Finding The Transformed Point P: A Step-by-Step Guide

Let's dive into a fascinating problem involving function transformations and coordinate geometry! We're given a point A(3, 2) that lies on the graph of a function f. A transformation then maps the graph of f to the graph of a new function g, defined as g(x) = (1/2)f(x-1). Crucially, the same transformation also maps point A to a new point P, and our mission is to find the coordinates of this transformed point P. This problem combines concepts of function transformations, coordinate geometry, and a bit of algebraic thinking, making it an excellent exercise in mathematical problem-solving. So, let's break it down step by step and unveil the solution together! This requires us to understand how transformations affect points on a graph, particularly horizontal shifts and vertical compressions. We'll need to carefully analyze the given transformation g(x) = (1/2)f(x-1) and apply it to the original point A. By understanding the effects of the transformation on the x and y coordinates, we can determine the precise location of point P. This exploration will not only solve the problem at hand but also deepen our understanding of function transformations in general. Remember, mathematics is not just about finding the right answer; it's about the journey of understanding the underlying principles and concepts. This problem is a perfect example of how different mathematical ideas connect and interact, creating a richer and more meaningful learning experience. So, let's embark on this mathematical adventure together and discover the coordinates of point P!

Understanding the Transformation

To find the coordinates of point P, we first need to dissect the transformation that maps f to g. The equation g(x) = (1/2)f(x-1) reveals two key transformations: a horizontal shift and a vertical compression. Let's break these down individually to fully grasp their impact. The term (x-1) inside the function f indicates a horizontal shift. Specifically, it represents a shift of the graph one unit to the right. Imagine the original graph of f; to obtain the graph corresponding to f(x-1), we essentially slide the entire graph one unit along the positive x-axis. This means every point on the original graph shifts one unit to the right. Now, consider the 1/2 multiplying the entire function f(x-1). This represents a vertical compression by a factor of 1/2. In simpler terms, it's like squishing the graph vertically towards the x-axis. Every y-coordinate on the graph is halved. Understanding these two transformations is crucial because they directly affect the coordinates of the points on the graph. A horizontal shift changes the x-coordinate, while a vertical compression changes the y-coordinate. By carefully tracking these changes, we can accurately determine the final position of point P after the transformation. This meticulous approach is essential in mathematics, where precision and clarity are paramount. We're not just blindly applying a formula; we're understanding the underlying principles that govern the transformation. This understanding allows us to tackle similar problems with confidence and develop a deeper appreciation for the elegance and logic of mathematical transformations. The ability to visualize and interpret these transformations is a valuable skill that extends beyond this specific problem and enhances our overall mathematical intuition. So, let's keep these concepts in mind as we move forward and apply them to find the coordinates of point P.

Applying the Transformation to Point A

Now that we've thoroughly understood the transformation g(x) = (1/2)f(x-1), let's apply it to point A(3, 2) to find the coordinates of point P. Remember, the transformation consists of two parts: a horizontal shift of one unit to the right and a vertical compression by a factor of 1/2. First, consider the horizontal shift. Since point A has an x-coordinate of 3, shifting it one unit to the right will increase the x-coordinate by 1. This means the new x-coordinate will be 3 + 1 = 4. So, after the horizontal shift, our point has temporarily moved to (4, 2). Next, we need to apply the vertical compression. The y-coordinate of point A is 2. Compressing it vertically by a factor of 1/2 means we multiply the y-coordinate by 1/2. This gives us a new y-coordinate of 2 * (1/2) = 1. Therefore, after the vertical compression, the y-coordinate becomes 1. Combining both transformations, we see that point A(3, 2) is first shifted horizontally to (4, 2) and then compressed vertically to (4, 1). This final point, (4, 1), is the location of point P. This step-by-step application of the transformation highlights the importance of understanding each component individually before combining them. By breaking down the complex transformation into simpler parts, we can accurately track the changes in the coordinates and avoid errors. This methodical approach is a hallmark of effective mathematical problem-solving. We're not just blindly following steps; we're understanding the underlying logic and applying it systematically. This understanding empowers us to tackle more challenging problems with confidence and clarity. So, with the coordinates of point P successfully determined, we've reached a significant milestone in our problem-solving journey. Let's now consolidate our understanding and appreciate the beauty of this mathematical transformation.

The Coordinates of Point P

After carefully analyzing the transformation and applying it step-by-step to point A(3, 2), we've successfully determined the coordinates of point P. As we discussed earlier, the transformation g(x) = (1/2)f(x-1) involves a horizontal shift of one unit to the right and a vertical compression by a factor of 1/2. Applying the horizontal shift to point A(3, 2) moved it to (4, 2), increasing the x-coordinate by 1. Subsequently, applying the vertical compression multiplied the y-coordinate by 1/2, transforming (4, 2) into (4, 1). Therefore, the coordinates of point P are (4, 1). This seemingly simple result encapsulates the essence of function transformations and their impact on points in the coordinate plane. It demonstrates how a combination of shifts and compressions can alter the position of a point while maintaining its relative relationship to the transformed graph. This understanding is fundamental in various areas of mathematics, including calculus, linear algebra, and differential equations. The ability to visualize and manipulate transformations is a powerful tool for solving a wide range of problems. Furthermore, this problem highlights the importance of precision and clarity in mathematical thinking. Each step must be carefully considered and executed to arrive at the correct answer. A small error in applying the transformation can lead to a completely different result. This underscores the need for a methodical approach and a deep understanding of the underlying principles. So, the coordinates of point P, (4, 1), are not just a numerical answer; they represent the culmination of our understanding of function transformations and our ability to apply them effectively. This achievement is a testament to our problem-solving skills and our growing mathematical maturity.

Conclusion

In conclusion, by meticulously analyzing the transformation g(x) = (1/2)f(x-1) and applying it step-by-step to point A(3, 2), we've successfully determined that the coordinates of point P are (4, 1). This journey has reinforced our understanding of horizontal shifts and vertical compressions, key components of function transformations. We've seen how a combination of these transformations can alter the position of a point in the coordinate plane, and we've developed a methodical approach to track these changes accurately. This problem serves as a valuable example of how different mathematical concepts intertwine and how a clear understanding of these concepts is crucial for effective problem-solving. The ability to visualize and manipulate transformations is a fundamental skill in mathematics, and this exercise has provided us with a practical application of this skill. Furthermore, we've emphasized the importance of precision and clarity in mathematical thinking. Each step must be carefully considered and executed to arrive at the correct answer, highlighting the need for a systematic approach and a deep understanding of the underlying principles. The coordinates of point P, (4, 1), represent more than just a numerical answer; they signify our successful application of mathematical principles and our growing problem-solving abilities. This experience equips us to tackle more complex challenges with confidence and clarity. We encourage you to explore further into the world of function transformations and discover their applications in various mathematical contexts. There's a wealth of knowledge and understanding to be gained from delving deeper into this fascinating area of mathematics. For more information on function transformations, consider visiting Khan Academy's section on Transformations of Functions. This resource provides comprehensive explanations, examples, and practice problems to further enhance your understanding.