A step graph is a type of function that takes constant values over intervals on the $x$x-axis. 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 in household bills, where a certain amount is charged for the first block of energy and then a different amount is charged for the next block and so forth.

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

Filled circles are used to represent that the value is included in the interval
Hollow circles are used to represent that the value is not included in the interval

Here is an example of a step graph:

It represents the following energy tariff structure:

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

Worked example

The Perth 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 filled 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 filled circle at the start of the interval and a hollow circle at the end of the interval.

Practice questions

Question 1

The graph shows the cost (in $dollars$ ) of a mobile phone call as a function of the length of the call.

Loading Graph...

How much does a call that lasts $4$4 minutes and $5$5 seconds cost?
How much does a $3$3-minute call cost?
What is the longest possible call that could be made for $\$1.50$$1.50?
What is the cost of each additional minute?

Question 2

At an indoor ski facility, the temperature is set to $-5$−5$^\circ$°C at $2$2 pm. At $3$3 pm, the temperature is immediately brought down to $-12$−12$^\circ$°C and left for $3$3 hours before immediately taking it down again to $-18$−18$^\circ$°C, where it stays for the rest of the day’s operation.

The facility operates until $10$10 pm.

Fill in the gaps to complete the stepwise function that models the indoor temperature, $y$y, at a certain time of the day, $x$x hours after midday.

$y$y $=$= $-5$−5$^\circ$°C $\editable{}$$\le x<3$≤x<3

$\editable{}$$^\circ$°C $3\le x<6$3≤x<6

$\editable{}$$^\circ$°C $6\le x\le10$6≤x≤10
By moving the endpoints of the intervals, create a graph of the step function relating time of day and temperature inside the ski facility.

Clicking the centre of each endpoint will change it from closed to open (or from open to closed).

Loading Graph...
$Lakota$ entered the ski facility at $3:30$3:30 pm. What was the temperature inside the facility at this time?
$Xavier$ wants to wait till the indoor temperature is $-7$−7$^\circ$°C or lower. When is the earliest he can enter the facility?

Question 3

An insurance company charges a fixed monthly premium depending on which age group you fall into.

The function outlines the premium, $y$y, for a given age, $x$x (take age to be what you turn on your birthday).

$y$`y`	$=$=	$\$112.17$$112.17	$18$18$<$<$x\le25$`x`≤25
		$\$128.72$$128.72	$25$25$<$<$x\le35$`x`≤35
		$\$151.73$$151.73	$35$35$<$<$x\le45$`x`≤45
		$\$176.27$$176.27	$45$45$<$<$x\le55$`x`≤55
		$\$184.52$$184.52	$55$55$<$<$x\le65$`x`≤65

How much will be the monthly premium for a $22$22 year old person?
What could be the maximum age of someone who is paying $\$176.27$$176.27 per month for their premium?
$Fikile$ held an insurance policy from when she turned $26$26 to when she turned $30$30. During this time, how much did she pay altogether in premiums?
If you have just turned $35$35 years old, how much will your annual premium increase by the following year?

Outcomes

ACMGM046

sketch piece-wise linear graphs and step graphs, using technology when appropriate

ACMGM047

interpret piece-wise linear and step graphs used to model practical situations; for example, the tax paid as income increases, the change in the level of water in a tank over time when water is drawn off at different intervals and for different periods of time, the charging scheme for sending parcels of different weights through the post

9.06 Step graphs

Step graphs