Function Transformations Horizontal Stretch And Reflection Of F(x)

In the realm of mathematics, understanding how functions transform is crucial for solving a wide array of problems. Transformations allow us to manipulate the graph of a function, changing its size, shape, and position in the coordinate plane. Among the most common transformations are horizontal stretches and reflections across the x-axis. This article provides a detailed exploration of these transformations, focusing on how they affect the algebraic representation of a function. Specifically, we will dissect the question of how the function f(x) changes when stretched horizontally by a factor of 2 and reflected across the x-axis. This topic is not just a theoretical exercise; it is a fundamental concept with applications in various fields, including physics, engineering, and computer graphics. Mastering function transformations enhances your ability to analyze and model real-world phenomena mathematically.

Horizontal Stretching

Horizontal stretching is a transformation that alters the width of a function's graph. When we say a function f(x) is stretched horizontally by a factor of k, where k > 1, it means the graph is widened away from the y-axis. Conversely, if 0 < k < 1, the graph is compressed towards the y-axis. The key to understanding horizontal stretches lies in how the input x is affected. To stretch f(x) horizontally by a factor of k, we replace x with x/k. This seemingly counterintuitive substitution is what causes the horizontal dilation. Let's consider a simple example to illustrate this. Suppose we have the function f(x) = x². If we want to stretch this function horizontally by a factor of 2, we replace x with x/2, resulting in a new function g(x) = (x/2)² = x²/4. Notice how the graph of g(x) is wider than the graph of f(x). Each point on the original graph is moved twice as far from the y-axis. The horizontal stretch affects the x-coordinates of the points on the graph, while the y-coordinates remain unchanged. This is a crucial distinction from vertical stretches, where the y-coordinates are altered. Understanding the relationship between the stretch factor k and the resulting transformation is essential for accurately manipulating functions. Whether you are working with polynomials, trigonometric functions, or exponential functions, the principle of replacing x with x/k holds true for horizontal stretches. This concept is fundamental in various applications, such as image processing, where stretching and compressing images are common operations. In essence, horizontal stretching allows us to manipulate the domain of a function, providing a powerful tool for analyzing and transforming mathematical models.

Reflection Across the x-axis

A reflection across the x-axis is a transformation that flips the graph of a function over the x-axis. This means that for every point (x, y) on the original graph, there is a corresponding point (x, -y) on the reflected graph. The y-coordinates change sign, while the x-coordinates remain the same. To achieve a reflection across the x-axis algebraically, we multiply the entire function by -1. That is, if we start with a function f(x), the reflection across the x-axis is represented by -f(x). This simple operation effectively flips the graph vertically. For example, consider the function f(x) = x³. Its reflection across the x-axis is -f(x) = -x³. The original function has a positive slope in the first quadrant and a negative slope in the third quadrant, while the reflected function has a negative slope in the first quadrant and a positive slope in the third quadrant. This visual flip is the essence of reflection across the x-axis. This transformation is particularly useful in understanding the symmetry of functions. Even functions, which are symmetric about the y-axis, remain unchanged when reflected across the x-axis if they are also symmetric about the origin (odd functions). Odd functions, on the other hand, change sign when reflected across the x-axis. This property is a direct consequence of their symmetry about the origin. The reflection across the x-axis is a fundamental transformation that helps us analyze and manipulate functions. It is widely used in various fields, including physics, where reflections are used to model phenomena such as the reflection of light or sound waves. In computer graphics, reflections are used to create mirror images and other visual effects. Therefore, mastering the concept of reflection across the x-axis is crucial for a comprehensive understanding of function transformations. Remember, multiplying the function by -1 is the key to achieving this transformation, effectively flipping the graph vertically and changing the sign of the y-coordinates.

