Point On Graph G(x) = F(2x) If F(2,4)

by Alex Johnson 38 views

Let's dive into the world of exponential functions and transformations! This article will walk you through finding a point on a transformed exponential function, given a point on the original function. We'll break down the steps with a clear example, so you can tackle similar problems with confidence. Let's consider the question: If the graph of exponential function f passes through the point (2,4), and g(x) = f(2x), which point lies on the graph of function g?

Understanding Exponential Functions and Transformations

Before we jump into the solution, let's recap some key concepts. Exponential functions are functions where the variable appears in the exponent, generally expressed in the form f(x) = ax, where a is a constant base. These functions exhibit rapid growth or decay, depending on the value of a. A transformation of a function involves altering its graph by shifting, stretching, compressing, or reflecting it. In our case, we're dealing with a horizontal compression, where the input to the function is multiplied by a constant. This article delves into finding points on transformed exponential functions, where understanding these functions and their behavior is paramount. The question at hand involves an exponential function f that passes through a specific point, and we're tasked with identifying a corresponding point on a transformed function g(x) = f(2x). This transformation involves a horizontal compression, and to solve this, we need to understand how this compression affects the coordinates of points on the graph. By understanding the nature of exponential functions and the effects of transformations, we can accurately determine which point lies on the graph of the transformed function g. This requires careful consideration of how the input and output values change under the transformation.

Solving the Problem: Step-by-Step

So, let's solve the problem step by step. Our main keyword here is the point on the transformed exponential functions. We're given that f(2) = 4. This is our starting point, the heart of the matter. We want to find a point (x, y) that satisfies g(x) = f(2x). Remember, g(x) is the transformed function. The key here is to recognize the transformation happening to the input x. In g(x) = f(2x), the input x is being multiplied by 2 before it's fed into the function f. This multiplication inside the function's argument indicates a horizontal compression. The graph of g(x) is the graph of f(x) compressed horizontally by a factor of 2. To find a point on the graph of g, we need to think about how this compression affects the x-coordinate. If f(2) = 4, we want to find an x value such that 2x = 2. Why? Because if 2x = 2, then g(x) = f(2x) = f(2) = 4. It's like we're trying to reverse the compression to match the known point on f. Solving 2x = 2 gives us x = 1. This is a crucial step. We've found the x-coordinate for a point on g. Now we know that when x = 1, g(1) = f(2 * 1) = f(2) = 4. Therefore, the point (1, 4) lies on the graph of the function g. This step-by-step approach highlights the importance of understanding transformations and how they affect the coordinates of points on a function's graph. By carefully analyzing the transformation and working through the equations, we can confidently identify the correct point on the transformed function.

The Solution and Why It Works

Therefore, the answer is B. (1,4). This is the point that lies on the graph of function g. The main reason for this is because g(x)=f(2x), when x = 1, we have g(1) = f(21) = f(2) = 4*. Let’s solidify why this solution works. The transformation g(x) = f(2x) represents a horizontal compression of the graph of f(x). This means that the x-coordinates are effectively halved. If a point (a, b) lies on the graph of f(x), then the point (a/2, b) will lie on the graph of g(x). We know (2, 4) is on f(x). Applying the compression, we divide the x-coordinate by 2: 2 / 2 = 1. The y-coordinate remains the same. Thus, (1, 4) is on g(x). This principle underscores the nature of horizontal compressions: they squeeze the graph towards the y-axis, altering the x-coordinates while preserving the y-coordinates. Understanding this concept is vital for solving such problems efficiently and accurately. Furthermore, this approach reinforces the link between transformations and their graphical effects, providing a solid foundation for more advanced mathematical concepts. The step-by-step breakdown allows for a clear comprehension of the problem-solving process, making it easier to apply the same logic to other similar problems involving function transformations.

Common Pitfalls to Avoid

When dealing with function transformations, several common pitfalls can trip up even seasoned problem-solvers. Let's highlight some key areas to watch out for. A frequent mistake is confusing horizontal and vertical transformations. Remember, transformations inside the function's argument (like the 2x in f(2x)) affect the x-coordinates, leading to horizontal stretches or compressions. Transformations outside the function (like adding a constant to f(x)) affect the y-coordinates, leading to vertical shifts or stretches. Another error is misinterpreting the direction of the transformation. For horizontal compressions, a factor greater than 1 (like our 2) compresses the graph, not stretches it. It's counterintuitive but crucial. Similarly, for horizontal stretches, a factor between 0 and 1 stretches the graph. Another common area of concern is the order of transformations. If multiple transformations are applied, the order matters. For instance, a horizontal compression followed by a vertical shift will yield a different result than a vertical shift followed by a horizontal compression. Therefore, careful attention to the order is vital. Finally, students often make algebraic errors when solving for the new coordinates. Double-checking your calculations is always a good practice. Avoiding these pitfalls requires a solid grasp of transformation principles and careful attention to detail in the problem-solving process. By understanding these common mistakes, you can significantly improve your accuracy and confidence in tackling function transformation problems.

Practice Problems

To really nail down your understanding of points on the graph of transformed exponential functions, practice is key. Let's look at a couple of example problems. These practice problems are crucial for reinforcing the concepts discussed and for developing problem-solving skills in a practical context. They provide an opportunity to apply the learned principles and identify any areas that may require further review. Practice problems help solidify the understanding of how horizontal transformations affect the coordinates of points on a graph and improve the ability to accurately determine corresponding points on transformed functions. Furthermore, engaging with practice problems enhances critical thinking and analytical skills, which are essential for success in mathematics. By working through these examples, you can gain confidence in your ability to handle similar problems and deepen your understanding of function transformations. Let's begin:

  1. The graph of an exponential function h passes through the point (3, 9). If k(x) = h(3x), what point is on the graph of function k?
  2. The graph of an exponential function p passes through the point (-1, 2). If q(x) = p(4x), what point is on the graph of function q?

Try solving these problems using the steps we discussed earlier. Remember to focus on the horizontal compression and how it affects the x-coordinate. These exercises will provide hands-on experience and help solidify your grasp of the topic. Consistent practice is essential for mastery, and these problems offer a valuable opportunity to reinforce the concepts and techniques discussed in this article.

Conclusion

Finding points on transformed exponential functions might seem tricky at first, but with a clear understanding of transformations and a systematic approach, it becomes much more manageable. Remember to focus on how the transformation affects the x-coordinate, and you'll be well on your way to solving these types of problems with ease. This article has provided a step-by-step guide, highlighting the key concepts and potential pitfalls to avoid. By practicing with example problems, you can further solidify your understanding and gain confidence in your problem-solving abilities. Remember, mathematics is a skill that improves with practice, so keep exploring and challenging yourself! For further exploration of exponential functions and their applications, you might find resources at Khan Academy's Exponential Functions Section helpful. This link provides a valuable external resource for deepening your understanding of exponential functions.