Transformations Of Cube Root Functions How Y=∛x Changes To Y=∛(1/2 X)

In mathematics, understanding function transformations is crucial for analyzing and manipulating graphs effectively. A parent function serves as the foundational graph from which other related graphs are derived through transformations such as stretches, compressions, reflections, and shifts. In this comprehensive guide, we will delve into the specific transformation applied to the parent cube root function, $y = \sqrt[3]{x}$, to produce the graph of $y = \sqrt[3]{\frac{1}{2}x}$. We will explore the concept of horizontal stretches and compressions, and how they affect the original graph. By the end of this discussion, you will have a clear understanding of how to identify and describe these transformations, enabling you to analyze and manipulate various functions with confidence.

The cube root function is a fundamental concept in algebra and calculus, and understanding its transformations is essential for various applications in mathematics and other fields. The parent function, $y = \sqrt[3]{x}$, has a distinct shape that extends infinitely in both the positive and negative directions. Transforming this parent function involves altering its graph by stretching, compressing, reflecting, or shifting it. Among these transformations, horizontal stretches and compressions are particularly important, as they change the graph's width relative to the y-axis. A horizontal stretch expands the graph away from the y-axis, while a horizontal compression squeezes it closer to the y-axis. Recognizing these transformations is key to interpreting and manipulating graphs of cube root functions effectively. Throughout this guide, we will dissect the specific transformation applied in the given problem, providing a step-by-step explanation and illustrative examples to solidify your understanding.

To fully grasp the transformation from $y = \sqrt[3]{x}$ to $y = \sqrt[3]{\frac{1}{2}x}$, we need to understand the concept of horizontal stretches. A horizontal stretch or compression occurs when the input variable, $x$, is multiplied by a constant inside the function. The general form for this transformation is $y = f(kx)$, where $k$ is a constant. If $|k| < 1$, the graph is stretched horizontally by a factor of $\frac{1}{|k|}$. Conversely, if $|k| > 1$, the graph is compressed horizontally by a factor of $\frac{1}{|k|}$. In our case, the function is transformed from $y = \sqrt[3]{x}$ to $y = \sqrt[3]{\frac{1}{2}x}$. Here, $k = \frac{1}{2}$, which is less than 1. Therefore, the graph undergoes a horizontal stretch. The stretch factor is $\frac{1}{|\frac{1}{2}|} = 2$. This means the graph of $y = \sqrt[3]{\frac{1}{2}x}$ is stretched horizontally by a factor of 2 compared to the graph of $y = \sqrt[3]{x}$. This concept is crucial for accurately interpreting and applying transformations to various functions. In the following sections, we will further explore the specific details of this transformation and compare it to other possible transformations.

When we analyze the transformation from the parent function $y = \sqrt[3]{x}$ to the transformed function $y = \sqrt[3]{\frac{1}{2}x}$, the key lies in understanding how the input variable $x$ is affected. The transformation involves replacing $x$ with $\frac{1}{2}x$ inside the cube root function. This type of modification directly impacts the horizontal aspect of the graph. To determine whether this represents a stretch or a compression, we must consider the coefficient of $x$, which in this case is $\frac{1}{2}$. As mentioned earlier, when the coefficient of $x$ is between 0 and 1 (i.e., $|k| < 1$), the transformation results in a horizontal stretch. The factor by which the graph is stretched is the reciprocal of this coefficient. In our scenario, the coefficient is $\frac{1}{2}$, so the stretch factor is $\frac{1}{\frac{1}{2}} = 2$. This means that the graph of $y = \sqrt[3]{\frac{1}{2}x}$ is stretched horizontally by a factor of 2 relative to the graph of $y = \sqrt[3]{x}$. This horizontal stretch effectively widens the graph along the x-axis, making it appear broader than the parent function.

To further illustrate this horizontal stretch, consider specific points on the parent function $y = \sqrt[3]{x}$. For instance, the point (8, 2) lies on the parent function because $\sqrt[3]{8} = 2$. Now, let's see how this point transforms on the new function $y = \sqrt[3]{\frac{1}{2}x}$. To achieve the same y-value of 2, we need to solve the equation $\sqrt[3]{\frac{1}{2}x} = 2$. Cubing both sides gives $\frac{1}{2}x = 8$, and multiplying by 2 yields $x = 16$. Thus, the corresponding point on the transformed graph is (16, 2). Notice that the x-coordinate has doubled from 8 to 16, which confirms the horizontal stretch by a factor of 2. Similarly, if we consider the point (-8, -2) on the parent function, we find that the corresponding point on the transformed function is (-16, -2), again demonstrating the horizontal stretch. By examining several such points, the effect of the transformation becomes clear: the graph is being stretched away from the y-axis, widening it horizontally. This detailed analysis of specific points provides a concrete understanding of the horizontal stretch and its impact on the graph.

