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Step graphs


Stepwise Linear Functions

A stepwise linear function is a type of function that applies a different rule depending on the input variable. Because of this, the graph of a stepwise linear function doesn't change gradually but has distinct steps  (called a 'step graph'). The ends of each step in the graph need to indicate whether that value is included in that particular step, or not included, as indicated in the graph below.

Stepwise linear function

Stepwise linear function

Stepwise linear functions feature in many practical contexts, particularly when looking at charges for certain things, like car park rates, postage charges, time constrained charges, and bulk purchases.

The example below goes through constructing a stepwise linear function for parking charges at an airport car park.


Secure parking at an airport charges a flat rate of $\$14$$14 for the first hour and then $\$8$$8for every hour after that. Construct the graph that represents this relationship for up to 5 hours.

Part 1

For the first hour, the charge is $\$14$$14. However, as soon as the car has been parked for a minute over 1 hour, the rate switches to $\$14$$14 plus the next hour charge of $\$8$$8. Therefore we will need to indicate a 'not included' end to this first portion of the graph.


Following this, the charge increases by $\$8$$8 for each subsequent hour. This means for the second step, the charge will be $\$22$$22. Note again that at the end of the second hour, the charge increases by another $\$8$$8 so the end of the second step is not included.

Part 3

Continue this pattern, increasing the steps by $\$8$$8 for each hour interval, ensuring that the ends of each step are 'not included' in that charge interval.

Completed stepwise linear function- Car Park charges

Completed stepwise linear function- Car Park charges

Worked Examples

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...

  1. How much does a call that lasts $4$4 minutes and $5$5 seconds cost?

  2. How much does a $3$3-minute call cost?

  3. What is the longest possible call that could be made for $\$1.50$$1.50?

  4. What is the cost of each additional minute?

question 2

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$$112.17   $18$18$<$<$x\le25$x25
$\$128.72$$128.72   $25$25$<$<$x\le35$x35
$\$151.73$$151.73   $35$35$<$<$x\le45$x45
$\$176.27$$176.27   $45$45$<$<$x\le55$x55
$\$184.52$$184.52   $55$55$<$<$x\le65$x65
  1. How much will be the monthly premium for a $22$22 year old person?

  2. What could be the maximum age of someone who is paying $\$176.27$$176.27 per month for their premium?

  3. 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?

  4. If you have just turned $35$35 years old, how much will your annual premium increase by the following year?

question 3

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.

  1. 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$3x<6
    $\editable{}$$^\circ$°C   $6\le x\le10$6x10
  2. 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...


  3. Lakota entered the ski facility at $3:30$3:30 pm. What was the temperature inside the facility at this time?

  4. Xavier wants to wait till the indoor temperature is $-7$7$^\circ$°C or lower. When is the earliest he can enter the facility?

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