What is a Graph of y = 1/x?

S3-SA5-0281

Grade Level:

Class 9

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

Definition

What is it?

The graph of y = 1/x is a special curve that shows how two quantities are inversely related. This means as one quantity (x) increases, the other quantity (y) decreases, and vice-versa, always keeping their product equal to 1.

Simple Example

Quick Example

Imagine you have 1 litre of juice to share equally among your friends. If you have 1 friend (x=1), each gets 1 litre (y=1). If you have 2 friends (x=2), each gets 0.5 litres (y=0.5). If you have 4 friends (x=4), each gets 0.25 litres (y=0.25). The graph of y=1/x shows this sharing pattern visually.

Worked Example

Step-by-Step

Let's plot some points for the graph of y = 1/x:

Step 1: Choose some values for x.
Let's pick x = -2, -1, -0.5, 0.5, 1, 2.

---
Step 2: Calculate the corresponding y values using y = 1/x.
If x = -2, y = 1/(-2) = -0.5
If x = -1, y = 1/(-1) = -1
If x = -0.5, y = 1/(-0.5) = -2

---
Step 3: Notice what happens near x = 0. The value 1/0 is undefined, so the graph will never touch the y-axis (where x=0).

---
Step 4: Continue calculating y values for positive x.
If x = 0.5, y = 1/(0.5) = 2
If x = 1, y = 1/1 = 1
If x = 2, y = 1/2 = 0.5

---
Step 5: List the points (x, y) you have found:
(-2, -0.5), (-1, -1), (-0.5, -2), (0.5, 2), (1, 1), (2, 0.5).

---
Step 6: If you plot these points on a graph paper and connect them smoothly (but not crossing the y-axis), you will see two separate curves, one in the top-right section and one in the bottom-left section of the graph.

---
Answer: The points show that as x gets larger (positive), y gets smaller (positive), and as x gets smaller (negative), y gets larger (negative). The graph has two branches, never crossing the x or y axes.

Why It Matters

Understanding inverse relationships is super important in many fields! Engineers use it to design circuits, economists use it to understand supply and demand, and even computer scientists use it in algorithms. It helps you see how things balance each other out in the real world.

Common Mistakes

MISTAKE: Thinking the graph passes through the origin (0,0) or crosses the axes. | CORRECTION: The graph of y=1/x never touches the x-axis (because y can never be 0) and never touches the y-axis (because x can never be 0, as 1/0 is undefined).

MISTAKE: Connecting the two separate branches of the graph. | CORRECTION: The graph of y=1/x has two distinct parts (branches) that are in diagonally opposite quadrants and are never connected. They are separated by the x and y axes.

MISTAKE: Assuming y increases as x increases. | CORRECTION: For y=1/x, y *decreases* as x *increases* (for positive x), and y *increases* as x *decreases* (for negative x). This is an inverse relationship.

Practice Questions

Try It Yourself

QUESTION: If x = 0.2, what is the value of y for the equation y = 1/x? | ANSWER: y = 5

QUESTION: What happens to the value of y in y = 1/x as x becomes a very large positive number (e.g., 1000)? | ANSWER: y becomes a very small positive number, approaching 0.

QUESTION: In which two quadrants of the coordinate plane will the graph of y = 1/x lie? Explain why. | ANSWER: Quadrants I and III. Because if x is positive, y must also be positive (1/positive = positive), placing points in Quadrant I. If x is negative, y must also be negative (1/negative = negative), placing points in Quadrant III.

MCQ

Quick Quiz

Which statement is true about the graph of y = 1/x?

It passes through the origin (0,0).

It has two separate branches.

As x increases, y also increases.

It crosses the y-axis at y=1.

The Correct Answer Is:

The graph of y=1/x has two distinct branches in opposite quadrants. It never passes through the origin, never crosses the axes, and shows an inverse relationship where y decreases as x increases.

Real World Connection

In the Real World

Think about how quickly a task gets done. If one person can paint a wall in 10 hours, two people (x=2) might paint it in 5 hours (y=5). This is an inverse relationship. Many apps like Swiggy or Zomato use algorithms that manage delivery times based on the number of available delivery partners – more partners (x) means less delivery time (y), often following a pattern similar to y=1/x.

Key Vocabulary

Key Terms

INVERSE RELATIONSHIP: When one quantity increases, the other decreases proportionally. | HYPERBOLA: The name for the specific curve formed by the graph of y=1/x. | ASYMPTOTE: A line that the graph gets closer and closer to but never actually touches (for y=1/x, the x and y axes are asymptotes). | RECIPROCAL: The reciprocal of a number 'x' is 1/x.

What's Next

What to Learn Next

Now that you understand inverse relationships, you can explore other types of graphs like y = x^2 (parabola) or y = |x| (absolute value function). These will help you build a strong foundation for understanding more complex functions in higher classes!