What is a Horizontal Stretch?

S3-SA5-0056

Grade Level:

Class 9

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

Definition

What is it?

A horizontal stretch is a type of transformation that pulls or pushes a graph away from or towards the y-axis. Imagine you are stretching a rubber band sideways. This changes the x-coordinates of all points on the graph, making the graph wider or narrower while keeping its height the same.

Simple Example

Quick Example

Imagine your favourite cricket match highlights are playing on a TV. If you use a special setting that 'stretches' the video horizontally, all the players would look wider, and the cricket pitch would appear much longer than it actually is. The game itself doesn't change, just how it looks on screen.

Worked Example

Step-by-Step

Let's say we have a point (2, 4) on a graph. We want to apply a horizontal stretch by a factor of 2.

1. Identify the original coordinates: x = 2, y = 4.
2. For a horizontal stretch, only the x-coordinate changes. The y-coordinate remains the same.
3. To stretch horizontally by a factor of 'k', we multiply the x-coordinate by 'k'. Here, k = 2.
4. New x-coordinate = Original x-coordinate * k = 2 * 2 = 4.
5. The y-coordinate remains 4.
6. The new point after the horizontal stretch is (4, 4).

Answer: The point (2, 4) becomes (4, 4) after a horizontal stretch by a factor of 2.

Why It Matters

Understanding transformations like horizontal stretches is crucial in fields like AI/ML and data science, where models are 'stretched' or 'compressed' to fit data better. In computer graphics, it helps designers create different visual effects, like making a character look wider or taller without redrawing it completely. It’s also used in physics to model how materials deform under stress.

Common Mistakes

MISTAKE: Applying the stretch factor to the y-coordinate instead of the x-coordinate. | CORRECTION: Remember, 'horizontal' means side-to-side, so only the x-values are affected. The y-values (height) stay the same.

MISTAKE: Confusing a horizontal stretch with a horizontal shift (translation). | CORRECTION: A stretch changes the 'width' by multiplying, while a shift moves the entire graph left or right by adding or subtracting.

MISTAKE: Thinking a horizontal stretch by a factor of 1/2 makes the graph wider. | CORRECTION: A stretch factor greater than 1 makes it wider. A stretch factor between 0 and 1 (like 1/2) actually makes it narrower, also called a horizontal compression.

Practice Questions

Try It Yourself

QUESTION: A point (6, 3) is on a graph. What are its new coordinates after a horizontal stretch by a factor of 3? | ANSWER: (18, 3)

QUESTION: The graph of y = x^2 has a point (2, 4). If the graph undergoes a horizontal stretch by a factor of 0.5, what are the new coordinates of this point? | ANSWER: (1, 4)

QUESTION: A line segment connects points A(1, 5) and B(4, 5). If this segment is horizontally stretched by a factor of 2, what are the new coordinates of points A' and B'? How long is the new segment? | ANSWER: A'(2, 5), B'(8, 5). The new segment length is 6 units.

MCQ

Quick Quiz

Which of the following describes a horizontal stretch?

Multiplying y-coordinates by a constant

Adding a constant to x-coordinates

Multiplying x-coordinates by a constant

Adding a constant to y-coordinates

The Correct Answer Is:

A horizontal stretch affects the x-coordinates, making the graph wider or narrower. This is achieved by multiplying the x-coordinates by a constant factor. Options A and D relate to vertical changes, and option B relates to a horizontal shift.

Real World Connection

In the Real World

When you edit photos on your mobile phone, like stretching an image of your friend to make them look taller or wider for fun, you are applying a form of horizontal (or vertical) stretch. Similarly, in animation studios in India, animators use these transformations to change the proportions of characters or objects without redrawing them from scratch, saving a lot of time and effort.

Key Vocabulary

Key Terms

TRANSFORMATION: Changing the position, size, or shape of a figure | COORDINATE: A set of values that show an exact position on a graph | STRETCH FACTOR: The number by which coordinates are multiplied during a stretch | Y-AXIS: The vertical line on a coordinate plane

What's Next

What to Learn Next

Now that you understand horizontal stretches, you should explore 'Vertical Stretches and Compressions'. They are very similar but affect the height of the graph instead of the width, building on the same core ideas of transformation!