What is a Graph of y = kx?

S3-SA5-0258

Grade Level:

Class 9

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

Definition

What is it?

A graph of y = kx is a straight line that always passes through the origin (0,0) on a coordinate plane. It shows a direct relationship where one quantity (y) is directly proportional to another quantity (x), with 'k' being the constant of proportionality.

Simple Example

Quick Example

Imagine you buy samosas, and each samosa costs Rs. 10. If 'y' is the total cost and 'x' is the number of samosas, then y = 10x. If you buy 1 samosa (x=1), y=10. If you buy 2 samosas (x=2), y=20. Plotting these points (1,10), (2,20), etc., on a graph will give you a straight line passing through the origin.

Worked Example

Step-by-Step

Let's draw the graph of y = 2x.

STEP 1: Create a table of values for x and y. Choose a few simple values for x, including 0.

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STEP 2: When x = 0, y = 2 * 0 = 0. So, the first point is (0,0).

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STEP 3: When x = 1, y = 2 * 1 = 2. So, the second point is (1,2).

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STEP 4: When x = 2, y = 2 * 2 = 4. So, the third point is (2,4).

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STEP 5: Plot these points (0,0), (1,2), (2,4) on a graph paper.

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STEP 6: Draw a straight line passing through all these plotted points. Extend the line in both directions.

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ANSWER: The line you drew is the graph of y = 2x. It's a straight line passing through the origin.

Why It Matters

Understanding y = kx is fundamental because it describes direct relationships seen everywhere in science and technology. From predicting how much fuel a car uses (Physics) to calculating simple interest (Economics) or understanding data trends in AI/ML, this concept is crucial for many real-world applications and future careers.

Common Mistakes

MISTAKE: Assuming the line always passes through the origin, even if the equation is y = kx + c (where c is not 0). | CORRECTION: The graph of y = kx specifically passes through the origin (0,0). If there's an added constant (like y = 2x + 3), it will be a straight line but will NOT pass through the origin.

MISTAKE: Confusing 'k' with the x-intercept or y-intercept. | CORRECTION: 'k' is the slope (gradient) of the line, which tells us how steep it is. For y = kx, the y-intercept is always 0 (it passes through the origin).

MISTAKE: Plotting points incorrectly, especially negative values of x or y. | CORRECTION: Always double-check your calculations for y for each x value. Remember that for y = kx, if x is negative, y will also be negative (if k is positive), leading to points in the third quadrant.

Practice Questions

Try It Yourself

QUESTION: Which of these equations represents a straight line passing through the origin: a) y = 3x + 1, b) y = 5x, c) y = x^2? | ANSWER: b) y = 5x

QUESTION: If the graph of y = kx passes through the point (2, 8), what is the value of k? | ANSWER: k = 4 (Since 8 = k * 2, k = 8/2 = 4)

QUESTION: Draw the graph of y = -3x by finding three points. What is the coordinate of the point where it crosses the y-axis? | ANSWER: Points could be (0,0), (1,-3), (-1,3). It crosses the y-axis at (0,0).

MCQ

Quick Quiz

What is a key characteristic of the graph of y = kx?

It is a curved line.

It always passes through the point (0,0).

It is a horizontal line.

It never passes through the origin.

The Correct Answer Is:

The equation y = kx describes direct proportionality, and when x=0, y will always be 0 (k*0 = 0). This means the graph must pass through the origin (0,0). It is also always a straight line, not curved.

Real World Connection

In the Real World

This concept is used by delivery apps like Swiggy or Zomato! The distance an delivery rider travels (y) is roughly proportional to the time taken (x) at a constant speed (k). So, y = kx helps them estimate delivery times. Also, in banking, simple interest earned (y) is directly proportional to the principal amount (x) for a fixed rate and time, following a y=kx pattern.

Key Vocabulary

Key Terms

ORIGIN: The point (0,0) where the x and y axes intersect. | CONSTANT OF PROPORTIONALITY (k): The fixed ratio between two quantities in a direct variation. | DIRECT PROPORTION: A relationship where two quantities increase or decrease at the same rate. | COORDINATE PLANE: A 2D surface formed by the intersection of the x-axis and y-axis.

What's Next

What to Learn Next

Great job understanding y = kx! Next, you should learn about the graph of y = mx + c. This builds on what you've learned by adding a 'c' term, which shifts the straight line up or down, making it even more powerful for describing real-world situations.