Horizontal Compression Of Y=1/x By Factor Of 6 A Detailed Explanation

In the realm of mathematics, specifically within the study of functions and their transformations, understanding how different operations affect the graph of a function is crucial. Transformations can involve shifts, stretches, compressions, and reflections, each altering the function's graph in a predictable manner. Among these transformations, horizontal compression is a key concept that allows us to manipulate the graph of a function along the x-axis. This article delves into the specifics of horizontal compression, using the function y = 1/x as a primary example. We will explore how different algebraic manipulations result in the compression of this function and analyze the correct answer to the question: "Which results only in a horizontal compression of y = 1/x by a factor of 6?"

Defining Horizontal Compression

Before diving into the specifics, let's define what horizontal compression means in mathematical terms. A horizontal compression occurs when the graph of a function is squeezed towards the y-axis. This transformation effectively reduces the x-coordinates of the points on the graph by a certain factor. Mathematically, if we have a function y = f(x), a horizontal compression by a factor of k (where k > 1) is achieved by replacing x with kx in the function's equation, resulting in y = f(kx). This means that the new function's graph will appear to be compressed horizontally compared to the original function's graph. The larger the value of k, the greater the compression. Understanding this fundamental principle is crucial for identifying and applying horizontal compressions correctly.

Analyzing the Base Function: y = 1/x

The function y = 1/x, also known as the reciprocal function, is a classic example often used to illustrate transformations. Its graph is a hyperbola, consisting of two branches that lie in the first and third quadrants. The function has vertical and horizontal asymptotes at x = 0 and y = 0, respectively. These asymptotes are critical features of the graph and play a role in how the graph behaves under transformations. The original function y = 1/x serves as our baseline for comparison. When we apply a horizontal compression, we are essentially altering the x-values while observing the corresponding changes in the y-values. This change in x-values will visually compress the graph towards the y-axis. To better understand how this function transforms under compression, it's helpful to consider specific points on the graph. For instance, the point (1, 1) lies on the graph of y = 1/x. Under a horizontal compression, the x-coordinate of this point will change, while the y-coordinate may remain the same or change depending on the specific transformation applied.

Applying Horizontal Compression to y = 1/x

To compress the function y = 1/x horizontally by a factor of 6, we need to apply the rule x → 6x. This means we replace x in the original equation with 6x. So, the transformed function becomes y = 1/(6x). This transformation compresses the graph of y = 1/x towards the y-axis, making it appear narrower horizontally. Each point on the original graph is effectively squeezed closer to the y-axis by a factor of 6. For example, the point (1, 1) on the original graph is transformed to (1/6, 1) on the compressed graph. This illustrates how the x-coordinate is reduced by a factor of 6, while the y-coordinate remains the same. This new function, y = 1/(6x), is the result of a pure horizontal compression, without any other transformations like reflections or vertical stretches. Recognizing this direct application of the compression factor is key to correctly answering the problem. Let's examine why the other options are incorrect.

Analyzing the Given Options

Now, let's analyze the given options in the context of horizontal compression and determine which one correctly represents a horizontal compression of y = 1/x by a factor of 6.

Option A: y = 1/(6x)

As we discussed earlier, replacing x with 6x in the original function y = 1/x results in y = 1/(6x). This equation represents a pure horizontal compression by a factor of 6. The graph is squeezed towards the y-axis, and the x-coordinates of the points on the graph are reduced by a factor of 6. There are no other transformations involved, such as reflections or vertical stretches. Therefore, this option correctly represents the desired transformation. This makes option A the correct answer. This is a direct application of the horizontal compression transformation rule, making it a clear and concise representation of the compressed function. The absence of any other operations, such as a negative sign or a multiplier in the numerator, confirms that this is solely a horizontal compression.

Option B: y = -1/(6x)

This option introduces a negative sign in front of the function, changing the equation to y = -1/(6x). While the 6x in the denominator still indicates a horizontal compression by a factor of 6, the negative sign represents a reflection across the x-axis. This means that the graph is not only compressed horizontally but also flipped vertically. The combination of horizontal compression and reflection makes this option incorrect because it does not represent a pure horizontal compression. The graph of this function would be the horizontal compression of the original graph, followed by a reflection over the x-axis. This added reflection changes the fundamental shape and position of the graph, making it different from a simple horizontal compression.

Option C: y = 6/x

In this option, the equation is y = 6/x. This can be rewritten as y = 6 * (1/x). This form clearly shows that this transformation is a vertical stretch by a factor of 6. A vertical stretch affects the y-coordinates of the points on the graph, pulling the graph away from the x-axis. In this case, each y-coordinate is multiplied by 6, making the graph taller but not compressed horizontally. Since the question specifically asks for a horizontal compression, this option is incorrect. The graph of y = 6/x would appear to be stretched vertically compared to the original graph, not compressed horizontally.

Option D: y = -6/x

This option, y = -6/x, combines a vertical stretch by a factor of 6 (as in option C) with a reflection across the x-axis (as in option B). The equation can be seen as y = -6 * (1/x). The multiplication by 6 stretches the graph vertically, and the negative sign reflects it across the x-axis. This combination of transformations does not result in a pure horizontal compression. Therefore, this option is also incorrect. The graph of y = -6/x would be both stretched vertically and reflected over the x-axis, resulting in a different shape compared to a horizontally compressed graph.

Conclusion: The Correct Transformation

In conclusion, the only option that results solely in a horizontal compression of y = 1/x by a factor of 6 is Option A: y = 1/(6x). This transformation directly applies the horizontal compression rule by replacing x with 6x, without introducing any other transformations like reflections or stretches. Understanding the individual effects of different transformations is crucial for correctly manipulating and interpreting functions in mathematics. This analysis underscores the importance of recognizing how algebraic changes to a function's equation translate into graphical transformations. By systematically evaluating each option, we can confidently identify the one that accurately represents the desired horizontal compression. This exercise not only provides the correct answer but also reinforces a deeper understanding of function transformations and their mathematical representations.

Which equation represents a horizontal compression of the function y = 1/x by a factor of 6?

Horizontal Compression of y=1/x by Factor of 6 A Detailed Explanation