NCERT Solutions for Class 8 Maths Chapter 15 Exercise 15.3: Introduction to Graphs

Understanding Independent and Dependent Variables

Before we start drawing lines on graph paper, we must understand what we are plotting. In any real-world relationship, we deal with two types of variables:

1. Independent Variable: This is the quantity that changes freely or is chosen by us. For example, the time you study, the quantity of petrol you buy, or the side length of a square. We always plot the independent variable on the horizontal X-axis.

1. Dependent Variable: This variable changes as a direct result of the independent variable. For example, the marks you score depend on study hours, the total cost depends on the litres of petrol bought, and the perimeter depends on the side length of the square. We plot the dependent variable on the vertical Y-axis.

When we plot these paired values as coordinates $(x, y)$ and connect them, we get a visual representation of their relationship. If the resulting line is straight, we call it a Linear Graph. If it curves, it represents a non-linear relationship.

Step-by-Step Process to Draw a Graph

1. Identify the Axes and Variables — Determine which variable is independent (assign to X-axis) and which is dependent (assign to Y-axis).
2. Choose a Suitable Scale — Look at the maximum and minimum values in your data table. Choose a scale (e.g., 1 unit = ₹50, or 1 unit = 2 cm) that allows the graph to fit cleanly on your graph sheet without being too tiny or overflowing.
3. Plot the Points — Write down the data pairs as coordinates $(x, y)$. Locate each point on the grid by moving along the X-axis first, then vertically along the Y-axis, and mark it with a small dot or cross.
4. Connect the Points — Use a ruler to join the points. Check if they form a single straight line passing through the origin $(0,0)$.

CBSE Board Tips: Avoiding Common Graphing Mistakes

• Don't forget the scale: CBSE marking schemes assign specific marks just for writing down the scale on the top-right corner of your graph sheet (e.g., On X-axis: 1 unit = 1 cm).
• Label both axes clearly: Always write the variable name and units (like 'Time in hours' or 'Distance in km') on the respective axes. An unlabeled graph can cost you 1 to 1.5 marks.
• Verify if it passes through the origin: If you buy 0 litres of petrol, the cost is ₹0. Thus, graphs representing direct variation should pass through the origin $(0,0)$.

Practice Questions with Solutions

• Q: Draw a graph for the following table representing the side of a square and its perimeter: Side of Square (in cm): 2, 3, 3.5, 5, 6 Perimeter (in cm): 8, 12, 14, 20, 24 Is it a linear graph? A: Step 1: Identify variables. The side of the square is the independent variable (X-axis), and the perimeter is the dependent variable (Y-axis). Step 2: Choose a scale. On X-axis: 1 unit = 1 cm. On Y-axis: 1 unit = 4 cm. Step 3: Write points as coordinates: (2, 8), (3, 12), (3.5, 14), (5, 20), and (6, 24). Plot these on the graph paper. Step 4: Join all the points with a ruler. Final answer: Since all points lie on a single straight line, yes, it is a linear graph.
• Q: Plot a graph for the number of litres of petrol and its cost from the table below: Litres of petrol: 10, 15, 20, 25 Cost of petrol (₹): 500, 750, 1000, 1250 Use the graph to find the cost of 12 litres of petrol. A: Step 1: Let Litres of petrol be on X-axis and Cost (₹) on Y-axis. Step 2: Choose a scale. X-axis: 1 unit = 5 litres. Y-axis: 1 unit = ₹250. Step 3: Plot the coordinates: (10, 500), (15, 750), (20, 1000), (25, 1250). Join them to form a straight line passing through the origin. Step 4: To find the cost of 12 litres, locate 12 on the X-axis (between 10 and 15), draw a vertical line up to meet the graph line, then read the corresponding value on the Y-axis. Final answer: From the graph, the cost of 12 litres of petrol is ₹600.
• Q: Draw a graph for the side of a square and its area: Side of Square (in cm): 2, 3, 4, 5, 6 Area (in sq. cm): 4, 9, 16, 25, 36 Is this a linear graph? A: Step 1: Let Side of Square be on X-axis, and Area on Y-axis. Step 2: Choose scale. X-axis: 1 unit = 1 cm. Y-axis: 1 unit = 5 sq. cm. Step 3: Plot coordinates: (2, 4), (3, 9), (4, 16), (5, 25), (6, 36). Step 4: Connect the points. When we join these points, we see that they do not form a straight line but rather a curve. Final answer: No, this is not a linear graph because the area does not increase at a constant rate relative to the side length.
• Q: A car travels at a uniform speed. Plot a distance-time graph using this data: Time (in hours): 1, 2, 3, 4 Distance (in km): 60, 120, 180, 240 Find the time taken by the car to cover a distance of 150 km. A: Step 1: Plot Time on X-axis and Distance on Y-axis. Step 2: Scale: X-axis: 1 unit = 1 hour. Y-axis: 1 unit = 60 km. Step 3: Plot coordinates (1, 60), (2, 120), (3, 180), (4, 240) and draw a straight line through them. Step 4: To find time for 150 km, look at 150 on the Y-axis (midway between 120 and 180). Trace horizontally to the graph line, then move vertically down to read the X-axis value. Final answer: The time taken to cover 150 km is 2.5 hours.

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