Exponential Functions And Transformations Finding Points On Transformed Graphs
In the fascinating world of mathematics, exponential functions hold a special place, describing phenomena that grow or decay at a rate proportional to their current value. Understanding exponential functions is crucial in various fields, from finance and biology to physics and computer science. This article delves into the properties of exponential functions, focusing on how transformations affect their graphs. Specifically, we will explore how a horizontal compression impacts the coordinates of points on the graph of an exponential function. The problem at hand involves an exponential function f that passes through the point (2,4). We are then introduced to a new function g(x) = f(2x), which represents a transformation of f. Our goal is to determine which point lies on the graph of g. This problem allows us to explore the relationship between the input and output values of functions and how transformations alter these relationships. By understanding these concepts, we can gain a deeper appreciation for the behavior of exponential functions and their applications.
Understanding Exponential Functions
At their core, exponential functions are characterized by a constant base raised to a variable exponent. The general form of an exponential function is f(x) = a^x, where a is a positive constant not equal to 1. The value of a determines whether the function represents exponential growth (if a > 1) or exponential decay (if 0 < a < 1). The graph of an exponential function has a distinctive shape, either increasing rapidly (for growth) or decreasing rapidly towards zero (for decay). Key features of exponential function graphs include a horizontal asymptote at y = 0 and a y-intercept at (0,1). These fundamental properties form the basis for understanding more complex exponential functions and their transformations. When we talk about transformations of exponential functions, we're essentially looking at how we can shift, stretch, compress, or reflect the basic exponential graph. These transformations can be represented by modifying the function's equation. For instance, a vertical shift can be achieved by adding a constant to the function, while a horizontal shift can be achieved by adding or subtracting a constant from the input variable x. The most relevant transformation for our problem is the horizontal compression, which occurs when we multiply the input variable x by a constant. This transformation affects the x-coordinates of points on the graph, bringing them closer to the y-axis. Understanding how these transformations work is crucial for solving problems involving exponential functions and their graphs.
Problem Setup: Function f and the Point (2,4)
The problem introduces us to an exponential function f that passes through the point (2,4). This means that when x = 2, the function value f(2) is equal to 4. This crucial piece of information serves as the foundation for solving the problem. We can express this mathematically as f(2) = 4. This single point on the graph provides us with a specific relationship between the input and output of the function f. It allows us to think about the general form of f(x) = a^x and consider what value of the base a would satisfy this condition. While we don't necessarily need to determine the exact form of f(x) to solve the problem, understanding this relationship is essential. The fact that f(2) = 4 tells us something significant about the exponential function f. It implies that the base a, when raised to the power of 2, results in 4. This limits the possibilities for the base a and provides a concrete anchor point for our analysis. This understanding is crucial for visualizing the graph of f and how it might be transformed. The next step in the problem involves introducing a new function g(x), which is defined in terms of f(x). This is where the concept of function transformations comes into play, and the point (2,4) serves as a reference for understanding how the transformation affects the graph.
Introducing Function g and the Transformation
The problem introduces a new function, g(x) = f(2x). This function g is defined in terms of the original function f, but with a crucial difference: the input variable x is multiplied by 2 before being passed into the function f. This transformation, f(2x), represents a horizontal compression of the graph of f. A horizontal compression occurs when the input variable x is multiplied by a constant greater than 1. In this case, the constant is 2, which means the graph of g is compressed horizontally by a factor of 2 compared to the graph of f. This compression affects the x-coordinates of points on the graph. For any point (x, y) on the graph of f, the corresponding point on the graph of g will have an x-coordinate that is half of the original. Understanding this compression is key to determining which point lies on the graph of g. We know that f(2) = 4. To find a point on the graph of g, we need to find an x-value such that g(x) equals a specific value. Since g(x) = f(2x), we are looking for an x-value that, when multiplied by 2, gives us an input value for f that we already know. This connection between f and g is crucial for solving the problem. By carefully considering the transformation and the known point on the graph of f, we can deduce a corresponding point on the graph of g. This process highlights the importance of understanding how transformations affect function graphs and how to use this knowledge to solve problems.
Finding a Point on the Graph of g
Now, let's find a point on the graph of the function g. We know that g(x) = f(2x), and we also know that f(2) = 4. To utilize this information, we need to find an x-value such that the input to f in the expression f(2x) becomes 2. In other words, we need to solve the equation 2x = 2. Dividing both sides of the equation by 2, we get x = 1. This is a crucial step, as it connects the known value of f(2) to the function g. Since x = 1 satisfies the condition 2x = 2, we can substitute this value into the expression for g(x): g(1) = f(2 * 1) = f(2). We already know that f(2) = 4, so we can conclude that g(1) = 4. This tells us that the point (1, 4) lies on the graph of the function g. We have successfully found a point on the graph of g by leveraging the information about the point on the graph of f and the transformation that defines g. This process demonstrates the power of understanding function transformations and how they affect the coordinates of points on a graph. The horizontal compression in this case squeezed the graph of f towards the y-axis, effectively halving the x-coordinate of the point where the function reaches a certain value. This understanding of transformations is crucial for analyzing and manipulating functions in various mathematical contexts. Now, let's confirm our solution by examining the given answer choices and selecting the one that matches the point we found.
Solution: Identifying the Correct Point
Having determined that the point (1,4) lies on the graph of the function g, we can now compare this result to the given answer choices. The answer choices are:
A. (2,2) B. (4,4) C. (2,8) D. (1,4)
By direct comparison, we see that option D, (1,4), matches the point we calculated. Therefore, the correct answer is D. This confirms our understanding of the transformation and our ability to apply it to find a specific point on the transformed graph. The process of elimination could also be used here. We know that the horizontal compression by a factor of 2 will affect the x-coordinate. Since the original point on f was (2,4), we expect the x-coordinate on g to be smaller. This immediately rules out options B and C, which have x-coordinates greater than 2. Option A has an x-coordinate of 2, but the y-coordinate is different from the known value of 4, so it can also be ruled out. This leaves us with option D, which matches our calculated point. This exercise highlights the importance of both calculation and logical reasoning in solving mathematical problems. By combining our understanding of function transformations with careful analysis of the given information, we were able to efficiently arrive at the correct solution. The key takeaway is that transformations alter the coordinates of points on a graph in a predictable way, and by understanding these patterns, we can solve a variety of problems.
Conclusion
In conclusion, this problem provided a valuable opportunity to explore the properties of exponential functions and the impact of transformations on their graphs. By understanding the relationship between the functions f and g, and specifically the horizontal compression, we were able to determine that the point (1,4) lies on the graph of g. This exercise reinforces the importance of understanding function transformations and how they affect the coordinates of points on a graph. Furthermore, it highlights the power of using known information about a function to deduce properties of its transformed counterpart. The problem-solving approach involved a combination of analytical reasoning and direct calculation, demonstrating the multifaceted nature of mathematical problem-solving. By carefully considering the given information, applying the concept of horizontal compression, and verifying our solution against the answer choices, we successfully navigated the problem and arrived at the correct answer. This understanding of exponential functions and transformations is not only crucial for academic success in mathematics but also has broader applications in various scientific and engineering fields. The ability to analyze and manipulate functions is a fundamental skill that empowers us to model and understand real-world phenomena that exhibit exponential behavior. This problem serves as a stepping stone towards more complex mathematical concepts and their applications, fostering a deeper appreciation for the beauty and power of mathematics.
