18.05 Line graphs

There are many different types of graphs that can be used to display different types of data. In this lesson, we will focus on three common types: line graphs, conversion graphs and step graphs.

Common features of each of these graphs include:

• a clear title
• both axes should be labelled with the variable and units where appropriate
• clear scale on each axis

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

Question 2

Question 3

 

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

Question 5

Step graphs

A step graph displays that a different rule applies depending on the input variable. Because of this, the graph doesn't change gradually but has distinct steps. This kind of graph is common for fee structures, like the tariff structures we encountered in household bills, where a certain amount was charged for the first block of energy and then a different amount was charged for the next block and so forth. 

The ends of each step in the graph indicate whether that value is included or not:

• Closed circles are used to represent that the value is included in the interval
• Open circles are used to represent that the value is not included in the interval

Such as:

Representing the following energy tariff structure:

Energy used, $x$x (MJ)	Price per MJ (cents)
First $30\ MJ$30 MJ ($0\le x\le30)$0≤x≤30)	$2.2$2.2
Next $20\ MJ$20 MJ ($3030<x≤50)	$1.8$1.8
Next $20\ MJ$20 MJ ($5050<x≤70)	$1.4$1.4
Any remaining ($x\ge70$x≥70)	$1.2$1.2

 

Worked example

Example 3

The Melbourne Airport terminal car park charges the following fees for the first hour of parking.

(a) If a car is in the car park for $20$20 minutes, how much will they be charged?

Think: Run a line up from $20$20 minutes on the horizontal axis, which bar does this hit?

Do:

We can see the line hits the second fee bar and hence, the charge will be $\$8$$8.

(b) If a car is in the car park for $30$30 minutes, how much will they be charged?

Think: Run a line up from $30$30 minutes on the horizontal axis, which bar does this hit?

Do:

 

We can see the line touches the second bar but this circle is open so not included, if we continue the line up it hits the closed circle on the third bar, hence the charge will be $\$12$$12.

(c) If $\$24$$24 is charged for parking from $1$1 hour up to $3$3 hours and $\$34$$34 is charged for parking from $3$3 hours up to $4$4 hours, create a graph that displays charges for up to $4$4 hours of parking.

Think: We need to include the information on the first graph and add two new horizontal bars. The horizontal scale needs to now go up to $4$4 hours ($240$240 minutes) and the vertical scale needs to increase to include a fee of $\$34$$34.  

Do: The two new bars will be from $60$60 minutes to $180$180 minutes at $\$24$$24 and from $180$180 minutes to $240$240 minutes at $\$34$$34 - remember to note whether the ends of the interval are included or not. In this case we will draw a closed circle at the start of the interval and an open circle at the end of the interval.

 

Practice questions

Question 6

Question 7