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What is a Horizontal Compression?
Grade Level:
Class 9
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
Horizontal compression means squeezing a graph or shape inwards towards the y-axis. It makes the graph appear narrower or shorter along the x-axis, changing its horizontal dimensions.
Simple Example
Quick Example
Imagine you have a drawing of a cricket pitch on a rubber sheet. If you pull the sheet from the left and right sides, squeezing it inwards, the pitch will look narrower. This squeezing is like horizontal compression.
Worked Example
Step-by-Step
Let's say you have a function y = f(x). If you apply a horizontal compression by a factor of 2, the new function becomes y = f(2x).
Step 1: Consider a point on the original graph, for example, (4, 8) on the graph of y = 2x.
---Step 2: To find the new x-coordinate after compression by a factor of 2, divide the original x-coordinate by 2. So, new x = 4 / 2 = 2.
---Step 3: The y-coordinate remains the same. So, new y = 8.
---Step 4: The new point on the compressed graph is (2, 8).
---Step 5: Let's check this with the compressed function. If y = 2x is compressed by a factor of 2, the new function is y = 2(2x) = 4x. If x = 2, then y = 4(2) = 8. This matches our new point.
---Answer: A point (4, 8) on y=2x moves to (2, 8) after horizontal compression by a factor of 2.
Why It Matters
Understanding horizontal compression helps in fields like computer graphics to resize images and videos without distortion. In physics, it's used to model how waves change when they pass through different materials. Engineers use it to design structures that need to withstand varying forces.
Common Mistakes
MISTAKE: Multiplying the x-coordinate by the compression factor. | CORRECTION: Divide the x-coordinate by the compression factor. If the factor is 'k', the new x-coordinate is x/k, not x*k.
MISTAKE: Changing the y-coordinate during horizontal compression. | CORRECTION: Horizontal compression only affects the x-coordinates; the y-coordinates remain unchanged.
MISTAKE: Confusing horizontal compression with vertical compression. | CORRECTION: Horizontal compression makes the graph narrower (affects x). Vertical compression makes the graph shorter (affects y).
Practice Questions
Try It Yourself
QUESTION: If a point (6, 10) is on a graph, what are its new coordinates after a horizontal compression by a factor of 3? | ANSWER: (2, 10)
QUESTION: The graph of y = x^2 passes through the point (2, 4). What is the new point on the graph of y = (3x)^2 after horizontal compression? | ANSWER: (2/3, 4)
QUESTION: A function y = f(x) has a maximum point at (8, 5). If the function is horizontally compressed by a factor of 4, what are the coordinates of the new maximum point? Explain your steps. | ANSWER: The x-coordinate (8) is divided by the compression factor (4), so 8/4 = 2. The y-coordinate (5) remains the same. The new maximum point is (2, 5).
MCQ
Quick Quiz
Which transformation makes a graph narrower by squeezing it towards the y-axis?
Vertical stretch
Horizontal compression
Vertical compression
Horizontal stretch
The Correct Answer Is:
B
Horizontal compression squeezes the graph inwards towards the y-axis, making it narrower. Vertical stretch would make it taller, vertical compression shorter, and horizontal stretch wider.
Real World Connection
In the Real World
When you edit photos on your mobile phone and use a 'crop' or 'resize' tool to make an image fit a smaller screen size without distorting the height, you are applying a form of horizontal compression (or scaling). This is also how video editing software adjusts footage to different aspect ratios for various social media platforms like Instagram Reels or YouTube shorts.
Key Vocabulary
Key Terms
COMPRESSION: The act of squeezing or making something smaller | FACTOR: The number by which a quantity is multiplied or divided | Y-AXIS: The vertical line on a graph | X-COORDINATE: The first number in an ordered pair (x, y) that indicates horizontal position
What's Next
What to Learn Next
Next, explore 'Horizontal Stretch' to understand how graphs become wider. After that, you can learn about 'Vertical Compression' and 'Vertical Stretch' to see how graphs change in height. These concepts are fundamental to understanding function transformations!
