Find The Image Of Point (-3, 6) After Reflection Across X=0

Hey guys! Let's dive into a cool math problem today that involves reflections. Reflections might sound intimidating, but they're really just about creating mirror images. We're going to figure out what happens when we reflect a point across a specific line. So, let’s break it down step by step!

The Problem: Reflecting a Point

Our main goal here is to find the image, think of it as the mirrored twin, of the point $(-3, 6)$ when it's reflected across the line $x = 0$. Now, what exactly does this mean? Well, imagine you have a point on a graph, and you have a mirror standing vertically at the line $x = 0$. The reflection is the new point that appears on the other side of the mirror, at the same distance from it. Understanding this concept is super important for visualizing transformations in geometry. You know, geometry can be pretty fun once you get the hang of visualizing these things! We’ll be walking through the whole process, so no worries if it seems a bit tricky at first. Stick with me, and we'll get through it together!

Breaking Down the Reflection Concept

First off, what does the line $x = 0$ even mean? Picture your regular Cartesian coordinate system, with the x-axis running horizontally and the y-axis running vertically. The line $x = 0$ is actually the y-axis itself! Yep, it’s that simple. So, we're reflecting our point across the y-axis. Now, when we reflect a point across the y-axis, something interesting happens: the x-coordinate changes its sign, while the y-coordinate stays the same. Think about it like this – if you're standing 3 steps to the left of a mirror, your reflection will be 3 steps to the right. The distance to the mirror is the same, but the direction changes. That's essentially what's happening with the x-coordinate. So, if we start with the point $(-3, 6)$, the x-coordinate is $-3$. After the reflection, it will become $3$. The y-coordinate, which is $6$, will stay exactly where it is. This gives us our new point: $(3, 6)$.

Visualizing the Transformation

To really nail this down, let's visualize it. Imagine our point $(-3, 6)$ plotted on a graph. It’s in the second quadrant, which is the top-left part of the graph. Now, picture the y-axis as a mirror. The reflection of $(-3, 6)$ will be in the first quadrant, which is the top-right part. If you draw a straight line from $(-3, 6)$ to the y-axis, and then continue that line the same distance on the other side, you'll end up at the point $(3, 6)$. The distance from $(-3, 6)$ to the y-axis is 3 units, and the distance from $(3, 6)$ to the y-axis is also 3 units. The y-coordinate remains unchanged at 6 because the point’s height above the x-axis doesn’t change during a reflection across the y-axis. Visualizing these transformations can make them so much clearer and easier to remember. It’s like watching a little math movie in your head!

Step-by-Step Solution

Okay, let's get down to the nitty-gritty and solve this problem step by step. This is how we’ll approach it:

1. Identify the Original Point: Our starting point is $(-3, 6)$. This is the point we're going to reflect.
2. Understand the Line of Reflection: We're reflecting across the line $x = 0$, which is the y-axis.
3. Apply the Reflection Rule: When reflecting across the y-axis, the x-coordinate changes its sign, and the y-coordinate remains the same. This is our golden rule for this type of reflection.
4. Change the x-coordinate: The x-coordinate of our point is $-3$. Changing its sign gives us $3$.
5. Keep the y-coordinate: The y-coordinate is $6$, and it stays the same.
6. Write the New Point: Combining the new x-coordinate and the original y-coordinate, we get the reflected point $(3, 6)$.

So, after reflecting the point $(-3, 6)$ across the line $x = 0$, we get the point $(3, 6)$. See? It's not as scary as it looks when you break it down into simple steps!

Detailed Explanation of the Steps

Let's zoom in a bit on each of these steps to make sure we’re all on the same page. First, identifying the original point is crucial because this is where we’re starting our journey. It’s like knowing your starting location on a map. If you don’t know where you are, you can’t figure out how to get anywhere else! In our case, the starting point is $(-3, 6)$, which means we’re 3 units to the left of the y-axis and 6 units above the x-axis.

Next, understanding the line of reflection is just as important. The line $x = 0$ might seem like a simple thing, but it’s the key to our transformation. Remember, this line is the y-axis. When we reflect across it, we’re essentially creating a mirror image with the y-axis acting as the mirror. This understanding helps us visualize the transformation and predict what will happen to our point.

