Decoding The Equation Fill The Table With Y = -2x + 5

Hey guys! Today, we're diving into the world of functions and how to use them to solve problems. Specifically, we're going to tackle the function rule y = -2x + 5 and use it to fill in a table of values. Don't worry, it's not as intimidating as it sounds! We'll break it down step by step, and by the end, you'll be a pro at plugging in values and finding the corresponding outputs. So, grab your pencils and paper (or your favorite note-taking app) and let's get started!

Understanding Function Rules

Before we jump into filling the table, let's make sure we're all on the same page about what a function rule actually is. Think of a function rule as a mini-machine. You feed it an input (in this case, our 'x' values), and it follows a specific set of instructions to spit out an output (our 'y' values). The function rule y = -2x + 5 is telling us exactly what those instructions are. It says, "Take the input value ('x'), multiply it by -2, and then add 5 to the result." That's it! That's the whole secret. Understanding this core concept is crucial because it's the foundation for everything else we'll be doing. Imagine you have a vending machine. You put in your money (the input), press the button for your favorite snack, and the machine dispenses it (the output). The function rule is like the internal mechanism of the vending machine, dictating how the input leads to the output. To really solidify this understanding, let’s think about it in everyday terms. Let’s say you’re baking a cake, and the recipe is your function rule. The ingredients you put in (like flour, sugar, and eggs) are your inputs, and the delicious cake that comes out of the oven is your output. The recipe tells you exactly how much of each ingredient to use and what to do with them to get the desired result. In the same way, the function rule y = -2x + 5 tells us exactly what to do with our input ‘x’ to get the output ‘y’. The number -2 is what we call the coefficient of x, and it indicates the rate of change of y with respect to x. In simpler terms, it tells us how much y changes for every one unit change in x. The +5 is the constant term, and it represents the y-intercept, which is the point where the line crosses the y-axis on a graph. This constant term shifts the entire line up or down on the graph. So, when we talk about understanding function rules, we’re really talking about understanding the relationship between inputs and outputs, and how the rule defines that relationship. We're understanding the recipe, the vending machine mechanism, the mathematical instructions that connect 'x' and 'y'. Once you grasp this fundamental idea, working with functions becomes much less like memorizing steps and more like understanding a language. And just like with any language, the more you practice, the more fluent you become. We are going to be fluent in function language by the end of this, folks! So, let's keep this analogy in mind as we move forward and tackle the table-filling task. Think of each 'x' value as an ingredient we're feeding into our function-machine, and the rule y = -2x + 5 as the process that transforms it into the 'y' value. Now, let's put this understanding into action!

Filling the Table: Step-by-Step

Now that we've got a solid grasp on what function rules are all about, let's get down to business and fill in the table. Remember, our goal is to find the 'y' value that corresponds to each given 'x' value using the rule y = -2x + 5. We'll take each 'x' value one at a time, plug it into the equation, and solve for 'y'. It's like following a recipe, one ingredient at a time. The beauty of this process is that it's consistent and methodical. Once you understand the steps, you can apply them to any function rule, not just this one. So, let's take a deep breath, roll up our sleeves, and get to work!

1. When x = -4

Okay, let's start with the first 'x' value: -4. This is our first ingredient in the recipe. To find the corresponding 'y' value, we'll substitute -4 for 'x' in our function rule: y = -2x + 5. So, we get: y = -2(-4) + 5. Now, we just need to simplify this expression using the order of operations (PEMDAS/BODMAS). First, we multiply -2 by -4. Remember that a negative times a negative is a positive, so -2 * -4 = 8. Our equation now looks like this: y = 8 + 5. Finally, we add 8 and 5 to get y = 13. So, when x = -4, y = 13. We've successfully found our first pair of values! The key here is to be meticulous with the signs. Negative numbers can sometimes trip us up, so it's important to pay close attention to the rules of multiplication and addition with negative numbers. It's like carefully measuring your ingredients when baking – a slight mistake can change the whole outcome. Think of this as the first step on a journey. We've successfully navigated the first part of the equation, and we're well on our way to completing the table. Each 'x' value is a new leg of the journey, and we'll use the same process to find the corresponding 'y' values. Don't be afraid to double-check your work at each step. A quick mental calculation or a glance back at the previous step can save you from making a small error that could throw off the final answer. This is where that methodical approach really pays off. By breaking the problem down into smaller steps and checking our work along the way, we can ensure that we're on the right track. Alright, we've conquered the first 'x' value. Let's keep the momentum going and move on to the next one!

