Finding G(x) After Reflecting F(x) = X² - 2x + 4

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Hey guys! Let's dive into this math problem where we need to figure out what happens to a function when we reflect it over the X-axis. It might sound a bit intimidating, but trust me, we'll break it down step by step. We're given the function f(x) = x² - 2x + 4, and the key here is understanding how reflections work in the world of functions.

Understanding Reflections Over the X-Axis

So, what does it really mean to reflect a function over the X-axis? Think of the X-axis as a mirror. When you reflect something, you're essentially creating a mirror image of it. In the context of functions, this means that the y-values of our original function, f(x), will become their opposites. If a point on the graph of f(x) is at (x, y), the corresponding point on the reflected graph, g(x), will be at (x, -y). This is the core concept we need to grasp to solve this problem. Understanding this reflection principle is crucial not just for this specific question, but for a whole range of function transformations you might encounter. It's like learning a fundamental move in a dance – once you've got it down, you can use it in so many different routines!

In practical terms, when you reflect a function over the X-axis, you're essentially multiplying the entire function by -1. This is because every y-value in the original function gets flipped to its negative counterpart. This might sound a bit abstract, but it's actually a pretty straightforward operation. We're not changing the 'x' values at all; we're only focusing on how the 'y' values transform. To really make this stick, let's think about a simple example. Imagine a point on the graph of f(x) is (2, 3). After reflecting over the X-axis, that point will become (2, -3). See how the x-value stays the same, but the y-value changes sign? That's the magic of reflection in action! Visualizing this transformation can be incredibly helpful, especially when you're dealing with more complex functions. You can even sketch a quick graph to see the effect of the reflection. It's all about making the abstract concrete. Another thing to consider is how this reflection impacts the overall shape of the graph. If the original function had a peak above the X-axis, the reflected function will have a valley below it, and vice versa. This is because the reflection essentially flips the entire graph upside down. So, keep an eye on the turning points and general direction of the graph as you perform the reflection. It's a great way to double-check that your answer makes sense!

Applying the Reflection to f(x) = x² - 2x + 4

Now that we've got a solid understanding of reflections, let's apply this to our specific function, f(x) = x² - 2x + 4. Remember, to reflect this function over the X-axis, we need to multiply the entire function by -1. This means every term in the function will have its sign flipped. Think of it like distributing a negative sign across the whole expression. It's a simple algebraic step, but it's the key to finding our new function, g(x). Don't rush through this step! It's easy to make a small mistake with the signs, and that can throw off your entire answer. Take your time and double-check each term as you multiply by -1.

So, let's do it! We start with f(x) = x² - 2x + 4. To find g(x), which is the reflection of f(x), we multiply the entire expression by -1. This gives us g(x) = -1 * (x² - 2x + 4). Now, we distribute the -1 to each term inside the parentheses: -1 * x² becomes -x², -1 * (-2x) becomes +2x, and -1 * 4 becomes -4. So, after distributing, we have g(x) = -x² + 2x - 4. And there you have it! We've successfully reflected our function over the X-axis. But let's not stop here. It's always a good idea to verify your answer to make sure it makes sense. One way to do this is to think about the original function and the reflected function at a specific point. For example, if we plug in x = 0 into f(x), we get f(0) = 4. This means the point (0, 4) is on the graph of f(x). Now, if we plug in x = 0 into our g(x), we get g(0) = -4. This means the point (0, -4) is on the graph of g(x). Notice how the y-value has changed sign, just like we expected! This simple check can give you a lot of confidence in your solution.

Identifying the Correct Option

Okay, we've done the hard work of finding the reflected function, g(x) = -x² + 2x - 4. Now, let's take a look at the options provided in the problem and see which one matches our answer. The options are:

A. -x² + 2x - 4 B. x² + 2x - 4 C. -x² - 2x + 4 D. x² + 2x + 4 E. -x² - 2x - 4

Comparing our solution, g(x) = -x² + 2x - 4, to the options, we can clearly see that option A, -x² + 2x - 4, is the correct answer. Woohoo! We nailed it! But even though we've found the right answer, let's take a moment to think about the other options and why they're incorrect. This can help solidify our understanding and prevent us from making similar mistakes in the future.

Option B, x² + 2x - 4, is incorrect because it doesn't reflect the x² term. Remember, we need to multiply the entire function by -1, so the sign of the x² term should change. Option C, -x² - 2x + 4, is incorrect because it changes the sign of the -2x term incorrectly. When we multiply -2x by -1, it should become +2x, not -2x. Option D, x² + 2x + 4, is incorrect for similar reasons. It doesn't reflect any of the terms correctly. And finally, option E, -x² - 2x - 4, is incorrect because it changes the sign of the -2x term incorrectly and also changes the sign of the constant term incorrectly. By analyzing why the incorrect options are wrong, we can reinforce our understanding of the reflection process and avoid making these errors ourselves. It's all about learning from our mistakes (or potential mistakes!) and becoming more confident in our problem-solving skills.

Key Takeaways

So, what have we learned today, guys? We've tackled a problem about reflecting a quadratic function over the X-axis, and we've come out on top! But more importantly, we've learned some valuable concepts and strategies that we can apply to other problems as well. Let's recap the key takeaways from this exercise:

  1. Reflection Over the X-Axis: Remember that reflecting a function over the X-axis means multiplying the entire function by -1. This changes the sign of the y-values, creating a mirror image of the original graph.
  2. Distributing the Negative Sign: When multiplying a function by -1, make sure to distribute the negative sign to every term in the function. This is a common area for mistakes, so take your time and double-check your work.
  3. Verifying Your Answer: It's always a good idea to verify your answer. One way to do this is to plug in a specific x-value into both the original function and the reflected function and see if the y-values are opposites.
  4. Analyzing Incorrect Options: Understanding why the incorrect options are wrong can be just as valuable as finding the correct answer. It helps you solidify your understanding of the concepts and avoid making similar mistakes in the future.

By keeping these points in mind, you'll be well-equipped to tackle similar problems involving function transformations. Math might seem daunting at times, but with a clear understanding of the principles and a bit of practice, you can conquer any challenge! Keep up the great work, and remember to always break down problems into smaller, manageable steps. You've got this!

Therefore, the function g(x) is A. -x² + 2x - 4.