Finding Corresponding Points On The Transformed Function F(-2/3 X)

Hey guys! Let's dive into a cool problem about transformations of functions. We're given a point on the graph of a function f(x), and we want to figure out the corresponding point on the graph of a transformed version of that function, specifically f(-2/3 x). This might sound a bit intimidating at first, but don't worry, we'll break it down step by step so it's super easy to understand.

Understanding Function Transformations

Before we jump into the specific problem, let's quickly review what function transformations are all about. Function transformations are ways we can modify the graph of a function by shifting, stretching, compressing, or reflecting it. These transformations are often represented by changes to the function's equation. Let's consider a few common transformations to make sure we're all on the same page. Firstly, consider vertical shifts. Adding a constant to a function, like f(x) + c, shifts the graph vertically. If c is positive, the graph moves up, and if c is negative, it moves down. These shifts don't change the shape of the graph, just its vertical position. Secondly, let’s talk about horizontal shifts. Changes inside the function's argument, like f(x + c), cause horizontal shifts. It’s important to remember that these shifts work in the opposite direction of the sign. So, f(x + c) shifts the graph to the left, and f(x - c) shifts it to the right. These shifts also maintain the shape of the graph, only altering its horizontal position. Next, consider vertical stretches and compressions. Multiplying a function by a constant, a f(x), stretches or compresses the graph vertically. If |a| > 1, the graph is stretched vertically, making it taller. If 0 < |a| < 1, the graph is compressed vertically, making it shorter. The graph is also reflected across the x-axis if a is negative. Lastly, we have horizontal stretches and compressions. Changes to the argument of the function, like f(bx), cause horizontal stretches or compressions. If |b| > 1, the graph is compressed horizontally, making it narrower. If 0 < |b| < 1, the graph is stretched horizontally, making it wider. And just like vertical transformations, if b is negative, the graph is reflected across the y-axis. Understanding these basic transformations is super important because they help us visualize and manipulate functions in a variety of ways. We can combine these transformations to create complex changes to a function's graph, and by recognizing these transformations, we can easily predict how a function will behave. Now that we've recapped the basics, let's get back to our original problem and see how we can apply this knowledge to find corresponding points on transformed functions. Remember, the key is to identify the transformations and understand how they affect the coordinates of points on the graph. Let's tackle the problem together!

Applying Transformations to Points

Now, let's apply this to our specific problem. We know that the point (8, -1) lies on the graph of f(x). This means that f(8) = -1. We want to find the corresponding point on the graph of f(-2/3 x). To do this, we need to figure out what value of x will make the inside of the function, (-2/3 x), equal to 8. This is because we know the value of f(8), and we want to use that information to find a point on the new graph. Let's set up an equation. We have (-2/3)x = 8. To solve for x, we can multiply both sides of the equation by the reciprocal of -2/3, which is -3/2. So, x = 8 * (-3/2). Multiplying this out, we get x = -12. This tells us that when x = -12, the input to the function f in f(-2/3 x) is 8. Since f(8) = -1, we know that f(-2/3 * -12) = f(8) = -1. This means that the point (-12, -1) will be on the graph of f(-2/3 x). So, we've found our corresponding x-value! The original x-value of 8 has been transformed into -12 in the new function. But what about the y-value? In this case, the y-value remains the same. This is because the transformation we applied only affects the x-coordinate. There's no vertical stretching, compression, or shifting happening here, so the y-coordinate stays as -1. Remember, when we deal with transformations inside the function's argument, like f(bx), it affects the x-coordinate. If there's a coefficient multiplying the x inside the function, it results in a horizontal stretch or compression. If the coefficient is negative, it also includes a reflection across the y-axis. And that's exactly what we see here. The (-2/3) is causing both a horizontal stretch and a reflection. So, to summarize, we found the x-value by setting the inside of the transformed function equal to the x-value of the original point. Then, we solved for x, and the y-value remained unchanged because there was no vertical transformation. This approach works for any transformation that only affects the x-coordinate. If there were also a transformation outside the function, like adding a constant or multiplying the entire function by a value, we would need to adjust the y-coordinate as well. But for this problem, we've nailed it! We've successfully found the corresponding point on the transformed function's graph.

