What is a Vertical Stretch?

S3-SA5-0054

Grade Level:

Class 9

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

Definition

What is it?

A vertical stretch is a transformation that pulls a graph or a shape away from the x-axis, making it taller or more elongated. It happens when you multiply the 'y' values (or the output) of a function by a number greater than 1.

Simple Example

Quick Example

Imagine you have a rubber band lying flat on a table. If you hold its ends and pull it upwards from the middle, it stretches vertically. In math, if your original marks in a test were (x, y) and now you score double marks in a re-test, your new marks (x, 2y) show a vertical stretch.

Worked Example

Step-by-Step

Let's say we have a simple line graph with points (1, 2), (2, 4), and (3, 6).

Step 1: Identify the original points. Our points are P1(1, 2), P2(2, 4), P3(3, 6).
---Step 2: Decide the stretch factor. Let's apply a vertical stretch by a factor of 3.
---Step 3: Multiply only the y-coordinate of each point by the stretch factor.
---Step 4: For P1(1, 2), the new y-coordinate will be 2 * 3 = 6. So, the new point is P1'(1, 6).
---Step 5: For P2(2, 4), the new y-coordinate will be 4 * 3 = 12. So, the new point is P2'(2, 12).
---Step 6: For P3(3, 6), the new y-coordinate will be 6 * 3 = 18. So, the new point is P3'(3, 18).
---Step 7: The new points after a vertical stretch by a factor of 3 are (1, 6), (2, 12), and (3, 18). These points will form a line that looks 'taller' than the original line.

Why It Matters

Understanding vertical stretches is super important for fields like Physics, where you study how springs stretch, or in Data Science, where you might need to scale data. Engineers use this concept to design structures that can withstand forces, and in AI, it helps in image processing and computer vision to resize or manipulate images.

Common Mistakes

MISTAKE: Multiplying both x and y coordinates by the stretch factor. | CORRECTION: For a vertical stretch, only the y-coordinate (output value) is multiplied by the stretch factor. The x-coordinate (input value) remains unchanged.

MISTAKE: Confusing vertical stretch with horizontal stretch. | CORRECTION: A vertical stretch changes the height, while a horizontal stretch changes the width. Vertical stretch affects y-values, horizontal stretch affects x-values.

MISTAKE: Using a stretch factor less than 1 (like 0.5) and still calling it a 'stretch'. | CORRECTION: If the factor is between 0 and 1, it's actually a vertical 'compression' or 'shrink', not a stretch. A stretch requires a factor greater than 1.

Practice Questions

Try It Yourself

QUESTION: A point (4, 5) undergoes a vertical stretch by a factor of 2. What are its new coordinates? | ANSWER: (4, 10)

QUESTION: If the graph of y = x^2 is vertically stretched by a factor of 4, what is the equation of the new graph? | ANSWER: y = 4x^2

QUESTION: A rectangular photo has corners at (0,0), (5,0), (5,3), and (0,3). If it's vertically stretched by a factor of 1.5, what are the new coordinates of its top-right corner? | ANSWER: (5, 4.5)

MCQ

Quick Quiz

Which of these functions represents a vertical stretch of y = x by a factor of 5?

y = x + 5

y = 5x

y = x/5

y = x - 5

The Correct Answer Is:

A vertical stretch means multiplying the entire function's output (y-value) by the stretch factor. So, y = x becomes y = 5x. Options A, C, and D represent different types of transformations.

Real World Connection

In the Real World

Think about how video editors or Instagram filters work. When you 'stretch' an image vertically, you're applying a transformation similar to a vertical stretch in mathematics. In ISRO, scientists use these concepts to model how rocket trajectories change under different forces, or how satellite images can be scaled and adjusted for better viewing.

Key Vocabulary

Key Terms

TRANSFORMATION: Changing the position, size, or orientation of a shape or graph | X-AXIS: The horizontal number line in a coordinate system | Y-AXIS: The vertical number line in a coordinate system | STRETCH FACTOR: The number by which you multiply to stretch a graph | COORDINATES: A set of values that show an exact position on a graph

What's Next

What to Learn Next

Now that you understand vertical stretches, you should explore 'Vertical Compression' and 'Horizontal Stretch and Compression'. These concepts are closely related and will help you master how graphs and shapes can be manipulated in different ways.