Finding G(x) After Translating F(x) = X^2
Introduction
In the realm of mathematics, understanding the transformations of functions is a fundamental concept. Function transformations allow us to manipulate the graph of a function by shifting, stretching, compressing, or reflecting it. This article delves into a specific type of transformation: translations. We will explore how the function f(x) = x² is translated 9 units up and 4 units to the right to form a new function, g(x). Our goal is to determine the equation that represents g(x). This involves understanding how horizontal and vertical translations affect the original function's equation. By carefully analyzing the transformations, we can accurately identify the correct representation of g(x) from the given options.
Vertical Translations
Vertical translations involve shifting the graph of a function upwards or downwards along the y-axis. When a function f(x) is translated vertically, a constant is added to or subtracted from the function's output. If we add a positive constant, the graph shifts upwards, and if we subtract a positive constant, the graph shifts downwards. In mathematical terms, if we want to translate f(x) k units upwards, the new function becomes f(x) + k. Conversely, if we want to translate f(x) k units downwards, the new function becomes f(x) - k. Understanding vertical translations is crucial because they directly affect the y-values of the function, essentially moving the entire graph up or down without changing its shape or orientation. The constant k acts as a direct modifier to the function's output, providing a straightforward way to control the vertical positioning of the graph.
In our specific problem, the function f(x) = x² is translated 9 units up. This means that every point on the graph of f(x) is shifted 9 units vertically upwards. Applying the principle of vertical translations, we add 9 to the function's output, resulting in f(x) + 9. Therefore, the intermediate function after the vertical translation is x² + 9. This vertical shift changes the vertex of the parabola from (0, 0) to (0, 9), indicating that the entire parabola has moved upwards along the y-axis. Understanding this vertical transformation is a key step in determining the final equation of g(x), as it addresses one of the two translations applied to the original function.
Horizontal Translations
Horizontal translations, on the other hand, involve shifting the graph of a function left or right along the x-axis. These translations are somewhat counterintuitive because the direction of the shift is opposite to what the sign might suggest. Specifically, if we want to translate f(x) h units to the right, the new function becomes f(x - h). Notice that we subtract h from the input x. Conversely, if we want to translate f(x) h units to the left, the new function becomes f(x + h), where we add h to the input x. This inverse relationship between the direction of the shift and the sign of h is a critical aspect of horizontal translations. Horizontal translations affect the x-values of the function, causing the graph to slide along the x-axis without changing its vertical position or shape. The value of h determines the magnitude of the shift, and its sign determines the direction.
In the given problem, the function f(x) = x² is translated 4 units to the right. Following the rules of horizontal translations, we need to replace x with (x - 4) in the function. This means that the intermediate function x² + 9 (after the vertical translation) now becomes (x - 4)² + 9. This horizontal shift moves the vertex of the parabola 4 units to the right, changing its x-coordinate from 0 to 4. The combination of the vertical and horizontal translations results in a new parabola that is both shifted upwards and to the right compared to the original function f(x) = x². Understanding the effect of horizontal translations, particularly the sign convention, is essential for accurately determining the final equation of the transformed function.
Combining Vertical and Horizontal Translations
Combining both vertical and horizontal translations allows us to fully manipulate the position of a function's graph in the coordinate plane. By applying these transformations sequentially, we can move the graph both up or down and left or right. In our case, we first translated f(x) = x² 9 units up and then 4 units to the right. As we established earlier, translating f(x) 9 units up results in the function x² + 9. Subsequently, translating this new function 4 units to the right involves replacing x with (x - 4), leading to the final transformed function g(x) = (x - 4)² + 9. This combined transformation shifts the vertex of the parabola from (0, 0) to (4, 9), indicating a movement both horizontally and vertically. Understanding how to combine these translations is crucial for solving a wide range of problems involving function transformations.
In general, if a function f(x) is translated k units vertically and h units horizontally, the transformed function g(x) can be represented as g(x) = f(x - h) + k. This formula encapsulates the combined effect of both types of translations. The horizontal translation is represented by the (x - h) term, and the vertical translation is represented by the + k term. By correctly applying this formula, we can easily determine the equation of a transformed function after any combination of vertical and horizontal shifts. This understanding is not only essential for solving mathematical problems but also for visualizing and analyzing the behavior of functions under various transformations.
Determining the Correct Representation of g(x)
Now that we have a clear understanding of vertical and horizontal translations, we can definitively determine the correct representation of g(x). We started with the function f(x) = x², translated it 9 units up to get x² + 9, and then translated it 4 units to the right to get (x - 4)² + 9. This step-by-step process allows us to construct the equation of the transformed function accurately. By carefully tracking the effect of each translation, we can avoid common errors and arrive at the correct answer.
Comparing our derived equation, g(x) = (x - 4)² + 9, with the given options, we can see that it matches option A. The other options represent different transformations or incorrect applications of the translation principles. Option B, g(x) = (x + 4)² + 9, represents a translation 4 units to the left instead of to the right. Option C, g(x) = (x + 9)² - 4, and option D, g(x) = (x + 9)² + 4, incorrectly apply both the horizontal and vertical translations. Therefore, by systematically applying the principles of function transformations, we can confidently identify option A as the correct representation of g(x).
Conclusion
In conclusion, understanding function transformations, particularly translations, is essential for manipulating and analyzing functions effectively. By breaking down the transformations into vertical and horizontal components, we can systematically determine the equation of the transformed function. In this case, translating f(x) = x² 9 units up and 4 units to the right results in the function g(x) = (x - 4)² + 9. This process involves applying the principles of vertical translations (f(x) + k) and horizontal translations (f(x - h)) in the correct order. The key takeaway is that horizontal translations are counterintuitive, requiring the subtraction of the translation value from x for a rightward shift and addition for a leftward shift. By mastering these concepts, we can confidently tackle more complex problems involving function transformations and gain a deeper understanding of their graphical representations.
This problem highlights the importance of precise application of transformation rules and the ability to combine different types of transformations. The correct representation of g(x), (x - 4)² + 9, accurately reflects both the vertical and horizontal shifts applied to the original function f(x) = x². This underscores the significance of understanding the underlying principles of function transformations in mathematics.