Generalizing the Approach

Okay, let's zoom out for a second and think about how we can apply this method more generally. What if we had a different point on f(x) or a different transformation? The good news is that the core idea stays the same. First, it is important to identify the transformation. Look at how the function has been changed. Is it a horizontal stretch or compression? A reflection? A shift? Breaking down the transformation into its components will help you understand how it affects the points on the graph. If you have a transformation like f(bx), it primarily affects the x-coordinates. To find the new x-coordinate, set the expression inside the function, bx, equal to the original x-coordinate and solve for the new x. Keep in mind the transformations involving reflections. If b is negative, there's a reflection across the y-axis, which changes the sign of the x-coordinate. If the transformation is a f(x), it mainly affects the y-coordinates. To find the new y-coordinate, simply multiply the original y-coordinate by a. If a is negative, there's a reflection across the x-axis, which changes the sign of the y-coordinate. What about shifts? If you have f(x + c), it's a horizontal shift. To find the new x-coordinate, subtract c from the original x-coordinate. If you have f(x) + c, it's a vertical shift. To find the new y-coordinate, add c to the original y-coordinate. Now, what if you have a combination of transformations, like a f(bx + c) + d? Don't panic! Just tackle it step by step. First, handle the horizontal transformations (compression/stretch and shift), then the vertical transformations (stretch/compression and shift). It's essential to understand the order of operations. Transformations inside the function argument affect the x-coordinates, and transformations outside affect the y-coordinates. It's also helpful to keep in mind that horizontal transformations tend to do the “opposite” of what you might expect. For example, f(x + 2) shifts the graph to the left, not the right. Similarly, f(2x) compresses the graph horizontally, making it narrower, not wider. Finally, let’s talk about the big picture. When you encounter a transformation problem, the goal is to relate the points on the new graph to the points on the original graph. By understanding the transformations, you can predict how the coordinates will change. This skill is not only valuable for solving specific problems but also for gaining a deeper understanding of how functions behave. So, keep practicing, and you'll become a transformation master in no time! Remember, it's all about breaking it down, step by step, and understanding the effects of each transformation. With this approach, you'll be able to tackle even the most complex function transformations with confidence.

Solution for our Problem

Alright, let's wrap up our initial problem. We started with the point (8, -1) on the graph of f(x) and wanted to find the corresponding point on the graph of f(-2/3 x). We went through the process of identifying the transformation, which in this case was a horizontal stretch/compression and a reflection. By setting (-2/3)x = 8, we found that the new x-coordinate is -12. The y-coordinate remained unchanged because there was no vertical transformation. Therefore, the corresponding point on the graph of f(-2/3 x) is (-12, -1). So, to double-check our solution, we can think about the transformation again. The factor of -2/3 inside the function means that the graph is stretched horizontally by a factor of 3/2 (the reciprocal of 2/3) and reflected across the y-axis because of the negative sign. The original point (8, -1) is 8 units to the right of the y-axis. After the reflection, it should be to the left of the y-axis, and after the stretch, it should be further away from the y-axis. The x-coordinate of -12 confirms this, as it's 12 units to the left of the y-axis, which is indeed a stretch and a reflection. The y-coordinate remaining at -1 makes sense because there's no vertical transformation. We've successfully found the corresponding point by understanding the transformation and applying it to the original point's coordinates. This approach works consistently for similar problems, so you can use it with confidence. Remember, the key is to identify the transformation, figure out how it affects the x and y coordinates, and then apply those changes to the original point. With practice, you'll become super comfortable with these types of problems. And that's it for this problem, guys! I hope this explanation was helpful. Keep practicing, and you'll master function transformations in no time. If you have any more questions or topics you'd like to explore, just let me know. Happy transforming!