Now comes the fun part: applying the reflection rule. This is where the magic happens! The rule for reflecting across the y-axis is that the x-coordinate changes sign, and the y-coordinate stays put. This is a fundamental concept in reflections, and it’s worth memorizing. Why does this happen? Well, think back to the mirror analogy. When you look in a mirror, your left and right are flipped, but your up and down stay the same. That’s exactly what’s happening with the coordinates. The x-coordinate, which represents the horizontal position, gets flipped, while the y-coordinate, which represents the vertical position, remains unchanged.

Then, we change the x-coordinate. Our original x-coordinate is $-3$. To change its sign, we simply multiply it by $-1$, which gives us $3$. This means our new point will be 3 units to the right of the y-axis, instead of 3 units to the left. We keep the y-coordinate at $6$ because the reflection across the y-axis doesn’t affect the vertical position of the point. Finally, we write the new point by combining our new x-coordinate and the original y-coordinate. This gives us $(3, 6)$, which is the image of $(-3, 6)$ after reflection across the y-axis. Awesome, right?

The Correct Answer and Why

So, which of the answer choices is correct? Let's look at them again:

A. $(3, -6)$ B. $(-3, -6)$ C. $(3, 6)$ D. $(-6, 3)$

We’ve already worked through the problem, and we know that the correct answer is $(3, 6)$. This matches option C. Let's quickly break down why the other options are incorrect:

• Option A, $(3, -6)$: This answer changes the sign of the y-coordinate as well as the x-coordinate. Remember, when reflecting across the y-axis, only the x-coordinate changes sign. The y-coordinate stays the same.
• Option B, $(-3, -6)$: This answer keeps the x-coordinate the same and changes the sign of the y-coordinate. This would be the result of a reflection across the x-axis, not the y-axis.
• Option D, $(-6, 3)$: This answer seems to have swapped the coordinates and changed one of the signs. It doesn’t follow the rule for reflection across the y-axis at all.

Therefore, the correct answer is C, $(3, 6)$. We nailed it! Choosing the correct answer is more than just getting the right numbers; it’s about understanding the underlying principles and applying the correct rules. Each wrong answer represents a misunderstanding of the process, which is why it’s so important to break down each step and understand why we do what we do.

Why This Answer Is the Only Logical Choice

Let’s really drive this point home: why is $(3, 6)$ the ONLY logical answer here? It all boils down to the fundamental properties of reflections. Reflections are transformations that create a mirror image of a point or shape. This means the distance from the original point to the line of reflection (in our case, the y-axis) must be the same as the distance from the reflected point to the line of reflection. Furthermore, the line connecting the original point and its reflection must be perpendicular to the line of reflection.

Think about it this way: if you stand a certain distance from a mirror, your reflection appears to be the same distance away on the other side. Your height doesn’t change, just your apparent left-right position. This is exactly what’s happening with our point $(-3, 6)$. It’s 3 units away from the y-axis on the left side. Its reflection must be 3 units away from the y-axis on the right side. That’s why the x-coordinate changes from $-3$ to $3$. The y-coordinate, which represents the height, remains the same because reflections across the y-axis don’t change the vertical position.

Options A, B, and D simply don’t adhere to these basic principles. They either change the y-coordinate when it shouldn’t be changed, or they don’t flip the x-coordinate correctly, or they completely mix up the coordinates. Option C, $(3, 6)$, is the only option that correctly applies the rules of reflection across the y-axis. It maintains the same distance from the y-axis (3 units) but on the opposite side, and it keeps the y-coordinate (the height) unchanged. This logical consistency is what makes $(3, 6)$ the only valid answer.

Conclusion

So, we've successfully found the image of the point $(-3, 6)$ under a reflection in the line $x = 0$. The answer is $(3, 6)$. We walked through the steps, visualized the transformation, and even discussed why the other options don't make sense. Great job, everyone! Keep practicing, and reflections will become second nature. Remember, math is like building with blocks – each concept builds on the previous one. Once you understand the basics, the more complex stuff becomes much easier. Keep up the awesome work, guys!

Final Thoughts on Reflections

Reflections are just one type of geometric transformation, but they’re a fundamental concept that pops up in all sorts of areas, from simple geometry problems to more advanced topics like linear algebra and computer graphics. Understanding reflections helps you develop spatial reasoning skills, which are super useful in many fields, not just math. Think about architects designing buildings, artists creating symmetrical patterns, or even video game developers creating realistic environments – they all use the principles of geometric transformations.

So, mastering reflections is not just about getting the right answer on a test; it’s about building a foundation for future learning and problem-solving. Keep practicing, keep visualizing, and keep exploring the wonderful world of math! And remember, every problem you solve is like a little victory that builds your confidence and skills. You got this!