What is a Graph of a Linear Relationship?

S1-SA5-0999

Grade Level:

Class 5

Maths, Computing, AI, Physics, Economics, Data Science

Definition

What is it?

A graph of a linear relationship is a straight line drawn on a special grid (called a coordinate plane) that shows how two things are connected. It helps us see clearly how one quantity changes exactly with another, like how the cost of something increases with the quantity you buy.

Simple Example

Quick Example

Imagine you buy samosas for Rs. 10 each. If you buy 1 samosa, you pay Rs. 10. If you buy 2 samosas, you pay Rs. 20. If you buy 3, you pay Rs. 30. If you plot these points (1 samosa, Rs. 10), (2 samosas, Rs. 20), (3 samosas, Rs. 30) on a graph, they will all lie on a straight line. This straight line is the graph of the linear relationship between the number of samosas and the total cost.

Worked Example

Step-by-Step

Let's say an auto-rickshaw charges a fixed fare of Rs. 20 plus Rs. 5 for every kilometer traveled. We want to graph the total cost for different distances.
---Step 1: Understand the relationship. The cost (C) depends on the distance (D). C = 20 + 5 * D.
---Step 2: Find some points. If D = 0 km, C = 20 + 5 * 0 = Rs. 20. So, point is (0, 20).
---Step 3: If D = 1 km, C = 20 + 5 * 1 = Rs. 25. So, point is (1, 25).
---Step 4: If D = 2 km, C = 20 + 5 * 2 = Rs. 30. So, point is (2, 30).
---Step 5: If D = 3 km, C = 20 + 5 * 3 = Rs. 35. So, point is (3, 35).
---Step 6: Draw a grid (coordinate plane). Label the horizontal axis 'Distance (km)' and the vertical axis 'Cost (Rs)'.
---Step 7: Plot the points (0, 20), (1, 25), (2, 30), (3, 35) on your grid.
---Step 8: Connect these points with a straight line. This straight line is the graph of the linear relationship between distance and cost.
ANSWER: The graph is a straight line passing through points like (0, 20), (1, 25), (2, 30), (3, 35).

Why It Matters

Understanding linear graphs helps scientists predict outcomes, engineers design structures, and economists understand market trends. It's crucial for careers in data science, where you analyze information, and even in AI, where computers learn from patterns, often represented as lines. Doctors use it to track patient health over time.

Common Mistakes

MISTAKE: Not labeling the axes of the graph or choosing incorrect scales. | CORRECTION: Always label the horizontal (x-axis) and vertical (y-axis) with what they represent (e.g., 'Time (minutes)', 'Distance (meters)') and include units. Choose a scale that fits all your points clearly.

MISTAKE: Connecting points with a curvy line instead of a straight line when the relationship is linear. | CORRECTION: If the relationship is linear (meaning one quantity changes constantly with another), the graph MUST be a straight line. Use a ruler to draw a neat straight line through the plotted points.

MISTAKE: Plotting points incorrectly, mixing up x and y coordinates. | CORRECTION: Remember that points are always written as (x, y). The first number (x) tells you how far to go horizontally, and the second number (y) tells you how far to go vertically.

Practice Questions

Try It Yourself

QUESTION: A mobile data pack costs Rs. 50 for 1 GB, Rs. 100 for 2 GB, and Rs. 150 for 3 GB. What kind of line would you get if you plot 'Data (GB)' on the x-axis and 'Cost (Rs.)' on the y-axis? | ANSWER: A straight line.

QUESTION: Plot the points (1, 5), (2, 7), (3, 9) on a graph. What is the y-value when x is 4, assuming it's a linear relationship? | ANSWER: When x is 4, y is 11. (Each time x increases by 1, y increases by 2).

QUESTION: The temperature in a city decreases by 2 degrees Celsius every hour after 6 PM. At 6 PM, the temperature is 20 degrees Celsius. Plot the temperature for 6 PM, 7 PM, 8 PM, and 9 PM. What will be the temperature at 10 PM if this trend continues? | ANSWER: At 6 PM (hour 0), Temp = 20. At 7 PM (hour 1), Temp = 18. At 8 PM (hour 2), Temp = 16. At 9 PM (hour 3), Temp = 14. At 10 PM (hour 4), Temp = 12 degrees Celsius.

MCQ

Quick Quiz

Which of these describes a graph of a linear relationship?

A curved line that goes up and down

A straight line

A collection of random dots

A line that forms a circle

The Correct Answer Is:

A linear relationship means one quantity changes consistently with another, and when plotted on a graph, this always forms a straight line. Curved lines, random dots, or circles do not represent linear relationships.

Real World Connection

In the Real World

Many apps we use daily rely on linear relationships! For example, when you book an OLA or Uber cab, the fare calculation often involves a base fare plus a charge per kilometer – this is a linear relationship. The app uses this to show you an estimated cost. Even in cricket, if a batsman scores runs at a constant rate, plotting 'overs' vs 'runs' would give you a linear graph.

Key Vocabulary

Key Terms

COORDINATE PLANE: A flat surface with two number lines (axes) used to plot points | X-AXIS: The horizontal number line on a coordinate plane | Y-AXIS: The vertical number line on a coordinate plane | PLOT: To mark a point on a graph using its coordinates | LINEAR RELATIONSHIP: A relationship between two quantities where one changes by a constant amount for every unit change in the other.

What's Next

What to Learn Next

Great job understanding linear graphs! Next, you can explore 'Slope of a Line' and 'Equations of a Line'. These concepts will teach you how to measure the steepness of these straight lines and write mathematical rules for them, which is super useful for solving more complex problems.