What is a Direct Variation Graph?

S3-SA5-0270

Grade Level:

Class 9

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

Definition

What is it?

A Direct Variation Graph shows a relationship where two quantities increase or decrease together at a constant rate. When one quantity doubles, the other quantity also doubles. On a graph, this relationship always appears as a straight line passing through the origin (0,0).

Simple Example

Quick Example

Imagine you're buying ladoos. If one ladoo costs 10 rupees, then two ladoos cost 20 rupees, and three ladoos cost 30 rupees. Here, as the number of ladoos increases, the total cost also increases in direct proportion. If you plot 'number of ladoos' on the X-axis and 'total cost' on the Y-axis, you'll get a straight line passing through the origin.

Worked Example

Step-by-Step

QUESTION: The distance a car travels varies directly with the time it travels. If a car travels 60 km in 1 hour, what is the distance covered in 3 hours? Plot this on a graph.

STEP 1: Identify the relationship. Distance (D) varies directly with Time (T). So, D = kT, where k is the constant of variation.
---STEP 2: Use the given information to find 'k'. We know D = 60 km when T = 1 hour. So, 60 = k * 1, which means k = 60.
---STEP 3: Write the direct variation equation. D = 60T.
---STEP 4: Calculate the distance for 3 hours. D = 60 * 3 = 180 km.
---STEP 5: Identify points for plotting. We have (Time, Distance): (0,0) (since 0 hours means 0 km), (1, 60), and (3, 180).
---STEP 6: Plot these points on a graph. Time on the X-axis and Distance on the Y-axis.
---STEP 7: Draw a straight line connecting these points. You will see it passes through the origin.
---ANSWER: The car travels 180 km in 3 hours. The graph is a straight line through the origin (0,0), (1,60), and (3,180).

Why It Matters

Understanding direct variation graphs helps in fields like AI/ML to predict trends, in Physics to understand speed and distance, and in Economics to analyze supply and demand. Engineers use it to design systems where components scale proportionally, and data scientists use it to find simple relationships in data, helping them make better decisions.

Common Mistakes

MISTAKE: Assuming any straight line graph is direct variation. | CORRECTION: A direct variation graph MUST be a straight line AND pass through the origin (0,0). If it's a straight line but doesn't pass through (0,0), it's a linear relationship, but not direct variation.

MISTAKE: Confusing direct variation with inverse variation. | CORRECTION: In direct variation, if one quantity increases, the other increases. In inverse variation, if one quantity increases, the other decreases. The graphs look very different: direct variation is a straight line, inverse variation is a curve (hyperbola).

MISTAKE: Forgetting that the constant of variation (k) must be constant. | CORRECTION: Always check if the ratio Y/X is the same for all given pairs of values. If Y/X changes, it's not a direct variation.

Practice Questions

Try It Yourself

QUESTION: The cost of electricity (C) is directly proportional to the units consumed (U). If 10 units cost 50 rupees, how much will 25 units cost? | ANSWER: 125 rupees

QUESTION: A painter can paint 2 walls in 4 hours. If the number of walls painted (W) varies directly with the hours worked (H), how many walls can the painter paint in 10 hours? | ANSWER: 5 walls

QUESTION: The amount of rice (R) needed is directly proportional to the number of guests (G). If 5 kg of rice is needed for 20 guests, how much rice is needed for 30 guests? If you plot this, what would be the coordinates for 0 guests? | ANSWER: 7.5 kg of rice; (0,0)

MCQ

Quick Quiz

Which of the following graphs represents a direct variation?

A straight line passing through the point (2, 5) and (4, 10).

A straight line passing through the point (0, 3) and (1, 5).

A curved line passing through the origin.

A straight line passing through the origin (0,0).

The Correct Answer Is:

A direct variation graph is always a straight line that passes through the origin (0,0). Option A describes a direct variation because the ratio Y/X is constant (5/2 = 2.5, 10/4 = 2.5) and if extended, it would pass through the origin. However, the question asks for a graph, and D directly states 'straight line passing through the origin'. Option B is a straight line but doesn't pass through the origin. Option C is curved, not a direct variation. (Correction for A: A is also a direct variation, but D is the most fundamental definition of the graph itself.) Let's re-evaluate based on the question asking about the graph itself. The most defining characteristic of the graph is that it's a straight line through the origin.

Real World Connection

In the Real World

In India, think about your mobile data usage. The more data you use (GB), the faster your data pack gets consumed, often leading to higher costs if you exceed your limit. This is a direct variation. Also, in farming, the amount of fertilizer needed is directly proportional to the area of land, helping farmers calculate exact needs.

Key Vocabulary

Key Terms

Direct Variation: A relationship where two quantities change in the same direction at a constant rate. | Origin: The point (0,0) on a coordinate plane. | Constant of Variation (k): The constant ratio between two quantities in a direct variation (Y/X = k). | Linear Relationship: A relationship that, when graphed, forms a straight line.

What's Next

What to Learn Next

Great job understanding direct variation graphs! Next, you should explore 'Inverse Variation Graphs'. This will help you understand relationships where quantities change in opposite directions, which is another fundamental concept in mathematics and science.