Practice Maths

Step Graphs

← Back to Applications of Linear Equations and Their Graphs›General Maths

Key Terms

A step graph (step function) shows a value that is constant within an interval, then jumps to a new constant value at the boundary. There is no gradual change — only sudden steps.
Each horizontal segment represents one “step” or category.
Closed circle (•)
: the endpoint IS included in that step (the value applies at that exact point).
Open circle (ˆ)
: the endpoint is NOT included (the value does not apply at that exact boundary).
At every boundary, exactly ONE side must have a closed circle and the other an open circle — the value belongs to one step only.
Real-world contexts
parking fees (charged per commenced hour), postage (charged by weight category), admission by age group, public transport fare zones, tax brackets, shipping rates.
Reading a Step Graph

• Find your x-value on the horizontal axis.
• Move vertically up until you reach the horizontal segment that includes your x-value.
• Read the y-value (height) of that segment.
• At a boundary: the closed dot (•) tells you which segment includes that exact value.

Writing a step function as a table:

IntervalValueEndpoints
0 < t ≤ 1$3Open at 0, closed at 1
1 < t ≤ 2$6Open at 1, closed at 2
2 < t ≤ 3$9Open at 2, closed at 3

Parking fee step graph: $3 per commenced hour (charged per hour or part thereof)

hrs C ($) 0 1 2 3 4 $3 $6 $9 $12 $3 $6 $9 $12 closed (included) open (excluded)
Hot Tip Boundary values are the trickiest part of step graphs. At t = 1 in the parking example, the closed dot is on the step at $3 (not $6). This means someone who parks for exactly 1 hour pays $3. The next step ($6) only starts for time strictly greater than 1 hour. Always trace carefully along the correct horizontal segment.

Worked Example — Postage rates (step function)

Question: Postage rates are: 0–100 g: $2.50; 101–250 g: $4.00; 251–500 g: $6.50. Find the postage for: (a) 150 g; (b) exactly 100 g; (c) 260 g.

Identify which interval each weight falls in:

(a) 150 g: 101 ≤ 150 ≤ 250, so use the $4.00 step. Postage = $4.00

(b) 100 g: 0 ≤ 100 ≤ 100, so use the $2.50 step. Postage = $2.50
(The boundary 100 g belongs to the first step — the closed dot is at 100 g on the $2.50 line.)

(c) 260 g: 251 ≤ 260 ≤ 500, so use the $6.50 step. Postage = $6.50

Key insight: Step functions charge the same rate for everything within a range. A letter of 101 g costs the same as a letter of 249 g ($4.00 each). But a letter of exactly 100 g costs only $2.50.

A step function is not smooth or continuous — it jumps. The cost or value stays the same for a range of inputs, then suddenly changes to a new constant level at the boundary. This is fundamentally different from a piecewise linear graph (which has slope within each section).

How step functions differ from piecewise linear functions

Feature Piecewise Linear Step Function
ShapeDiagonal line segmentsHorizontal flat segments
GradientNon-zero within sectionsZero within each step
At boundaryMay be continuousAlways jumps (discontinuous)
ExampleTaxi fare increasing with kmParking charged per hour

Reading and using step functions

  1. Find the interval that contains your input value x.
  2. If x is at a boundary, check the open/closed circle to determine which interval it belongs to.
  3. The output is simply the constant value for that interval.
Example: Transport zone fares: Zone 1 (0–5 km): $3.20; Zone 2 (5–15 km): $5.80; Zone 3 (>15 km): $8.40. Find the fare for 5 km and for 16 km.

5 km: if the boundary at 5 km belongs to Zone 1 (closed at 5), fare = $3.20.
16 km: > 15, fare = $8.40.
Tip: In real-world problems, always read the boundary conditions carefully from the context. “First hour or part thereof” means the boundary goes to the higher rate: even 61 minutes counts as 2 hours. This means the boundary at 1 hour has a closed dot on the $3 step, and the $6 step starts with an open dot just after 1 hour.

Writing a step function from a description

List each category as an interval with its associated constant value. Be precise about whether boundary values go to the lower or upper category. Use the notation: f(x) = c&sub1; for a ≤ x ≤ b, f(x) = c&sub2; for b < x ≤ c, etc.

Tip: Tax brackets in Australia are a classic example of a step function. Each bracket has a fixed formula (flat rate on the amount in that bracket). The total tax combines the steps: fixed tax on lower brackets + marginal rate on the amount in the current bracket.

Mastery Practice

See Answers ➔
  1. Fluency

    A car park charges using the step graph shown in the Key Ideas above: $3 for any time up to 1 hour, $6 for over 1 up to 2 hours, $9 for over 2 up to 3 hours, $12 for over 3 up to 4 hours. Find the parking cost for each of the following durations.

