What is the Graph of y = 1/x?

S3-SA5-0065

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 called a hyperbola. It shows how two quantities are related when one is inversely proportional to the other. This means as one quantity gets bigger, the other gets smaller, and vice-versa, always keeping their product constant.

Simple Example

Quick Example

Imagine you have 100 laddus to share equally among your friends. If you have 1 friend, they get 100 laddus (y=100 when x=1). If you have 2 friends, each gets 50 laddus (y=50 when x=2). If you have 4 friends, each gets 25 laddus (y=25 when x=4). Here, 'y' is the number of laddus per friend, and 'x' is the number of friends. As 'x' (friends) increases, 'y' (laddus per friend) decreases, just like in y = 1/x.

Worked Example

Step-by-Step

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

Step 1: Choose some positive values for x. If x = 1, then y = 1/1 = 1. So, (1, 1) is a point.
---Step 2: If x = 2, then y = 1/2 = 0.5. So, (2, 0.5) is a point.
---Step 3: If x = 0.5, then y = 1/0.5 = 2. So, (0.5, 2) is a point.
---Step 4: Choose some negative values for x. If x = -1, then y = 1/(-1) = -1. So, (-1, -1) is a point.
---Step 5: If x = -2, then y = 1/(-2) = -0.5. So, (-2, -0.5) is a point.
---Step 6: If x = -0.5, then y = 1/(-0.5) = -2. So, (-0.5, -2) is a point.
---Step 7: Notice that when x is 0, y = 1/0, which is undefined. So, the graph never touches the y-axis (x=0) or the x-axis (since y can never be 0).
---Answer: Plotting these points and connecting them smoothly shows two separate curves, one in the top-right section (first quadrant) and one in the bottom-left section (third quadrant) of the graph.

Why It Matters

Understanding inverse relationships like y = 1/x is super important in many fields. Engineers use it to design circuits, economists use it to understand supply and demand, and data scientists use it to model complex systems. It helps in careers from building apps to analyzing stock markets.

Common Mistakes

MISTAKE: Thinking the graph touches the x-axis or y-axis. | 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). These are called asymptotes.

MISTAKE: Assuming the graph is a straight line or a parabola. | CORRECTION: The graph of y = 1/x is a hyperbola, which has two distinct curved branches. It's not a straight line like y = x, nor a U-shape like y = x^2.

MISTAKE: Plotting only positive values of x and forgetting negative values. | CORRECTION: The equation y = 1/x works for both positive and negative values of x (except x=0). Remember to plot points for negative x values too, which will form the second branch of the hyperbola.

Practice Questions

Try It Yourself

QUESTION: What happens to the value of y in y = 1/x if x becomes very, very large (e.g., x = 1000)? | ANSWER: y becomes very, very small (e.g., y = 1/1000 = 0.001).

QUESTION: If a point (a, b) lies on the graph of y = 1/x, what is the product of 'a' and 'b'? | ANSWER: The product a * b will always be 1 (because b = 1/a, so a * b = a * (1/a) = 1).

QUESTION: Plot the points for y = 1/x when x = 0.25, x = -0.25, x = 4, and x = -4. | ANSWER: (0.25, 4), (-0.25, -4), (4, 0.25), (-4, -0.25).

MCQ

Quick Quiz

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

It is a straight line passing through the origin.

It touches both the x-axis and the y-axis.

It has two separate branches and never touches the axes.

It is a U-shaped curve.

The Correct Answer Is:

The graph of y = 1/x is a hyperbola with two branches in opposite quadrants. It never touches the x or y axes because x cannot be 0, and y can never be 0.

Real World Connection

In the Real World

This inverse relationship is used in real life for things like mobile network signal strength. As you move further away from a mobile tower (increasing distance 'x'), the signal strength ('y') decreases. Similarly, in a car journey, if you want to cover a fixed distance, increasing your speed ('x') will decrease the time taken ('y'). Apps like Google Maps use these principles to calculate estimated arrival times.

Key Vocabulary

Key Terms

HYPERBOLA: A specific type of curve with two separate branches, seen in the graph of y = 1/x | INVERSE PROPORTION: A relationship where as one quantity increases, the other decreases, such that their product is constant | ASYMPTOTE: A line that a curve approaches as it heads towards infinity, but never actually touches | UNDEFINED: A mathematical expression that does not have a meaningful value, like division by zero.

What's Next

What to Learn Next

Great job understanding the graph of y = 1/x! Next, you can explore the graphs of other rational functions like y = 1/x^2 or y = k/x. This will help you see how small changes in the equation affect the shape and position of the graph, building your skills for higher math and science.