Reflections Of A Linear Function: A Step-by-Step Guide

Jan 13, 2026 by Tim Redaksi 55 views

Hey guys! Let's dive into the world of reflections in mathematics. Today, we're going to break down how to find the reflections of a linear function, specifically F(x) = -2/3x + 12, across different lines and axes. Don't worry, it might sound intimidating, but I'll walk you through it step-by-step. Understanding reflections is super important in geometry and helps you visualize how shapes and functions change in space. We'll be looking at reflections across vertical lines (like x = 3), horizontal lines (like y = 3), the x-axis, and the y-axis. By the end, you'll be a pro at this! Ready to get started?

1. Reflection Across the Line x = 3

Alright, first up, let's reflect our function across the vertical line x = 3. This is like looking at the function in a mirror that's standing up straight. The key here is to understand that the distance from any point on the original function to the line x = 3 must be the same as the distance from its reflected point to the line. Let's think about this visually. Imagine a point on your function, let's say (x, y). To find its reflection across x = 3, you'll need to find a new x-coordinate. The x-coordinate of the reflected point will be the same distance from x = 3 as the original point, but on the opposite side. The y-coordinate remains the same during this type of reflection. So the rule for reflection across the line x = 3 is: If your point is (x, y), then the reflected point will be (x', y), where x' = 2(3) - x = 6 - x. Now, since our original function is F(x) = -2/3x + 12, this means our y value is equal to -2/3x + 12. So, since x' = 6 - x, then x = 6 - x'. We'll substitute 6 - x' into the original equation to get our reflection, so F(x') = -2/3(6 - x') + 12. Let's simplify that! F(x') = -4 + 2/3x' + 12. F(x') = 2/3x' + 8. In conclusion, the equation of the reflected function across the line x = 3 is F(x) = 2/3x + 8. It's pretty cool, right? Basically, the slope changes sign, and the y-intercept shifts! This process shows how the function essentially flips over the vertical line, giving us a new line with different characteristics. This concept is fundamental in understanding how functions transform under different geometric operations. Remember that the x-coordinate is the one that gets affected in this type of reflection, while the y-coordinate remains unchanged. Keep in mind that the point (0, 12) on the original function will map to the point (6, 12), and the x-intercept of the original function can be computed, and it's (18, 0). After it is reflected, it will be (-12, 0).

2. Reflection Across the Line y = 3

Now, let's consider the reflection across the horizontal line y = 3. This time, we're looking at a horizontal mirror. The same principle applies here, but we’re focusing on the y-coordinate. The distance from any point on the original function to the line y = 3 must be the same as the distance from the reflected point to the line. The x-coordinate remains the same, and the y-coordinate changes. If your point is (x, y), then the reflected point will be (x, y'), where y' = 2(3) - y = 6 - y. Now, recall our original function: F(x) = -2/3x + 12. Since y = -2/3x + 12, then y' = 6 - (-2/3x + 12). Let's simplify this. So, y' = 6 + 2/3x - 12, then y' = 2/3x - 6. Therefore, the equation of the reflected function across the line y = 3 is F(x) = 2/3x - 6. In this case, notice that the slope remains the same, but the y-intercept changes! The function flips over the horizontal line, affecting the y-values. We basically applied the reflection transformation, which changed the vertical position of the line relative to the reflection line, while the overall trend of the line remains the same. Also, keep in mind that the y-intercept of the original function can be computed, and it's (0, 12). After it is reflected, it will be (0, -6). This process is very important for many real-world applications of reflections, such as optics and computer graphics. Furthermore, understanding the impact of reflection across a horizontal line is very important for any student that wants to get good grades in mathematics!

3. Reflection Across the x-axis

Next, let's explore reflection across the x-axis. This is like flipping the function upside down. Here, the x-coordinate stays the same, and the y-coordinate changes its sign. If your point is (x, y), then the reflected point will be (x, -y). Our original function is F(x) = -2/3x + 12. Since y = -2/3x + 12, then -y = -(-2/3x + 12). Let's simplify this: -y = 2/3x - 12. So, the equation of the reflected function across the x-axis is F(x) = 2/3x - 12. Notice how both the slope and the y-intercept change this time! The function is flipped over the x-axis, creating a mirror image. The original function has a negative slope, and after reflection, it becomes a positive slope. The x-intercept of the original function can be computed, and it's (18, 0). After it is reflected, it will remain as (18, 0), and the y-intercept of the original function is (0, 12). After the transformation, it will be (0, -12). This type of reflection is important for understanding the behavior of functions and their transformations in the coordinate plane. Understanding the x-axis reflection helps us analyze and compare the original function with its transformed counterpart, highlighting the effects of the reflection operation.

4. Reflection Across the y-axis

Finally, let's consider the reflection across the y-axis. This is like looking at the function in a mirror that's standing vertically. In this case, the y-coordinate remains the same, but the x-coordinate changes its sign. If your point is (x, y), then the reflected point will be (-x, y). Remember, our original function is F(x) = -2/3x + 12. To find the reflection, we'll replace every 'x' with '-x'. So, we get F(-x) = -2/3(-x) + 12. Simplifying this, we get F(-x) = 2/3x + 12. Therefore, the equation of the reflected function across the y-axis is F(x) = 2/3x + 12. In this instance, the y-intercept remains the same, but the slope changes its sign! The function flips over the y-axis. The y-intercept of the original function is (0, 12), and after the reflection, it is still (0, 12). The x-intercept of the original function is (18, 0), and after the reflection, it will be (-18, 0). This understanding is crucial for analyzing how the function's graphical properties change under reflection across the y-axis, providing a different perspective on the function's behavior. The transformation effectively mirrors the function across the y-axis. This transformation is very useful in various branches of mathematics, providing a detailed understanding of how a function behaves as it moves across the different axes.

Conclusion: Mastering Reflections

So there you have it, guys! We've covered how to find the reflections of the linear function F(x) = -2/3x + 12 across the lines x = 3 and y = 3, as well as the x-axis and y-axis. Remember the key rules: for reflections across vertical lines, the x-coordinate changes; for reflections across horizontal lines, the y-coordinate changes; for reflections across the x-axis, the y-coordinate changes sign; and for reflections across the y-axis, the x-coordinate changes sign. Keep practicing, and you'll become a pro at reflections in no time! Understanding these transformations is not just about getting the right answer; it's about building a strong foundation in geometry and functions. Keep up the awesome work!