Transforming Quadratic Functions Mapping F(x) Onto G(x)
In the fascinating realm of mathematics, transformations play a pivotal role in understanding the relationships between different functions. These transformations, like shifts and stretches, allow us to visualize how one function can be mapped onto another. This article delves into a captivating problem involving the functions f(x) = (x - 7)² - 1 and g(x) = (x + 6)² - 3. Our mission is to uncover the specific transformation that will seamlessly map the graph of f(x) onto the graph of g(x). This exploration will not only enhance our understanding of function transformations but also provide a valuable problem-solving experience.
Understanding the Functions f(x) and g(x)
To embark on this journey, we must first dissect the nature of the given functions. Both f(x) = (x - 7)² - 1 and g(x) = (x + 6)² - 3 are quadratic functions, characterized by their parabolic shapes when graphed. The general form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. The vertex is a crucial point as it signifies the minimum or maximum value of the function and serves as a reference point for transformations.
For the function f(x) = (x - 7)² - 1, we can identify that a = 1, h = 7, and k = -1. This tells us that the parabola opens upwards (since a is positive) and has its vertex at the point (7, -1). Similarly, for the function g(x) = (x + 6)² - 3, we have a = 1, h = -6, and k = -3. This parabola also opens upwards and has its vertex at (-6, -3). Understanding these vertices is key to determining the transformation that maps f(x) onto g(x). The vertex form of a quadratic equation is instrumental in visualizing transformations, as it directly reveals the horizontal and vertical shifts applied to the parent function, x². By analyzing the changes in the vertex coordinates, we can precisely identify the required translation to map one parabola onto another. Further, the coefficient a dictates the vertical stretch or compression of the parabola, but in this case, both functions have a = 1, simplifying our analysis to focus solely on the shifts. Analyzing the properties of these quadratic functions, including their vertices and the direction they open, forms the foundation for identifying the precise transformation required to map f(x) onto g(x). The vertex form not only aids in visualization but also provides a direct mathematical link between the equations and their graphical representations, making the transformation process more intuitive and understandable.
Identifying the Transformation
The task at hand is to determine the transformation that shifts the graph of f(x) to coincide perfectly with the graph of g(x). Function transformations involve altering the position, size, or orientation of a graph. Common transformations include translations (shifts), reflections, stretches, and compressions. In this specific problem, we are focusing on translations, which involve moving the graph horizontally and/or vertically without changing its shape or size.
To find the transformation, we need to analyze how the vertex of f(x), which is (7, -1), must be moved to reach the vertex of g(x), which is (-6, -3). The horizontal shift can be found by calculating the difference in the x-coordinates of the vertices: -6 - 7 = -13. This means the graph needs to be shifted 13 units to the left. The vertical shift can be found by calculating the difference in the y-coordinates of the vertices: -3 - (-1) = -2. This indicates a shift of 2 units downwards. Therefore, the transformation that maps f(x) onto g(x) is a shift of 13 units to the left and 2 units down. Understanding these shifts is crucial for visualizing the transformation process and confirming the answer. Each unit of shift represents a direct translation of the graph along the coordinate plane, maintaining its shape and orientation. By focusing on the movement of the vertex, which serves as a fixed point on the parabola, we can accurately determine the required horizontal and vertical translations. Furthermore, this approach highlights the connection between the algebraic representation of the functions and their geometric transformations, reinforcing the fundamental principles of function analysis. This step-by-step approach ensures clarity and precision in identifying the correct transformation.
Selecting the Correct Option
Now that we've determined the required transformation, we can compare it to the given options. We found that the graph of f(x) needs to be shifted 13 units to the left and 2 units down to map onto the graph of g(x). Let's examine the options:
A. Right 13 units, up 2 units B. Left 13 units, down 2 units
Comparing our findings with the options, we can see that option B, "Left 13 units, down 2 units," accurately describes the transformation we calculated. Therefore, option B is the correct answer. The other options either suggest shifts in the opposite direction or incorrect magnitudes of shifts. This highlights the importance of precise calculation and attention to detail in these types of problems. Selecting the correct option involves a careful comparison between the derived transformation and the provided choices, ensuring that both the direction and magnitude of the shifts are accurately matched. The process of elimination can also be used to rule out incorrect options, further reinforcing the correctness of the selected answer. This final step solidifies the understanding of the transformation and its application in mapping one function onto another. Choosing the correct answer is a culmination of the analytical steps taken, from understanding the functions to identifying the transformation, and confirms the successful navigation of the problem.
Verifying the Transformation
To ensure our solution is correct, we can verify the transformation algebraically. We know that f(x) = (x - 7)² - 1 and g(x) = (x + 6)² - 3. If we apply the transformation of shifting 13 units to the left and 2 units down to f(x), we should obtain g(x). Shifting 13 units to the left means replacing x with (x + 13) in f(x). Shifting 2 units down means subtracting 2 from the entire function. So, the transformed function, let's call it f'(x), would be:
f'(x) = ((x + 13) - 7)² - 1 - 2
Simplifying this, we get:
f'(x) = (x + 6)² - 3
This is exactly the same as g(x), which confirms that our transformation is correct. This algebraic verification provides a robust confirmation of our graphical and analytical approach. By applying the transformation to the function's equation and simplifying, we demonstrate that the transformed function is indeed identical to the target function. This process not only validates the solution but also reinforces the understanding of how transformations affect the algebraic representation of a function. Furthermore, this verification step underscores the interconnectedness between algebraic and graphical representations in mathematics, showcasing how they can be used in conjunction to solve problems and confirm solutions. The ability to verify solutions algebraically is a valuable skill in mathematics, fostering confidence in the accuracy of the results.
Conclusion
In this exploration, we successfully determined the transformation that maps the graph of f(x) = (x - 7)² - 1 onto the graph of g(x) = (x + 6)² - 3. By analyzing the vertices of the parabolas and calculating the horizontal and vertical shifts, we found that a translation of 13 units to the left and 2 units down is required. This transformation was then verified algebraically, confirming the accuracy of our solution. Understanding function transformations is crucial in mathematics as it allows us to analyze and relate different functions effectively. This problem serves as a valuable example of how transformations can be identified and applied to map one function onto another. The process involved not only mathematical calculations but also visualization and analytical reasoning, highlighting the multifaceted nature of mathematical problem-solving. The ability to identify transformations is a fundamental skill that extends beyond specific problem types and is applicable in various areas of mathematics and its applications. This exercise reinforces the importance of understanding function properties, algebraic manipulations, and graphical interpretations in solving mathematical problems. Moreover, it emphasizes the significance of verifying solutions to ensure accuracy and build confidence in one's mathematical abilities. This comprehensive approach to problem-solving is a key takeaway from this exploration.
