Line graphs
A line graph is a type of chart displaying points connected by straight line segments. This used to show how one value varies as another variable changes and is most often used to show trends in data over time.
For example a line graph may show temperature in a city over a day. The graph would show a quick visualisation of the data, we could find the temperature given a particular time of day and we can further interpret the graph to find the maximum or minimum temperature and look for trends such as, at what times was the temperature increasing and when was it decreasing.
Worked example
Example 1
The following shows average fuel prices in a city over a week.
(a) Which was the best day to buy fuel and what was the average cost on this day?
Think: Which day had the lowest(cheapest) average cost and what was the cost?
Do: The lowest point on the graph occurs on Wednesday and the average cost that day $\$1.38$$1.38.
(b) On which day(s) was the average fuel price $\$1.40$$1.40?
Think: Trace a line across at a price of $\$1.40$$1.40 and see if any day(s) has this average cost.
Do: The average cost is at $\$1.40$$1.40 on Monday and Thursday.
(c) By how much did the price change from Monday to Friday?
Think: Find the difference between the fuel prices on these two days and note whether the price increased or decreased.
Do: The fuel price increased by: $\$1.46-\$1.40=\$0.06$$1.46−$1.40=$0.06.
Practice questions
Question 1
The line graph shows the number of ice creams sold at certain times of the day.
At what time of day were the least amount of ice creams sold?
$6$6pm
A$4$4pm
B$12$12pm
C$10$10am
DWhat were the most amount of ice creams sold at a particular time?
There are two peak hours for ice cream sales, at lunch time ($1$1pm) and in the evening ($6$6pm).
What was the difference in sales between the lunch time peak and the evening peak?
Question 2
The line graph shows the amount of petrol in a car’s tank.
How much petrol was initially in the tank?
$\editable{}$ litres.
What happened at $9$9am and $1$1pm?
The driver filled the tank.
AThe amount of petrol being used increased.
BThe car was travelling at a fast speed.
CHow much petrol was used between $1$1pm and $5$5pm?
To the nearest hour, when did the petrol in the tank first fall below $18$18 litres?
Approximately $\editable{}$$:$:$00$00
Question 3
From the beginning of 2012, the number of new houses built in the suburb of Woodford was recorded and figures are released every four months.
The following table contains the data from the beginning of 2012 to the end of 2015:
|
Time Period |
Houses Built |
| April 2012 | $103$103 |
| August 2012 | $92$92 |
| December 2012 | $105$105 |
| April 2013 | $99$99 |
| August 2013 | $88$88 |
| December 2013 | $104$104 |
| April 2014 | $93$93 |
| August 2014 | $85$85 |
| December 2014 | $103$103 |
| April 2015 | $93$93 |
| August 2015 | $83$83 |
| December 2015 | $96$96 |
Plot the number of new houses built in the suburb of Woodford for every $4$4 months that the data was released.
Sketch segments through each point to represent the overall time series.
What is the overall trend?
Downwards only
AUpwards only
BSeasonal and Upwards
CSeasonal and Downwards
D
Conversion graphs
Conversion graphs are line graphs which are used to convert one unit into another. We can find equivalent values between two different scales by looking at a point on the graph and comparing the values on the horizontal axis the vertical axis. This can include conversions between units of length, conversions between different currencies and conversions between different temperature scales.
We have encountered converting units in length, area, volume, mass as well as converting currency using exchange rates. When doing this we were given a conversion factor. When using a graph many values may be converted by reading points directly from the graphs. For graphs passing through the origin the conversion factor can be found by calculating the gradient of the line.
Worked example
Example 2
The following conversion graph displays the conversion between the units of speed kilometres per hour and miles per hour. Use the graph to answer the following questions.
(a) Find the equivalent speed of $80$80 km/h in miles per hour.
Think: Trace a line up from $80$80 kilometres per hour on the horizontal axis to meet the line and then across to find the equivalent speed on the vertical axis.
Do:
$80$80 km/h is equivalent to approximately $50$50 mi/h.
(b) A highway speed sign in the United States indicates a maximum speed of $70$70 mi/h, what would this speed be equivalent to in kilometres per hour?
Think: Trace a line across from $70$70 miles per hour on the vertical axis to meet the line and then down to find the equivalent speed on the horizontal axis.
Do:
The line is just over the $110$110 km/h mark on the horizontal axis, so $70$70 miles per hour is equivalent to approximately $112$112 km/h.
(c) What is $1$1 mile per hour equivalent to in kilometres per hour? (What is the conversion factor between miles per hour and kilometres per hour?)
Think: This would be very hard to accurately read from the graph, but we can use that $80$80 km/h was equivalent to $50$50 mi/h to get a reasonable approximation.
Do:
From part a) we have $50$50 mi/h $=80$=80 km/h, dividing both sides of the equation by $50$50 we obtain:
$1$1 mi/h $=1.6$=1.6 km/h
This is close to the actual conversion factor of $1.609344$1.609344.
Practice questions
Question 4
Attached is a conversion graph of Celsius to Fahrenheit.
Water freezes at $0^\circ$0°C. What is this temperature in Fahrenheit?
$\editable{}$ °F
Would $80^\circ$80°F be above or below normal body temperature (approximately $37^\circ$37°C)?
Above
ABelow
BIf the temperature increases by $1^\circ$1°C, how many degrees Fahrenheit does it increase by? Give your answer as a decimal.
Complete the rule for conversion between Celsius (C) and Fahrenheit (F):
F $=$= $1.8$1.8C $+$+ $\editable{}$
Finally, convert $35^\circ$35°C into Fahrenheit.
Question 5
The graph shows the amount of Euros that can be bought with Australian Dollars.
How many Euros can $20$20 Australian Dollars buy?
AUD$$20$20 can buy € $\editable{}$
How much Australian currency is required to buy $6$6 Euros?
AUD$ $\editable{}$
How many Euros does $$1$1 Australian buy? Leave your answer to two decimal places.