    1. (a) 45 minutes
    2. (b) Exactly 2 hours
    3. (c) 2 hours 15 minutes
    4. (d) 3 hours 58 minutes
  2. Fluency

    A post office uses the following postage rates based on parcel weight. Find the postage cost for each parcel.

    WeightPostage
    0–50 g$1.20
    51–100 g$2.40
    101–200 g$3.80
    201–500 g$5.50
    1. (a) A birthday card weighing 45 g
    2. (b) A brochure weighing exactly 100 g
    3. (c) A small book weighing 145 g
    4. (d) A clothing item weighing 380 g
  3. Fluency

    Using the postage rate table from Question 2, write the step function as a formal set of intervals using correct notation (include open and closed endpoints).

  4. Understanding

    A cinema charges the following ticket prices based on age category.

    Age groupCategoryPrice
    Under 12Child$15
    12–17Youth$22
    18–64Adult$38
    65 and overSenior$28

    A family buys tickets: 2 adults (ages 38 and 42), 1 senior (age 70), 1 child (age 9), and 1 youth (age 16).

    1. (a) State the ticket price for each person.
    2. (b) Find the total cost for the family.
    3. (c) The cinema offers a 10% group discount for groups of 5 or more. Find the final cost after the discount.
  5. Understanding

    A family’s monthly internet data usage and charges are shown below.

    Monthly dataMonthly cost
    0–10 GB$35
    10–30 GB$55
    30–60 GB$75
    Over 60 GB$95

    The family uses 8 GB, 22 GB, 45 GB, and 65 GB in four consecutive months.

    1. (a) Find the cost for each month.
    2. (b) Find the average monthly cost over the four months.
    3. (c) A competing provider offers a flat $70/month for unlimited data. Would switching to the flat plan save money over these four months?
  6. Understanding

    Bus fare zones are determined by journey distance as shown below.

    ZoneDistanceFare
    Zone 10–3 km$3.20
    Zone 23–10 km$5.50
    Zone 310–20 km$7.80
    Zone 4Over 20 km$10.60
    1. (a) Find the single-trip fare for journeys of: (i) 2 km; (ii) 5 km; (iii) 15 km; (iv) 28 km.
    2. (b) A student travels 4 km each way to school, 5 days per week for 4 weeks. Find their total monthly transport cost.
    3. (c) A monthly bus pass costs $90. Would buying the monthly pass save the student money compared to paying per trip?
  7. Understanding

    A movie theatre uses the following age-based ticket prices: under 3 years: free; 3–12: $14; 13–17: $18; 18–60: $24; 61 and over: $20.

    1. (a) Construct a table showing each age category, the ticket price, and correct interval notation (using ≤ and <).
    2. (b) Sketch a step graph for this pricing structure. Clearly label both axes and show open/closed circles correctly at each boundary.
    3. (c) A group has ages: 2, 7, 14, 25, 42, 65. Find the total cost for the group.
    4. (d) If the group qualifies for a bulk concession at 15% off when there are 6+ people, find the final price.
  8. Understanding

    Calculate the daily earnings for a factory worker paid at the following rates: standard rate $28/hr for first 8 hours; overtime weekday rate $42/hr for hours worked beyond 8 on a weekday; weekend rate $56/hr for all hours worked on a Saturday or Sunday; night shift rate $35/hr for hours worked between 10 pm and 6 am.

    1. (a) The worker does a regular 6-hour weekday shift.
    2. (b) The worker does a 10-hour weekday shift.
    3. (c) The worker does an 8-hour Saturday shift.
    4. (d) The worker starts a weekday shift at 9 pm and works for 5 hours (2 hours at standard time before 10 pm, then 3 hours on night shift).
  9. Problem Solving

    The simplified Australian income tax brackets (based on ATO rates) are as follows for a financial year.

    Taxable incomeTax payable
    $0 – $18 200Nil
    $18 201 – $45 00019c for each $1 over $18 200
    $45 001 – $120 000$5 092 + 32.5c for each $1 over $45 000
    $120 001 – $180 000$29 467 + 37c for each $1 over $120 000

    Calculate the income tax payable on each of the following annual incomes.

    1. (a) $15 000
    2. (b) $32 000
    3. (c) $85 000
    4. (d) $140 000
  10. Problem Solving

    A shipping company charges based on the weight of each parcel, rounded up to the nearest 500 g. The rates are: 500 g: $8.00; 1.0 kg: $12.00; 1.5 kg: $15.00; 2.0 kg: $18.00; 2.5 kg: $21.00; 3.0 kg and above: $21.00 + $3.00 per additional 500 g (or part thereof).

    1. (a) Find the cost to ship a parcel weighing 800 g.
    2. (b) Find the cost to ship a parcel weighing exactly 2 000 g.
    3. (c) Find the cost to ship a parcel weighing 4.3 kg.
    4. (d) What is the maximum weight that can be shipped for $30?