Combining transformations allows us to create more complex manipulations of functions. When applying multiple transformations, the order in which they are performed matters. In general, horizontal stretches and compressions should be applied before reflections and vertical stretches/compressions. This is because horizontal transformations affect the input x, while vertical transformations affect the output f(x). Let's consider the specific case of stretching a function f(x) horizontally by a factor of 2 and then reflecting it across the x-axis. First, to stretch f(x) horizontally by a factor of 2, we replace x with x/2, resulting in f(x/2). This widens the graph of the function. Next, to reflect this stretched function across the x-axis, we multiply the entire function by -1. Therefore, the final transformed function is -f(x/2). This combined transformation effectively stretches the graph horizontally and then flips it over the x-axis. Understanding the order of operations is crucial. If we were to reflect f(x) across the x-axis first, we would get -f(x). Then, stretching this reflected function horizontally by a factor of 2 would result in -f(x/2), which is the same as our previous result. However, in general, changing the order of transformations can lead to different outcomes. For instance, if we had a vertical stretch instead of a horizontal stretch, the order would significantly impact the final function. Let's illustrate this with an example. Suppose f(x) = √x. Stretching it horizontally by a factor of 2 gives f(x/2) = √(x/2). Reflecting this across the x-axis gives -√(x/2). If we reflect first, we get -√x, and then stretching horizontally gives -√(x/2), which is the same. However, if we had a vertical stretch by a factor of 2 instead, stretching horizontally first would give 2*√(x/2), while reflecting first and then stretching vertically would give -2√x*. These are different functions. Thus, the principle of applying horizontal transformations before vertical ones is a valuable guideline. Combining transformations like horizontal stretching and reflections allows for versatile manipulation of functions, but careful attention to the order of operations is essential. This understanding is crucial in various mathematical and applied contexts, where complex transformations are used to model and analyze diverse phenomena.

Analyzing the Given Choices

Now, let's apply our understanding of transformations to analyze the given choices and determine the correct representation of f(x) after being stretched horizontally by a factor of 2 and reflected across the x-axis. The question asks us to identify the correct algebraic expression for the transformed function. We know that a horizontal stretch by a factor of 2 involves replacing x with x/2, resulting in f(x/2). We also know that a reflection across the x-axis involves multiplying the entire function by -1, resulting in -f(x). Combining these transformations, we first stretch horizontally to get f(x/2) and then reflect across the x-axis to get -f(x/2). Now let's examine the given choices:

A. -f(1/2 x) B. f(-1/2 x) C. -f(2x) D. f(-2x)

Choice A, -f(1/2 x), correctly represents a horizontal stretch by a factor of 2 (replacing x with x/2) and a reflection across the x-axis (multiplying the function by -1). This matches our derived transformation -f(x/2). Therefore, choice A is the correct answer. Choice B, f(-1/2 x), represents a horizontal stretch by a factor of 2 combined with a reflection across the y-axis (replacing x with -x). This is not the transformation we are looking for. Choice C, -f(2x), represents a horizontal compression by a factor of 2 (replacing x with 2x) and a reflection across the x-axis. This is also not the correct transformation. Choice D, f(-2x), represents a horizontal compression by a factor of 2 combined with a reflection across the y-axis. Again, this does not match our desired transformation. By systematically applying the rules of transformations and carefully analyzing each choice, we can confidently identify the correct answer. This process highlights the importance of understanding the individual effects of each transformation and how they combine to alter the function's graph and algebraic representation. Therefore, careful analysis is crucial when dealing with function transformations.

In conclusion, understanding transformations of functions is a fundamental concept in mathematics with wide-ranging applications. This article has provided a detailed exploration of horizontal stretching and reflections across the x-axis, demonstrating how these transformations affect the algebraic representation of a function. We have seen that stretching f(x) horizontally by a factor of 2 involves replacing x with x/2, and reflecting across the x-axis involves multiplying the function by -1. Combining these transformations, we arrive at the expression -f(x/2). Analyzing the given choices, we identified that -f(1/2 x) correctly represents the combined transformation of a horizontal stretch by a factor of 2 and a reflection across the x-axis. This analysis underscores the importance of understanding the order of operations and the individual effects of each transformation. The ability to manipulate functions through transformations is not only a valuable mathematical skill but also a powerful tool for modeling and analyzing real-world phenomena. Whether you are working with graphical representations or algebraic expressions, a solid grasp of function transformations will enhance your problem-solving capabilities. Mastering these concepts is essential for success in advanced mathematics and related fields. By understanding how functions behave under various transformations, you can gain deeper insights into their properties and applications. This knowledge will empower you to tackle more complex problems and appreciate the elegance and versatility of mathematical functions. Therefore, investing time in understanding function transformations is a worthwhile endeavor that will pay dividends in your mathematical journey.