What is a Stretch/Compression on y = x^2? | Simple Explanation for Class 10

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What is a Stretch/Compression on y = x²?

Grade Level:

Class 10

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

Definition

What is it?

A stretch or compression on the graph of y = x^2 changes how 'wide' or 'narrow' the parabola looks. It happens when we multiply the x^2 term by a number, making the graph either stretch upwards (become narrower) or compress downwards (become wider) compared to the original y = x^2.

Simple Example

Quick Example

Imagine you have a rubber band shaped like y = x^2. If you pull it upwards from the ends, it becomes thinner and taller – that's a stretch. If you push it down from the sides, it becomes fatter and shorter – that's a compression. This is similar to how the graph of y = x^2 changes when you multiply it by a number.

Worked Example

Step-by-Step

Let's compare y = x^2 and y = 3x^2 to see a stretch.

Step 1: Create a table of values for y = x^2.
If x = -2, y = (-2)^2 = 4
If x = -1, y = (-1)^2 = 1
If x = 0, y = (0)^2 = 0
If x = 1, y = (1)^2 = 1
If x = 2, y = (2)^2 = 4

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Step 2: Create a table of values for y = 3x^2.
If x = -2, y = 3*(-2)^2 = 3*4 = 12
If x = -1, y = 3*(-1)^2 = 3*1 = 3
If x = 0, y = 3*(0)^2 = 3*0 = 0
If x = 1, y = 3*(1)^2 = 3*1 = 3
If x = 2, y = 3*(2)^2 = 3*4 = 12

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Step 3: Compare the y-values for the same x-values.
For x=1, y=x^2 gives y=1, but y=3x^2 gives y=3. The y-value is 3 times larger.
For x=2, y=x^2 gives y=4, but y=3x^2 gives y=12. The y-value is 3 times larger.

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Step 4: Conclusion. Since the y-values for y = 3x^2 are 3 times greater than for y = x^2 (except at x=0), the graph of y = 3x^2 is stretched vertically, making it look narrower than y = x^2.

Why It Matters

Understanding stretches and compressions helps in AI/ML to scale data for better model performance, and in Physics to model how springs stretch or compress under force. Engineers use this to design structures that can withstand various loads, ensuring safety and efficiency in buildings and bridges.

Common Mistakes

MISTAKE: Thinking a large number multiplying x^2 makes the parabola wider. | CORRECTION: A number greater than 1 multiplying x^2 makes the parabola narrower (a vertical stretch).

MISTAKE: Confusing a vertical stretch/compression with a horizontal one. | CORRECTION: For y = ax^2, 'a' directly affects the vertical stretch/compression. Horizontal changes are different transformations.

MISTAKE: Assuming negative 'a' values also cause a stretch/compression. | CORRECTION: A negative 'a' value (like in y = -2x^2) reflects the parabola across the x-axis. The stretch/compression effect is determined by the absolute value of 'a'.

Practice Questions

Try It Yourself

QUESTION: Describe the transformation from y = x^2 to y = 0.5x^2. | ANSWER: The graph of y = x^2 is compressed vertically (it becomes wider).

QUESTION: Which graph is narrower: y = 4x^2 or y = 2x^2? Explain why. | ANSWER: y = 4x^2 is narrower because the coefficient 4 is greater than 2, causing a greater vertical stretch.

QUESTION: If the graph of y = x^2 passes through the point (3, 9), what point will the graph of y = 2x^2 pass through for x=3? What kind of transformation is this? | ANSWER: For x=3, y = 2*(3)^2 = 2*9 = 18. So, the point is (3, 18). This is a vertical stretch.

MCQ

Quick Quiz

Which of the following equations represents a parabola that is wider than y = x^2?

y = 5x^2

y = 2x^2

y = 0.2x^2

y = -3x^2

The Correct Answer Is:

A parabola y = ax^2 is wider (compressed) if the absolute value of 'a' is between 0 and 1. Here, 0.2 is between 0 and 1. Options A, B, and D (considering |-3|=3) all represent narrower parabolas.

Real World Connection

In the Real World

In cricket analytics, data scientists use parabolas to model the trajectory of a ball after a shot. If a batsman hits the ball harder, the 'a' value in the parabolic path equation might change, showing a stretch or compression that affects how high or far the ball travels, helping coaches analyze performance.

Key Vocabulary

Key Terms

PARABOLA: The U-shaped graph of a quadratic equation like y = x^2 | COEFFICIENT: The number multiplying the variable (e.g., 'a' in y = ax^2) | VERTICAL STRETCH: When the graph becomes narrower, y-values increase faster | VERTICAL COMPRESSION: When the graph becomes wider, y-values increase slower

What's Next

What to Learn Next

Next, explore 'Translations of y = x^2' to learn how to move the parabola up/down or left/right. This will help you understand all basic transformations and graph any quadratic function.