In contrast to the horizontal stretch, it's important to clarify what other types of transformations might look like and why they are not applicable in this case. For instance, a vertical stretch or compression would involve multiplying the entire function by a constant, such as $y = a\sqrt[3]{x}$. This would stretch or compress the graph vertically, altering its height rather than its width. A vertical stretch by a factor of $a$ (where $a > 1$) would make the graph taller, while a vertical compression (where $0 < a < 1$) would make it shorter. However, in our given transformation, the constant $\frac{1}{2}$ is inside the cube root function, directly affecting the input $x$, rather than the output $y$. This distinction is crucial for identifying the correct type of transformation. Additionally, reflections and shifts involve different modifications to the function. A reflection over the y-axis would involve replacing $x$ with $-x$, resulting in $y = \sqrt[3]{-x}$, while a reflection over the x-axis would involve negating the entire function, resulting in $y = -\sqrt[3]{x}$. Shifts, on the other hand, involve adding or subtracting a constant to either $x$ (horizontal shift) or the entire function (vertical shift). Understanding these alternative transformations helps in accurately diagnosing and describing the specific transformation applied in a given problem. In our case, the presence of $\frac{1}{2}x$ inside the cube root clearly indicates a horizontal stretch.

To reinforce the correct answer and prevent misunderstandings, it's crucial to explain why the other options presented are incorrect. Let's examine the options: "It is vertically stretched by a factor of $\frac{1}{2}$" and "It is compressed horizontally by a factor of $\frac{1}{2}$." The first incorrect option, a vertical stretch by a factor of $\frac{1}{2}$, would be represented by the equation $y = \frac{1}{2}\sqrt[3]{x}$. In this scenario, the entire cube root function is multiplied by $\frac{1}{2}$, which would compress the graph vertically, making it shorter but not wider. This is fundamentally different from the transformation we are analyzing, where the constant $\frac{1}{2}$ is inside the cube root function, directly impacting the input $x$. A vertical stretch or compression affects the y-values, while our transformation affects the x-values, indicating a horizontal change. Therefore, this option is incorrect because it misidentifies the direction of the transformation.

The second incorrect option, a horizontal compression by a factor of $\frac{1}{2}$, might seem plausible at first glance, but it misinterprets the effect of the constant inside the function. A horizontal compression would occur if we had a function of the form $y = \sqrt[3]{kx}$, where $|k| > 1$. For example, $y = \sqrt[3]{2x}$ would represent a horizontal compression by a factor of $\frac{1}{2}$. In our case, we have $y = \sqrt[3]{\frac{1}{2}x}$, where the coefficient of $x$ is $\frac{1}{2}$, which is less than 1. As discussed earlier, this indicates a horizontal stretch, not a compression. A compression would squeeze the graph towards the y-axis, making it narrower, whereas our transformation stretches the graph away from the y-axis, making it wider. Thus, understanding the reciprocal relationship between the coefficient of $x$ and the stretch/compression factor is crucial. Mistaking a stretch for a compression, or vice versa, can lead to incorrect interpretations of function transformations. By carefully analyzing the position of the constant and its value relative to 1, we can accurately identify the type and direction of the transformation.

In summary, the transformation of the parent function $y = \sqrt[3]{x}$ to the graph of $y = \sqrt[3]{\frac{1}{2}x}$ involves a horizontal stretch by a factor of 2. This is because the $x$ variable inside the cube root function is multiplied by $\frac{1}{2}$, and the reciprocal of this factor, which is 2, determines the extent of the horizontal stretch. Understanding this transformation requires a clear grasp of how coefficients within a function affect its graph's horizontal and vertical dimensions. Horizontal stretches and compressions occur when the input variable $x$ is multiplied by a constant, while vertical stretches and compressions involve multiplying the entire function by a constant. Recognizing the distinction between these types of transformations is essential for accurately interpreting and manipulating functions in mathematics. By mastering these concepts, you can effectively analyze and describe various transformations applied to different types of functions.

To further solidify your understanding, it is beneficial to practice identifying transformations on other parent functions, such as quadratic functions, exponential functions, and trigonometric functions. Each type of function has its own characteristic shape, and understanding how these shapes change under various transformations is a fundamental skill in mathematics. For instance, consider the quadratic function $y = x^2$. A horizontal stretch would be represented by $y = (\frac{1}{2}x)^2$, which stretches the parabola horizontally. Similarly, an exponential function like $y = 2^x$ can be transformed horizontally by changing the exponent, such as $y = 2^{\frac{1}{2}x}$. By exploring different functions and their transformations, you will develop a deeper intuition for how changes in the equation correspond to changes in the graph. This comprehensive approach will enhance your problem-solving abilities and your overall understanding of function transformations.

In conclusion, the correct answer is that the graph of $y = \sqrt[3]{x}$ is horizontally stretched by a factor of 2 to produce the graph of $y = \sqrt[3]{\frac{1}{2}x}$. This understanding is crucial for anyone studying algebra, calculus, or any field that involves mathematical modeling and analysis. By focusing on the position and value of the constant within the function, you can accurately identify and describe horizontal stretches and compressions. This skill is not only valuable for answering specific questions about function transformations but also for developing a deeper understanding of mathematical relationships and their graphical representations. Remember, practice and careful analysis are key to mastering these concepts and applying them effectively in various mathematical contexts.