Vertical vs. Horizontal Stretches vs. Compressions: What's the Difference?

📚 Understanding Transformations: Vertical vs. Horizontal Stretches vs. Compressions

In mathematics, transformations alter the size or position of a graph. Stretches and compressions specifically affect the dimensions of a function's graph, either vertically or horizontally. Let's dive into each of these transformations.

📈 Vertical Stretches and Compressions

A vertical stretch or compression affects the y-values of a function. The graph is stretched or compressed away from or towards the x-axis.

• 📏 Vertical Stretch: Occurs when you multiply the function by a constant greater than 1. If $y = f(x)$, then $y = af(x)$ where $a > 1$ is a vertical stretch.
• 📉 Vertical Compression: Occurs when you multiply the function by a constant between 0 and 1. If $y = f(x)$, then $y = af(x)$ where $0 < a < 1$ is a vertical compression.

↔️ Horizontal Stretches and Compressions

A horizontal stretch or compression affects the x-values of a function. The graph is stretched or compressed away from or towards the y-axis.

• 📐 Horizontal Compression: Occurs when you multiply the x-value inside the function by a constant greater than 1. If $y = f(x)$, then $y = f(bx)$ where $b > 1$ is a horizontal compression.
• растянуть Horizontal Stretch: Occurs when you multiply the x-value inside the function by a constant between 0 and 1. If $y = f(x)$, then $y = f(bx)$ where $0 < b < 1$ is a horizontal stretch.

📊 Comparison Table: Vertical vs. Horizontal

🔑 Key Takeaways

• 💡 Vertical transformations affect the output (y-values) of the function, while horizontal transformations affect the input (x-values).
• 📝 Stretches increase the dimension along the corresponding axis, while compressions decrease it.
• 🧮 Understanding these transformations is crucial for analyzing and manipulating functions in algebra and calculus.