What is Vertical Compression? | Simple Explanation for Class 9

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What is a Vertical Compression?

Grade Level:

Class 9

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

Definition

What is it?

Vertical compression is a transformation that shrinks a graph towards the x-axis. It makes the graph appear 'flatter' or 'squished' vertically. This happens when you multiply the entire function by a fraction between 0 and 1.

Simple Example

Quick Example

Imagine a tall stack of 'ladoos'. If you gently press down on the stack from the top, the stack becomes shorter but wider. Similarly, if a function y = f(x) is vertically compressed, its y-values become smaller, making the graph 'squish' downwards.

Worked Example

Step-by-Step

Let's vertically compress the function y = x^2 by a factor of 1/2.
---The original function is y = x^2.
---To compress it vertically by 1/2, we multiply the entire function by 1/2.
---The new function becomes y = (1/2) * x^2.
---Let's see some points:
---For original y = x^2: If x = 2, y = 2^2 = 4. If x = -2, y = (-2)^2 = 4.
---For compressed y = (1/2)x^2: If x = 2, y = (1/2)*2^2 = (1/2)*4 = 2. If x = -2, y = (1/2)*(-2)^2 = (1/2)*4 = 2.
---Notice how the y-values are halved for the same x-values, making the graph flatter.
---Answer: The vertically compressed function is y = (1/2)x^2.

Why It Matters

Understanding vertical compression is crucial in fields like Computer Science and Engineering for designing efficient algorithms and structures. In Data Science, it helps in scaling data, which is vital for training machine learning models effectively. Engineers use it to model how materials deform under stress or how signals change in communication systems.

Common Mistakes

MISTAKE: Multiplying only 'x' by the compression factor instead of the entire function. For y = x^2, students might write y = (1/2x)^2. | CORRECTION: The compression factor multiplies the *output* (y-value) of the function. So, it should be y = (1/2)x^2.

MISTAKE: Confusing vertical compression with horizontal compression. | CORRECTION: Vertical compression changes the y-values, making the graph shorter. Horizontal compression changes the x-values, making the graph narrower.

MISTAKE: Using a number greater than 1 for compression. | CORRECTION: Vertical compression happens when the factor is between 0 and 1 (a fraction). If the factor is greater than 1, it's a vertical stretch, making the graph taller.

Practice Questions

Try It Yourself

QUESTION: What is the new function if y = 2x + 1 is vertically compressed by a factor of 1/4? | ANSWER: y = (1/4)(2x + 1) or y = (1/2)x + 1/4

QUESTION: If the point (3, 9) is on the graph of y = f(x), what will be its new coordinates after a vertical compression by a factor of 1/3? | ANSWER: (3, 3)

QUESTION: A parabola defined by y = x^2 - 4 has its lowest point (vertex) at (0, -4). If it undergoes a vertical compression by a factor of 0.5, what will be the new equation and the new vertex? | ANSWER: New equation: y = 0.5(x^2 - 4) = 0.5x^2 - 2. New vertex: (0, -2)

MCQ

Quick Quiz

Which transformation applies a vertical compression to the function y = f(x)?

y = f(2x)

y = 2f(x)

y = (1/2)f(x)

y = f(x) + 2

The Correct Answer Is:

Option C, y = (1/2)f(x), multiplies the entire function's output by 1/2, which is a fraction between 0 and 1, causing vertical compression. Option B is a vertical stretch, A is horizontal compression, and D is a vertical shift.

Real World Connection

In the Real World

Think about how video editors adjust the 'aspect ratio' of a video. If you take a wide video and try to fit it into a narrow screen without cropping, you might 'compress' it vertically to make it fit, making everything look a bit squashed. Similarly, in graphic design, images are often scaled or compressed vertically to fit specific layouts or reduce file size without losing too much detail, like when you upload a profile picture on social media.

Key Vocabulary

Key Terms

TRANSFORMATION: A change made to a graph or function's position, size, or shape | X-AXIS: The horizontal line in a coordinate system | Y-VALUES: The output values of a function, representing vertical position | COMPRESSION FACTOR: The fraction by which a graph is shrunk

What's Next

What to Learn Next

Great job understanding vertical compression! Next, you should explore 'Vertical Stretch' to see how graphs can become taller. After that, dive into 'Horizontal Compression and Stretch' to learn how graphs can become narrower or wider along the x-axis. These concepts are fundamental for understanding all types of function transformations!