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What is a Compression of a Graph?
Grade Level:
Class 9
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
A compression of a graph makes the graph appear 'squished' or narrower along an axis. It happens when you multiply the input (x-value) or output (y-value) of a function by a number greater than 1, making the graph closer to an axis.
Simple Example
Quick Example
Imagine a graph showing your daily steps over a week. If you compress it horizontally, it means you're showing the same number of steps but in a shorter timeframe, making the graph look denser. For instance, showing a week's steps in half the width on the graph paper.
Worked Example
Step-by-Step
Let's compress the graph of y = x^2 horizontally.
Step 1: Start with the basic function y = x^2.
Step 2: To compress horizontally, we multiply the x-value inside the function. Let's choose a compression factor of 2. So, we replace 'x' with '2x'. The new function becomes y = (2x)^2.
Step 3: Simplify the new function: y = 4x^2.
Step 4: Compare points. For y = x^2, when x = 1, y = 1. When x = 2, y = 4.
Step 5: For y = 4x^2, when x = 1, y = 4. Notice that for the same y-value (e.g., y=4), x is 2 for y=x^2, but x is 1 for y=4x^2. This means the graph of y = 4x^2 reaches the same height (y=4) at half the x-distance (x=1 vs x=2) compared to y=x^2.
Step 6: This makes the graph of y = 4x^2 look narrower or 'squished' towards the y-axis compared to y = x^2.
Answer: Compressing y = x^2 horizontally by a factor of 2 results in the graph of y = (2x)^2 or y = 4x^2.
Why It Matters
Understanding graph compression helps in AI/ML to scale data, in Physics to analyze wave patterns, and in Data Science to visualize large datasets efficiently. Engineers use it to model how systems respond to different inputs, helping them design better products.
Common Mistakes
MISTAKE: Confusing horizontal compression with horizontal stretch. | CORRECTION: Horizontal compression happens when you multiply 'x' by a number GREATER than 1. Horizontal stretch happens when you multiply 'x' by a number BETWEEN 0 and 1.
MISTAKE: Applying the compression factor to the entire function for horizontal compression. | CORRECTION: For horizontal compression, the factor is applied ONLY to the 'x' variable inside the function (e.g., f(x) becomes f(kx) for k > 1).
MISTAKE: Thinking that multiplying the entire function by a number greater than 1 compresses the graph. | CORRECTION: Multiplying the entire function (e.g., y = k * f(x) where k > 1) results in a VERTICAL stretch, not a compression.
Practice Questions
Try It Yourself
QUESTION: If the graph of y = sin(x) is compressed horizontally by a factor of 3, what is the new equation? | ANSWER: y = sin(3x)
QUESTION: Describe the transformation if the graph of y = x^3 becomes y = (5x)^3. | ANSWER: Horizontal compression by a factor of 5.
QUESTION: A graph of daily chai sales (y) over time (x) is given by y = f(x). If the owner wants to show the sales for a month in half the usual width on the graph, what transformation should be applied to x? | ANSWER: Replace x with 2x (i.e., y = f(2x)), which is a horizontal compression by a factor of 2.
MCQ
Quick Quiz
Which transformation represents a horizontal compression of the graph y = f(x)?
y = f(x/2)
y = 2 * f(x)
y = f(2x)
y = f(x) - 2
The Correct Answer Is:
C
Option C, y = f(2x), compresses the graph horizontally by a factor of 2 because the input 'x' is multiplied by a number greater than 1. Options A is a horizontal stretch, B is a vertical stretch, and D is a vertical shift.
Real World Connection
In the Real World
When you use apps like Google Maps, the data for distances and routes is often compressed to load faster and fit on your phone screen. Similarly, in cricket analytics, complex data from a match might be compressed to highlight key moments or trends in a compact graph, making it easier for commentators or coaches to understand quickly.
Key Vocabulary
Key Terms
HORIZONTAL COMPRESSION: Making a graph narrower by multiplying the x-value by a number greater than 1 | VERTICAL COMPRESSION: Making a graph shorter by multiplying the entire function by a number between 0 and 1 | TRANSFORMATION: Changing the position, size, or orientation of a graph | FUNCTION: A rule that assigns exactly one output for each input
What's Next
What to Learn Next
Now that you understand graph compression, explore 'Graph Stretches'. This will show you how to expand a graph horizontally or vertically, which is the opposite of compression, and help you master graph transformations completely.
