Finding The Right Point: Does It Fit The Equation?
Hey everyone! Let's dive into a cool algebra problem. We've got a function, y = 2x² - 4, and we need to figure out which point from the options provided actually belongs to the graph of this function. Think of it like this: the graph is a visual representation of all the possible solutions to the equation. So, if a point is on the graph, it means the x and y values of that point will make the equation true. Let's break it down and see how we can solve this, step by step, it's pretty straightforward, trust me!
Understanding the Function and the Points
First off, let's understand what we're working with. Our function, y = 2x² - 4, is a quadratic equation. This means its graph is a parabola – a U-shaped curve. Now, we're given a few points: A (0, 0), B (0, -4), and C (0, 4). Each point has an x-coordinate and a y-coordinate, represented as (x, y). Our mission? To substitute the x-coordinate of each point into the equation and see if we get the corresponding y-coordinate. If the equation holds true, then that point lies on the graph. It's like a puzzle: we're checking if the pieces fit perfectly. The key here is careful substitution and calculation.
Now, before we get started, remember that the equation y = 2x² - 4 is the star of our show. We'll be using it to test each point. The goal is to see if the y-value we get from the equation matches the y-value of the point. If it does, we've found our match! If it doesn't, we know that point isn't on the graph. This is a fundamental concept in algebra, and getting a good grasp on this will help you with a lot of future problems. Think about it: if the point is on the graph, it means the equation is satisfied when you plug in those x and y values. It's like the point is saying, "Hey, I'm a solution!"
So let's get our hands dirty and start testing those points! It might sound intimidating at first, but once you start plugging in those values, it's just a matter of following the math. I promise, it's easier than it looks. You'll be a pro in no time, and the feeling of figuring it out is so satisfying. The next step is really just a series of small, individual calculations, one for each point. We'll take each point, plug the x-coordinate into our equation, and see if the resulting y-coordinate matches the point's y-coordinate. Let's do this!
Testing Point A: (0, 0)
Alright, let's start with point A, which is (0, 0). This means x = 0 and y = 0. We'll plug x = 0 into our equation y = 2x² - 4 and see if we get y = 0. So, we have y = 2(0)² - 4. First, square 0, which is still 0. Then, multiply by 2, which is still 0. Finally, subtract 4. So, we end up with y = -4. Now, does this match the y-coordinate of point A? No, it doesn't! Point A has a y-coordinate of 0, but our equation gave us -4. This means point A (0, 0) does not belong to the graph of the function. Easy, right? We've successfully eliminated one option.
It is important to understand what the numbers mean, right? The x-coordinate tells us where the point is located horizontally, and the y-coordinate tells us where the point is located vertically. When we substitute the x-value into the equation, we're essentially asking the equation, "Hey, if x is this value, what is the corresponding y-value?" If the answer matches the y-coordinate of the point, then that point is on the graph. If it doesn't match, the point isn't on the graph. It's like a coordinate treasure hunt, where the equation is the map and the points are the potential treasures. Only the points that satisfy the equation are the treasures we're looking for.
Now, even though the math seems simple, it's super important to be careful with the order of operations, guys. Always remember to square first, then multiply, and then add or subtract. One small mistake, like forgetting to square, can throw off the whole answer. And trust me, I've made plenty of those little mistakes! But hey, that's how we learn, isn't it? So, don't worry if you slip up – just double-check your work and you'll catch it. Getting the order of operations correct is a fundamental part of solving math equations, and it will serve you well in all sorts of problems.
Testing Point B: (0, -4)
Next up, let's test point B, which is (0, -4). Again, we'll plug in x = 0 into our equation y = 2x² - 4. So we get y = 2(0)² - 4. As we saw with point A, 0 squared is 0, then we multiply by 2, still 0. Finally, subtract 4. We get y = -4. Does this match the y-coordinate of point B? Yes, it does! Point B has a y-coordinate of -4, and our calculation also gave us -4. This means point B (0, -4) does belong to the graph of the function. Boom! We've found our answer. But hey, for the sake of thoroughness, let's check the last point as well.
So, by substituting the x-value into the equation and getting the exact same y-value as the point, we confirmed that it is indeed a point on the graph. It is like a matching puzzle! This is a core concept in algebra, and by solving such simple problems, you are sharpening your analytical thinking and ability to handle equations. Remember, the graph is a visualization of the equation, and every single point on it will satisfy the function. It is important to know that functions can be represented in multiple forms – as an equation, as a table of values, or as a graph, as in our case. The same values that satisfy the equation will also be represented on the graph and will appear in a table. All three representations are interconnected and show us the behavior of the function.
Now, take a moment to pat yourself on the back, guys! We've done it! We found a point on the function's graph. Feel good about understanding the logic behind it, because we're not just blindly following instructions, we are truly understanding the concept. Every point on the graph is a solution, and that's something really important to remember. Keep practicing these problems, and your confidence will grow with each one you solve!
Testing Point C: (0, 4)
Okay, let's quickly check point C, which is (0, 4). We'll plug x = 0 into our trusty equation: y = 2x² - 4. Again, we get y = 2(0)² - 4, which simplifies to y = -4. Does this match the y-coordinate of point C? Nope! Point C has a y-coordinate of 4, but our equation gave us -4. So, point C (0, 4) does not belong to the graph. Just to confirm we were correct when finding point B earlier.
It is so important to understand that not all points will necessarily lie on the graph. In a lot of problems, you may be provided with several options, and you need to determine which one works. This is what you have just done. The beauty of this approach is that it applies to all equations and functions. So the more you practice, the more confident you'll become in solving these types of problems. Remember that the x and y coordinates are like clues, and the equation is your tool to unlock the answer. Always be careful with your calculations, and you'll find that these problems are not as scary as they initially seem. It is all about following the steps methodically and double-checking your work.
Now, at the end, let's recap the whole process. We have learned how to find out which point belongs to a function. We plugged in the x-coordinate of the point, and the function gave us the y-coordinate. And if that number matched the y-coordinate of the point, it meant that the point belongs to the function's graph. Keep these steps in mind, and you will be able to solve these types of problems with ease in the future. Now go on, and show off your new skills.
Conclusion
So, to wrap it all up, the only point that belongs to the graph of the function y = 2x² - 4 is point B (0, -4). We confirmed this by substituting the x-coordinate of the point into the equation and checking if the resulting y-coordinate matched the point's y-coordinate. Remember, this is a fundamental concept in algebra, and understanding this will help you with more complex problems down the line. Keep practicing, and you'll become a pro in no time! Keep up the amazing work, guys, you're doing awesome!