Function Transformations Horizontal Stretch And Reflection
#f(x)# is stretched horizontally by a factor of 2 and reflected across the #x-axis#. This article delves into the process of identifying the correct representation of #f(x)# after these transformations. Function transformations are a fundamental concept in mathematics, particularly in algebra and calculus, allowing us to manipulate the graph of a function in various ways, including stretching, compressing, reflecting, and shifting. Understanding these transformations is crucial for analyzing and interpreting functions and their graphs. In this article, we will focus on two specific transformations: horizontal stretching and reflection across the x-axis. We will explore how these transformations affect the equation of a function and how to represent them mathematically. By understanding the rules and principles behind function transformations, you'll be better equipped to solve mathematical problems, analyze graphical representations, and apply these concepts in various fields such as physics, engineering, and computer science. Let's start by defining what horizontal stretching and reflection across the x-axis mean and how they visually affect a function's graph.
Horizontal Stretch and Reflection Across the X-Axis: A Detailed Explanation
In the realm of function transformations, horizontal stretch and reflection across the x-axis are two key concepts that manipulate the graph of a function in distinct ways. A horizontal stretch by a factor of #k# (where #k > 1#) expands the graph away from the y-axis. This means that every x-coordinate on the original graph is multiplied by #k#. Conversely, a horizontal compression (or shrink) occurs when #0 < k < 1#, squeezing the graph towards the y-axis. Think of it as stretching or compressing the graph like an elastic band horizontally. The larger the value of #k#, the more the graph is stretched horizontally.
On the other hand, a reflection across the x-axis flips the graph vertically. This transformation takes every point #(x, y)# on the original graph and maps it to #(x, -y)#. In simpler terms, the part of the graph above the x-axis is mirrored below it, and vice versa. This transformation is achieved by multiplying the entire function by -1. So, if you have a function #f(x)#, reflecting it across the x-axis results in #-f(x)#. Understanding the difference between horizontal stretching and reflection across the x-axis is crucial for accurately interpreting and manipulating functions. These transformations can be combined, creating more complex transformations that require careful attention to the order in which they are applied. In the next section, we will explore how to represent these transformations mathematically, which will help us solve the given problem.
Mathematical Representation of Transformations
To accurately represent function transformations, it's essential to understand how they translate into mathematical notation. When a function #f(x)# is stretched horizontally by a factor of 2, the transformation affects the input #x#. To achieve a horizontal stretch by a factor of 2, we replace #x# with #\frac{1}{2}x# in the function. This might seem counterintuitive, but remember that a horizontal stretch means the graph is being pulled away from the y-axis. Therefore, the function's argument needs to change more slowly, which is achieved by multiplying #x# by a fraction less than 1.
So, the horizontally stretched function becomes #f(\frac{1}{2}x)#. This transformation essentially doubles the x-values, causing the graph to stretch horizontally. Now, let's consider the reflection across the x-axis. As mentioned earlier, reflecting a function across the x-axis involves multiplying the entire function by -1. This changes the sign of the y-values, effectively flipping the graph over the x-axis. Therefore, to reflect #f(x)# across the x-axis, we obtain #-f(x)#.
When these transformations are combined, the order in which they are applied matters. In this case, #f(x)# is first stretched horizontally and then reflected across the x-axis. This means we first apply the horizontal stretch, resulting in #f(\frac{1}{2}x)#, and then apply the reflection, which means multiplying the entire transformed function by -1. Combining these steps gives us #-f(\frac{1}{2}x)#. Understanding these mathematical representations is crucial for correctly interpreting and applying function transformations. In the following sections, we'll delve into how to apply these concepts to the given problem and identify the correct answer choice.
Applying Transformations to the Given Problem
The original question asks us to identify the correct representation of the function #f(x)# after it undergoes two transformations: a horizontal stretch by a factor of 2 and a reflection across the x-axis. As we've discussed, a horizontal stretch by a factor of 2 is achieved by replacing #x# with #\frac{1}{2}x# in the function, resulting in #f(\frac{1}{2}x)#. Following this, a reflection across the x-axis involves multiplying the entire function by -1. Thus, we take #f(\frac{1}{2}x)# and multiply it by -1, which gives us #-f(\frac{1}{2}x)#. Now, let's analyze the given answer choices:
- A. #-f(\frac{1}{2} x)#
- B. #f(-2 x)#
- C. #f(-\frac{1}{2} x)#
- D. #-f(2 x)#
By comparing our derived expression #-f(\frac{1}{2}x)# with the answer choices, we can clearly see that option A, #-f(\frac{1}{2} x)#, matches our result. The other options represent different transformations or apply the transformations in the wrong order. For instance, option B, #f(-2 x)#, represents a horizontal compression by a factor of 2 and a reflection across the y-axis, not the x-axis. Option C, #f(-\frac{1}{2} x)#, represents a horizontal stretch by a factor of 2 and a reflection across the y-axis. Option D, #-f(2 x)#, represents a horizontal compression by a factor of 2 and a reflection across the x-axis. Therefore, only option A correctly represents the given transformations.
Step-by-Step Solution and Explanation
To further solidify our understanding, let's break down the solution step-by-step:
- Identify the transformations: The problem states that #f(x)# is stretched horizontally by a factor of 2 and reflected across the x-axis.
- Represent the horizontal stretch: A horizontal stretch by a factor of 2 means replacing #x# with #\frac{1}{2}x#. This transforms #f(x)# into #f(\frac{1}{2}x)#.
- Represent the reflection across the x-axis: Reflecting a function across the x-axis means multiplying the entire function by -1. So, we multiply #f(\frac{1}{2}x)# by -1, resulting in #-f(\frac{1}{2}x)#.
- Combine the transformations: Applying the horizontal stretch first and then the reflection, we get #-f(\frac{1}{2}x)#.
- Match with the answer choices: Comparing our result with the given options, we find that option A, #-f(\frac{1}{2} x)#, is the correct representation.
This step-by-step approach ensures that we correctly apply each transformation in the proper order. By understanding the individual transformations and their mathematical representations, we can confidently solve similar problems. It's crucial to remember the order in which transformations are applied, as changing the order can lead to different results. In this case, the horizontal stretch is applied before the reflection across the x-axis.
Conclusion: Mastering Function Transformations
In conclusion, the correct representation of #f(x)# after a horizontal stretch by a factor of 2 and a reflection across the x-axis is #-f(\frac{1}{2} x)#, which corresponds to option A. This problem highlights the importance of understanding function transformations and their mathematical representations. By breaking down the transformations into individual steps and applying them in the correct order, we can accurately determine the resulting function.
Function transformations are a fundamental concept in mathematics and have wide-ranging applications in various fields. Mastering these transformations allows us to analyze and manipulate functions effectively, providing valuable insights into their behavior and properties. Whether you're studying algebra, calculus, or any other related field, a solid understanding of function transformations is essential for success. Remember, practice is key to mastering these concepts. Work through various examples, try different combinations of transformations, and challenge yourself to apply these concepts in different contexts. With consistent effort and a clear understanding of the principles, you'll be well-equipped to tackle any function transformation problem that comes your way.
By understanding the underlying principles and practicing regularly, you can master function transformations and apply them effectively in various mathematical contexts. This problem serves as a great example of how breaking down complex transformations into simpler steps can lead to a clear and accurate solution.
