Vertically Compressed By A Factor Of 1/2

Vertically Compressed by a Factor of 1/2: A Deep Dive into Transformations

Understanding transformations in mathematics, particularly those involving scaling, is crucial for a strong foundation in various fields like geometry, calculus, and computer graphics. This article explores the concept of vertical compression, specifically focusing on a compression factor of 1/2. We'll delve into the mathematical principles, illustrative examples, and practical applications of this transformation. This comprehensive guide will provide a clear and intuitive understanding of how a graph or function is affected when vertically compressed by a factor of 1/2.

Introduction: Understanding Vertical Compression

In the world of functions and graphs, transformations allow us to manipulate the visual representation of a function without altering its fundamental properties. One such transformation is vertical compression, which essentially "squishes" the graph towards the x-axis. When we say a graph is vertically compressed by a factor of 1/2, it means every y-coordinate is halved, resulting in a graph that's half as tall as the original. This transformation affects the vertical scale of the graph, making it appear shorter and wider.

The Mathematical Principle: Transforming the Function

Let's consider a function f(x). To vertically compress this function by a factor of 1/2, we apply the transformation:

g(x) = (1/2)f(x)

This simple equation encapsulates the entire process. Every y-value of the original function f(x) is multiplied by 1/2 to obtain the corresponding y-value in the compressed function g(x). This means that if a point (a, b) lies on the graph of f(x), then the point (a, b/2) will lie on the graph of g(x). The x-coordinates remain unchanged; only the y-coordinates are affected by the compression.

Step-by-Step Guide: Applying the Transformation

Let's walk through a practical example to solidify our understanding. Consider the function f(x) = x². This is a simple parabola. To vertically compress it by a factor of 1/2, we follow these steps:

1. Identify the original function: We have f(x) = x².

2. Apply the compression factor: We multiply the function by 1/2: g(x) = (1/2)f(x) = (1/2)x².

3. Analyze the transformation: The new function g(x) = (1/2)x² represents the vertically compressed parabola. Every y-value of the original parabola is now half its original value. For example, if f(2) = 4, then g(2) = 2.

4. Visualize the change: By plotting points or using graphing software, you can visually observe how the parabola has been compressed vertically. The parabola retains its basic shape (a parabola), but its height is reduced. The vertex remains at the origin (0,0).

Illustrative Examples: Various Function Types

The vertical compression by a factor of 1/2 applies to various function types. Let's examine a few more examples:

• Linear Function: Consider f(x) = 2x + 1. Applying the compression, we get g(x) = (1/2)(2x + 1) = x + 1/2. The slope becomes half its original value, and the y-intercept is also halved.

• Exponential Function: Let f(x) = eˣ. The compressed function becomes g(x) = (1/2)eˣ. The exponential growth is slower; the curve is closer to the x-axis than the original function.

• Trigonometric Function: Take f(x) = sin(x). The compressed function is g(x) = (1/2)sin(x). The amplitude of the sine wave is reduced to half its original value (from 1 to 1/2).

In each case, the fundamental characteristics of the function are preserved; however, the vertical scaling is altered. The shape remains similar, but its vertical extent is compressed.

Visual Representation: Graphing the Transformation

The most effective way to understand vertical compression is through visual representation. Graphing both the original function and its compressed counterpart reveals the transformation clearly. You can use graphing calculators or software like Desmos or GeoGebra to plot these functions and see the effect of the compression firsthand. Observe how the y-values change while the x-values remain constant. This visual comparison will reinforce your understanding of the transformation.

Explaining the Transformation with Calculus

From a calculus perspective, the vertical compression affects the derivative and integral of the function. Let's consider the derivative:

If g(x) = (1/2)f(x), then g'(x) = (1/2)f'(x). The derivative of the compressed function is simply half the derivative of the original function at each point. This means the slope of the tangent line at any point on the compressed graph is half the slope at the corresponding point on the original graph.

Similarly, the integral is also affected:

The definite integral of g(x) from a to b will be half the definite integral of f(x) from a to b. This signifies that the area under the compressed curve is half the area under the original curve between the same limits.

Applications in Real-World Scenarios

The concept of vertical compression has numerous applications in various fields:

• Computer Graphics: Image scaling and resizing often involve vertical compression (and horizontal compression/scaling). Reducing the height of an image while maintaining the aspect ratio is a direct application.

• Physics: In modeling physical phenomena, scaling factors are frequently used. Vertical compression might represent a reduction in amplitude of a wave or a decrease in the height of an object.

• Engineering: Designing scaled-down models of structures often involves vertical compression to maintain proportions.

• Economics: Economic models may use scaling factors to adjust parameters, and vertical compression could represent a scaling down of production or consumption.

Frequently Asked Questions (FAQ)

• Q: What happens if the compression factor is greater than 1? A: If the factor is greater than 1, it becomes a vertical stretch, not a compression. The graph is elongated vertically.

• Q: Can vertical compression be combined with other transformations? A: Yes, vertical compression can be combined with other transformations like horizontal shifts, vertical shifts, and horizontal compressions/stretches. The order of operations matters when applying multiple transformations.

• Q: Does vertical compression change the x-intercepts? A: No, vertical compression does not affect the x-intercepts (roots) of the function. The x-coordinates where the graph intersects the x-axis remain unchanged.

• Q: What is the difference between vertical compression and horizontal compression? A: Vertical compression affects the y-coordinates, scaling the graph vertically. Horizontal compression affects the x-coordinates, scaling the graph horizontally. They are distinct transformations with different effects on the graph.

Conclusion: Mastering Vertical Compression

Understanding vertical compression by a factor of 1/2, and transformations in general, is a fundamental concept in mathematics with far-reaching applications. By grasping the mathematical principles, applying the transformation step-by-step, and visualizing the results through graphs, you can develop a comprehensive understanding of this crucial concept. Remember, practice is key. Experiment with different functions and observe the effects of vertical compression to solidify your understanding. This thorough understanding will prove invaluable as you progress in your mathematical journey and encounter more complex concepts. Remember that mastering this fundamental transformation will build a strong base for understanding more advanced topics in mathematics and related fields.