Graph Transformations Finding Transformed Coordinates For Y=-1/2f(x)

Introduction to Graph Transformations

Hey guys! Let's dive into the fascinating world of graph transformations. These transformations allow us to manipulate the graphs of functions, giving us a powerful tool for understanding and visualizing mathematical relationships. Today, we’re going to tackle a problem that involves transformations of a function’s graph. Specifically, we'll look at what happens when we have a point on the graph of a function $y = f(x)$ and we apply a transformation to get a new function, like $y = -\frac{1}{2} f(x)$. This type of problem might seem a bit abstract at first, but by breaking it down step by step, you'll see it's totally manageable. Transformations are a fundamental concept in mathematics, especially when dealing with functions and their graphical representations. They allow us to take a basic function and modify its graph by shifting, stretching, compressing, or reflecting it. Understanding these transformations helps in visualizing and analyzing functions more effectively. It's like having a superpower to reshape and understand graphs! So, let's get started and explore how these transformations work in action. We'll focus on how the coordinates of a point on the original graph change when we apply a transformation to the function. This skill is super useful not only in math class but also in various real-world applications where functions and graphs are used to model phenomena.

The Problem: Transforming a Point on a Graph

Okay, so here's the problem we're going to solve. We know that the point $(6, -4)$ lies on the graph of the function $y = f(x)$. The big question is: where does this point end up if we transform the graph to $y = -\frac{1}{2} f(x)$? This is a classic transformation problem, and it's a great way to understand how transformations affect specific points on a graph. When we talk about transforming a graph, we're essentially talking about changing its shape and position in the coordinate plane. This can involve stretching, compressing, reflecting, or shifting the graph. Each of these transformations affects the coordinates of the points on the graph in a specific way. For instance, multiplying a function by a constant can stretch or compress the graph vertically. Similarly, multiplying the input variable (x) by a constant can stretch or compress the graph horizontally. Reflections occur when we multiply the function or the input variable by -1. These reflections flip the graph across the x-axis or the y-axis, respectively. Shifts, on the other hand, involve adding or subtracting a constant to the function or the input variable, which moves the graph horizontally or vertically. Understanding these basic transformations is key to solving more complex problems involving function transformations. Now, let's dig deeper into this specific problem and see how we can apply our knowledge of transformations to find the new coordinates of the transformed point. We'll break down the transformation step by step, making sure we understand the effect of each operation on the original point.

Breaking Down the Transformation

Let's break down this transformation $y = -\frac{1}{2} f(x)$ step by step. There are two key components here: the multiplication by $-\frac{1}{2}$. First, let’s consider the factor of $\frac{1}{2}$. This is a vertical compression. Think of it as squishing the graph vertically. If we multiply a function by a fraction between 0 and 1, like $\frac{1}{2}$, we're making the y-values smaller. In this case, each y-value will be halved. This means that if a point on the original graph has a y-coordinate of -4, the corresponding point on the transformed graph will have a y-coordinate of -4 multiplied by $\frac{1}{2}$, which is -2. This vertical compression brings the points closer to the x-axis. The next component is the negative sign. The negative sign in front of the $\frac{1}{2}$ indicates a reflection across the x-axis. This means the graph is flipped upside down. If a point is below the x-axis (i.e., has a negative y-coordinate), it will now be above the x-axis after the reflection, and vice versa. In our example, the point (6, -4) has a negative y-coordinate. After the reflection, the y-coordinate will change its sign, becoming positive. So, the combination of the vertical compression and the reflection across the x-axis is what we need to consider to find the new coordinates of the transformed point. Understanding these individual transformations and their effects is crucial for solving this type of problem. Now, let’s put these two transformations together and see how they affect the point (6, -4).

Applying the Transformation to the Point

So, we have the point $(6, -4)$ on the graph of $y = f(x)$, and we want to find its new coordinates on the graph of $y = -\frac{1}{2} f(x)$. Remember, the transformation involves two steps: a vertical compression by a factor of $\frac{1}{2}$ and a reflection across the x-axis. First, let's apply the vertical compression. This means we multiply the y-coordinate of the original point by $\frac{1}{2}$. So, the new y-coordinate becomes $-4 \times \frac{1}{2} = -2$. This compression brings the point closer to the x-axis, halving its distance from the axis. Next, we apply the reflection across the x-axis. This means we change the sign of the y-coordinate. So, the y-coordinate of -2 becomes +2. The reflection flips the point from below the x-axis to above the x-axis. The x-coordinate remains unchanged because this transformation only affects the y-values. The x-coordinate stays the same because we are only vertically transforming the graph. The horizontal position of the point is not affected by this particular transformation. Therefore, the new x-coordinate is still 6. Putting it all together, the transformed point has coordinates (6, 2). This means that the original point (6, -4) on the graph of $y = f(x)$ is transformed to the point (6, 2) on the graph of $y = -\frac{1}{2} f(x)$.

Final Answer and Conclusion

Alright, guys! We’ve successfully navigated this transformation problem. The final answer is that the point (6, -4) on the graph of $y = f(x)$ is transformed to the point (6, 2) on the graph of $y = -\frac{1}{2} f(x)$. We achieved this by breaking down the transformation into two steps: a vertical compression by a factor of $\frac{1}{2}$ and a reflection across the x-axis. By applying these transformations sequentially, we were able to determine the new coordinates of the point. Understanding graph transformations is a crucial skill in mathematics. It allows us to analyze and manipulate functions and their graphs, providing a deeper insight into their behavior. This knowledge is not only useful in academic settings but also in various real-world applications, such as modeling physical phenomena, analyzing data, and designing engineering systems. Remember, the key to mastering graph transformations is to understand the effect of each transformation individually and then combine them as needed. Practice is essential, so try tackling similar problems to solidify your understanding. Keep exploring and keep learning, and you'll become a graph transformation pro in no time! So, next time you encounter a graph transformation problem, remember the steps we discussed: identify the transformations, apply them one by one, and keep track of how the coordinates of the points change. You've got this!