What is a Step Graph?

S3-SA5-0092

Grade Level:

Class 9

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

A step graph is a special type of graph that looks like a series of steps. It is used to show how a value changes suddenly at certain points, but stays constant in between those points. Think of it like walking up a staircase, where each step is flat for a while before you go up to the next level.

Simple Example

Quick Example

Imagine the cost of sending a parcel at the post office. If a parcel weighs up to 1 kg, it costs Rs 50. If it weighs more than 1 kg but up to 2 kg, it costs Rs 80. And if it weighs more than 2 kg but up to 3 kg, it costs Rs 110. The cost jumps up only when the weight crosses a certain limit, staying the same for all weights within a specific range.

Worked Example

Step-by-Step

Let's plot a step graph for mobile data charges. Suppose a telecom company charges Rs 100 for up to 1 GB of data. For more than 1 GB up to 2 GB, it's Rs 150. For more than 2 GB up to 3 GB, it's Rs 200.

Step 1: Identify the x-axis (data used in GB) and y-axis (cost in Rs).
---Step 2: For 0 GB to 1 GB (not including 1 GB), the cost is Rs 100. We draw a horizontal line from (0, 100) to (1, 100) with an open circle at (1, 100) to show it doesn't include 1 GB.
---Step 3: At exactly 1 GB, the cost is Rs 100. We place a closed circle at (1, 100) on the first step.
---Step 4: For more than 1 GB (but not including 2 GB), the cost is Rs 150. We draw a horizontal line from (1, 150) with an open circle at (1, 150) to (2, 150) with an open circle at (2, 150).
---Step 5: At exactly 2 GB, the cost is Rs 150. We place a closed circle at (2, 150) on the second step.
---Step 6: For more than 2 GB (but not including 3 GB), the cost is Rs 200. We draw a horizontal line from (2, 200) with an open circle at (2, 200) to (3, 200) with an open circle at (3, 200).
---Step 7: At exactly 3 GB, the cost is Rs 200. We place a closed circle at (3, 200) on the third step.

This creates a graph with horizontal 'steps' that jump up at 1 GB and 2 GB.

Why It Matters

Step graphs are super useful in fields like Computer Science and Data Science for showing how things change in 'chunks' rather than smoothly. They are used in economics to model taxes or tariffs, and even in physics to represent certain types of signals. Understanding them can open doors to careers in software development, financial analysis, or engineering.

Common Mistakes

MISTAKE: Connecting the horizontal steps with vertical lines. | CORRECTION: Step graphs should only have horizontal segments and 'jumps' (represented by open and closed circles), not continuous vertical lines between steps.

MISTAKE: Confusing open circles and closed circles, or using them incorrectly. | CORRECTION: A closed circle (•) means the point IS included in that step, while an open circle (o) means the point IS NOT included. Pay close attention to whether the interval is inclusive or exclusive.

MISTAKE: Assuming the function is continuous. | CORRECTION: A step graph represents a discontinuous function, meaning there are sudden jumps in value. The value doesn't smoothly transition between steps.

Practice Questions

Try It Yourself

QUESTION: A taxi charges Rs 50 for the first 2 km, and then Rs 20 for every additional kilometer (or part thereof). What would be the charge for a 3.5 km ride? | ANSWER: Rs 90 (Rs 50 for first 2 km + Rs 20 for 3rd km + Rs 20 for 4th km (part thereof))

QUESTION: Draw a step graph for the following scenario: A library charges a late fee of Rs 5 per day for the first 3 days, and then Rs 10 per day for every day after that. Assume the fee is for a full day. Plot for up to 5 days. | ANSWER: (0,0) to (1,5) (open at 0), (1,5) to (2,10), (2,10) to (3,15), (3,15) to (4,25), (4,25) to (5,35). Each step would have a closed circle at the right end and an open circle at the left end, except for day 0 where it starts at (0,0).

QUESTION: A courier company charges Rs 100 for parcels up to 500g, Rs 150 for parcels over 500g up to 1kg, and Rs 200 for parcels over 1kg up to 2kg. If a customer sends two parcels, one weighing 400g and another weighing 1.2kg, what is the total cost? | ANSWER: Rs 300 (Rs 100 for 400g parcel + Rs 200 for 1.2kg parcel)

MCQ

Quick Quiz

Which of the following scenarios is best represented by a step graph?

The height of a plant growing over several weeks.

The temperature of water heating up on a stove.

The cost of movie tickets based on age groups (e.g., child, adult, senior).

The distance covered by a car travelling at a constant speed.

The Correct Answer Is:

Option C represents discrete categories (age groups) where the price changes suddenly at certain age cut-offs, remaining constant within each group. The other options describe continuous changes, which are better shown by smooth lines.

Real World Connection

In the Real World

You see step graphs in real life when checking your electricity bill! Many electricity companies in India use a 'slab' system. For example, the first 100 units might cost Rs 3 per unit, the next 100 units Rs 5 per unit, and so on. The total cost jumps up when you cross into a new consumption slab, creating a step-like pattern in the pricing structure.

Key Vocabulary

Key Terms

DISCONTINUOUS: Not continuous; having breaks or jumps. | INTERVAL: A set of numbers between two given numbers. | CLOSED CIRCLE: A point on a graph that is included in the interval. | OPEN CIRCLE: A point on a graph that is NOT included in the interval. | SLAB SYSTEM: A pricing method where rates change at specific thresholds.

What's Next

What to Learn Next

Great job understanding step graphs! Next, you can explore piecewise functions, which are mathematical functions defined by multiple sub-functions, each applying to a different interval. This will help you write the equations for the 'steps' you've learned to graph.