What is the Graph of y = x?

S3-SA5-0098

Grade Level:

Class 9

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

Definition

What is it?

The graph of y = x is a straight line that passes through the origin (0,0) of a coordinate plane. It shows that for every point on the line, the y-coordinate is exactly equal to the x-coordinate.

Simple Example

Quick Example

Imagine you are buying samosas, and each samosa costs 10 rupees. If 'x' is the number of samosas you buy and 'y' is the total cost, then y = 10x. But if each samosa magically cost exactly 1 rupee, then y = x. So, if you buy 2 samosas, you pay 2 rupees. If you buy 5 samosas, you pay 5 rupees.

Worked Example

Step-by-Step

Let's plot the graph of y = x.

Step 1: Choose some values for x. Let's pick x = -2, -1, 0, 1, 2.
---Step 2: Since y = x, the corresponding y-values will be the same as the x-values.
---Step 3: Create a table of (x, y) coordinates:
x | y
---|---
-2 | -2
-1 | -1
0 | 0
1 | 1
2 | 2
---Step 4: Draw a coordinate plane with an x-axis and a y-axis.
---Step 5: Plot each point from the table: (-2, -2), (-1, -1), (0, 0), (1, 1), (2, 2).
---Step 6: Connect these points with a straight line.
---Step 7: Extend the line in both directions with arrows to show it continues infinitely.

Answer: The line passes through the origin (0,0) and makes a 45-degree angle with the positive x-axis.

Why It Matters

Understanding y = x is crucial for many fields like AI/ML and Data Science, where it helps in basic data visualization and understanding relationships. Engineers use it to model simple direct relationships, and it's fundamental for creating visual representations of data in Computer Science.

Common Mistakes

MISTAKE: Students sometimes think y = x means the line will be horizontal or vertical. | CORRECTION: Remember y = x means x and y are always equal. This only happens on a diagonal line passing through the origin.

MISTAKE: Plotting points incorrectly, for example, confusing (1,1) with (1,0) or (0,1). | CORRECTION: Always check the x-coordinate first (how far left/right), then the y-coordinate (how far up/down) from the origin.

MISTAKE: Not extending the line beyond the plotted points. | CORRECTION: A graph represents all possible solutions, not just the few points you plotted. Always extend the line with arrows.

Practice Questions

Try It Yourself

QUESTION: Which point lies on the graph of y = x? (3, -3) or (-5, -5) | ANSWER: (-5, -5)

QUESTION: If the point (k, 7) lies on the graph of y = x, what is the value of k? | ANSWER: k = 7

QUESTION: A line passes through the points (2, 2) and (-4, -4). Is this the graph of y = x? Explain why. | ANSWER: Yes, because for both points, the y-coordinate is equal to the x-coordinate, which is the definition of y = x. Also, it's a straight line.

MCQ

Quick Quiz

Which of these statements is true about the graph of y = x?

It is a horizontal line.

It passes through the origin (0,0).

The y-coordinate is always twice the x-coordinate.

It is a vertical line.

The Correct Answer Is:

The equation y = x means that when x is 0, y must also be 0, so it passes through the origin. Options A and D describe lines where x or y is constant, and option C describes y = 2x.

Real World Connection

In the Real World

Imagine a digital payment app like PhonePe or Google Pay. If you send 100 rupees to a friend, they receive exactly 100 rupees (ignoring any small fees). This is a direct 1:1 relationship, similar to y = x. In data analysis for cricket, if a player scores 'x' runs in 'x' balls, their strike rate is 100, showing a y=x relationship between runs and balls.

Key Vocabulary

Key Terms

COORDINATE PLANE: A 2D surface where points are located using x and y coordinates | ORIGIN: The point (0,0) where the x-axis and y-axis intersect | LINEAR EQUATION: An equation whose graph is a straight line | X-AXIS: The horizontal number line in a coordinate plane | Y-AXIS: The vertical number line in a coordinate plane

What's Next

What to Learn Next

Next, you can explore graphs of other linear equations like y = x + c (where 'c' is a constant) or y = mx (where 'm' is the slope). This will help you understand how changing parts of the equation affects the line's position and steepness.