each \latex{ day? }
e) Which part of the journey did he cover the fastest?
{{exercise_number}}. One day it took Pete \latex{ 15 } \latex{ minutes } to get to football practice instead of the usual \latex{ 7 }. Think about what could have happened and illustrate it with a graph.
{{exercise_number}}. The coordinate system shows the time-distance history of a cycling trip.
- How many \latex{ kilometres } did the participants cycle?
- How many \latex{ kilometres } did they cycle between the \latex{ 5 th} and the \latex{ 7 th} \latex{ hour }\latex{ ? } What might be the explanation?
- On which part of the route were they the fastest?
- How many breaks did they take? How long were the breaks?
\latex{distance} (\latex{ km })
\latex{time\;(hour)}
\latex{ 100 }
\latex{ 90 }
\latex{ 80 }
\latex{ 70 }
\latex{ 60 }
\latex{ 50 }
\latex{40 }
\latex{30 }
\latex{20 }
\latex{10 }
\latex{0 }
\latex{1 }
\latex{2 }
\latex{3 }
\latex{4 }
\latex{5 }
\latex{6 }
\latex{7 }
\latex{8 }
\latex{9 }
\latex{10 }
\latex{11 }
\latex{12 }
{{exercise_number}}. The graph shows the change in a baby’s weight. (→)
- How many \latex{ kilograms } did the baby gain in the first \latex{ year? }
- How many \latex{ months } old was the baby when his weight doubled?
- Was there a \latex{ month } when the baby's weight did not change?
- On average, how many \latex{ kilograms } \latex{ per } \latex{ month } did the baby gain?
- How much will he weigh by his \latex{ 10 }th birthday?
\latex{weight} (\latex{ kg })
\latex{time\;(month)}
\latex{ 11 }
\latex{ 10}
\latex{ 9}
\latex{ 8}
\latex{ 7}
\latex{ 6}
\latex{ 5}
\latex{ 4}
\latex{ 3}
\latex{ 2}
\latex{ 1}
\latex{ 0}
\latex{ 1}
\latex{ 2}
\latex{ 3}
\latex{ 4}
\latex{ 5}
\latex{ 6}
\latex{ 7}
\latex{ 8}
\latex{ 9}
\latex{ 10}
\latex{ 11}
\latex{ 12}
{{exercise_number}}. The table shows the change in temperature on a summer day from \latex{ 6 } in the morning to \latex{ 6 } in the evening.
\latex{ time (hour) }
\latex{ temperature (ºC) }
\latex{ 6:00 }
\latex{ 7:00 }
\latex{ 8:00 }
\latex{ 9:00 }
\latex{ 10:00 }
\latex{ 11:00 }
\latex{ 12:00 }
\latex{ 13:00 }
\latex{ 14:00 }
\latex{ 15:00 }
\latex{ 16:00 }
\latex{ 17:00 }
\latex{ 18:00 }
\latex{ 19 }
\latex{ 19 }
\latex{ 20 }
\latex{ 22 }
\latex{ 24 }
\latex{ 27 }
\latex{ 31 }
\latex{ 32 }
\latex{ 33 }
\latex{ 32 }
\latex{ 30 }
\latex{ 28 }
\latex{ 25 }
- Draw a graph of the changes on a coordinate system.
- What was the highest temperature?
- What is the most likely temperature at \latex{ 10:30 ?}
- When was the temperature most likely to be \latex{ 26 } \latex{ degrees } \latex{ Celsius? }
{{exercise_number}}. A pedestrian, a cyclist and a motorcyclist set off at the same time for the nearest village, \latex{ 20 } \latex{ km } away. Draw their progress on a coordinate system where the pedestrian walks at \latex{ 5 } \latex{ km } \latex{ per } \latex{ hour }, the cyclist pedals at \latex{ 20 } \latex{ km } \latex{ per } \latex{ hour } and the motorcyclist rides at \latex{ 40 } \latex{ km } \latex{ per } \latex{ hour. } What can you conclude from the resulting graph?
{{exercise_number}}. What is the perimeter of squares whose sides are integers less than \latex{ 6 ?} Make a table and draw a graph of the side \latex{ length - } \latex{ perimeter } value pairs of the squares.
Quiz
How many points are there in the Cartesian coordinate system where both coordinates are integers and both coordinates have an absolute value of no more than \latex{ 50 ?}
