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What is a Stretch/Compression on a Generic Graph?
Grade Level:
Class 10
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
A stretch or compression on a graph changes the shape of a function by making it wider or narrower, or taller or shorter. It's like pulling or pushing the graph horizontally or vertically, without changing its basic form.
Simple Example
Quick Example
Imagine you have a rubber band shaped like a 'U' (a parabola). If you pull the ends horizontally, the 'U' becomes wider – that's a horizontal stretch. If you push the ends closer, it becomes narrower – that's a horizontal compression. Similarly, if you stretch it upwards, it becomes taller (vertical stretch), and pushing it down makes it shorter (vertical compression).
Worked Example
Step-by-Step
Let's see how the graph of y = x^2 changes with a stretch/compression.
Step 1: Start with the basic function y = x^2. Plot points like (0,0), (1,1), (-1,1), (2,4), (-2,4).
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Step 2: Consider a vertical stretch: y = 2x^2. For each x-value, the y-value is now twice as large. So, (1,1) becomes (1,2), and (2,4) becomes (2,8). The graph becomes 'thinner' or 'taller'.
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Step 3: Consider a vertical compression: y = (1/2)x^2. For each x-value, the y-value is now half as large. So, (1,1) becomes (1, 0.5), and (2,4) becomes (2,2). The graph becomes 'wider' or 'shorter'.
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Step 4: Consider a horizontal compression: y = (2x)^2. This is equivalent to y = 4x^2. If you want y to be 1, then 2x must be 1, so x = 0.5. The graph becomes 'thinner'.
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Step 5: Consider a horizontal stretch: y = (x/2)^2. This is equivalent to y = (1/4)x^2. If you want y to be 1, then x/2 must be 1, so x = 2. The graph becomes 'wider'.
Answer: Vertical stretches/compressions affect the y-values (making the graph taller/shorter), while horizontal stretches/compressions affect the x-values (making the graph wider/narrower).
Why It Matters
Understanding stretches and compressions helps in predicting how changing numbers in a formula will alter a graph's shape. This is crucial in AI/ML for scaling data, in Physics for modeling wave behavior, and in Economics for understanding how price changes affect demand curves. Many engineers use this to design structures or model systems.
Common Mistakes
MISTAKE: Thinking that multiplying x by a number (like y = (2x)^2) will make the graph wider (stretch). | CORRECTION: Multiplying x by a number greater than 1 actually causes a horizontal compression, making the graph narrower. It's the opposite of what you might expect.
MISTAKE: Confusing vertical and horizontal transformations. | CORRECTION: A number multiplying the entire function (like y = 2f(x)) causes a vertical stretch/compression. A number multiplying only the 'x' inside the function (like y = f(2x)) causes a horizontal stretch/compression.
MISTAKE: Assuming a negative sign causes a stretch or compression. | CORRECTION: A negative sign (e.g., y = -f(x) or y = f(-x)) causes a reflection, not a stretch or compression. It flips the graph across an axis.
Practice Questions
Try It Yourself
QUESTION: How does the graph of y = 3x^2 compare to y = x^2? | ANSWER: It's a vertical stretch, making the graph 'thinner' or 'taller' by a factor of 3.
QUESTION: Describe the transformation from y = sin(x) to y = sin(x/2). | ANSWER: It's a horizontal stretch by a factor of 2, making the wave twice as wide.
QUESTION: If you have the graph of y = sqrt(x), how would you write the equation for a graph that is vertically stretched by a factor of 4 and horizontally compressed by a factor of 2? | ANSWER: y = 4 * sqrt(2x)
MCQ
Quick Quiz
Which transformation makes the graph of y = x^2 appear wider?
y = 2x^2
y = (x/3)^2
y = (3x)^2
y = -x^2
The Correct Answer Is:
B
Multiplying the entire function by a number less than 1 (like 1/3) or dividing x by a number greater than 1 (like x/3) causes the graph to widen. Options A and C make it narrower, and D reflects it.
Real World Connection
In the Real World
Imagine a sound engineer using software to adjust the 'pitch' or 'timbre' of a song. They are often applying stretches or compressions to the sound waves (which can be represented as graphs) to make the sound higher/lower or longer/shorter. In cricket analytics, if you plot a batsman's scoring rate over time, a horizontal stretch might show a slower scoring phase, while a vertical stretch might show a sudden burst of boundaries.
Key Vocabulary
Key Terms
TRANSFORMATION: A change made to a graph's position, size, or shape. | VERTICAL STRETCH/COMPRESSION: Changes the height of the graph. | HORIZONTAL STRETCH/COMPRESSION: Changes the width of the graph. | FACTOR: The number by which a stretch or compression occurs.
What's Next
What to Learn Next
Great job learning about stretches and compressions! Next, you should explore 'Translations on a Generic Graph'. This will teach you how to shift graphs up/down or left/right, which, combined with stretches, will give you a complete picture of graph transformations!