2. When x = -2

Alright, let's tackle the next 'x' value in our table: -2. We're going to follow the same process we used for x = -4, plugging this value into our function rule y = -2x + 5 and solving for 'y'. This consistent approach is what makes working with functions so manageable. Think of it like following a map – you know the route, you just need to follow the steps. So, let's substitute -2 for 'x': y = -2(-2) + 5. Just like before, we start with the multiplication. We have -2 multiplied by -2, which, as we know, equals a positive 4. So, our equation becomes: y = 4 + 5. Now, we simply add 4 and 5, and we get y = 9. Therefore, when x = -2, y = 9. We've added another piece to our puzzle! Notice how the process is becoming more familiar? With each value we calculate, we're reinforcing our understanding of the function rule and how it works. This repetition is key to mastering any mathematical concept. It's like practicing a musical instrument – the more you play, the more natural it becomes. One important thing to remember is to stay organized. Writing out each step clearly, as we've been doing, helps to prevent errors and makes it easier to double-check your work. It's like having a well-organized workspace – you can find what you need quickly and easily, and you're less likely to make mistakes. Also, it’s worth noting the pattern that’s starting to emerge. As ‘x’ increases, ‘y’ decreases, which makes sense given the negative coefficient in front of the ‘x’ in our function rule. This kind of observation can be a valuable tool for checking your work – if your calculated values don't seem to fit the pattern, it might be a sign that you've made an error somewhere. Okay, we're making great progress! Let's keep that momentum going and move on to the next 'x' value. We're halfway there!

3. When x = 0

Now, let's move on to x = 0. This one's often a little easier to work with, as multiplying by zero has a special property. We're still following the same trusty process, though: plugging the 'x' value into our function rule y = -2x + 5 and solving for 'y'. So, we substitute 0 for 'x': y = -2(0) + 5. Now comes the magic of zero! Anything multiplied by zero is zero, so -2 * 0 = 0. Our equation simplifies to: y = 0 + 5. And finally, 0 + 5 = 5. So, when x = 0, y = 5. We've found another piece of the puzzle, and this one was particularly straightforward! The beauty of x = 0 is that it often helps us to isolate the constant term in our function rule. In this case, the 'y' value when x = 0 is directly equal to the constant term, +5. This is a useful observation to keep in mind, as it can sometimes provide a shortcut for finding the y-intercept of a linear function. Think of this as a little mathematical trick that can make your life easier. It's like knowing a secret code that unlocks a part of the problem. It's important to recognize these patterns and shortcuts as you work with functions. They not only save you time and effort, but they also deepen your understanding of the underlying concepts. The more you recognize these patterns, the more confident you'll become in your ability to solve problems quickly and efficiently. Also, consider this an opportunity to mentally check your work so far. Do the values we’ve calculated make sense in the context of our function rule? Is there a consistent trend? Asking these questions as you go can help you catch errors early on and ensure that your final answer is accurate. Alright, we're on a roll! Let's tackle the final 'x' value and complete our table. We're almost there!

4. When x = 2

Alright, team, we've reached the final 'x' value in our table: 2. Let's bring it home! We're going to use the same process that's served us so well so far: substitute 2 for 'x' in our function rule y = -2x + 5 and solve for 'y'. So, here we go: y = -2(2) + 5. First up, we multiply -2 by 2, which gives us -4. So, our equation becomes: y = -4 + 5. Now, we add -4 and 5. Remember that adding a negative number is the same as subtracting, so we're essentially doing 5 - 4, which equals 1. Therefore, when x = 2, y = 1. We've done it! We've successfully calculated the 'y' value for the last 'x' in our table. Give yourselves a pat on the back – you've earned it! The satisfaction of completing a problem like this comes from the methodical approach and the consistent application of the rules. It's like building a house, brick by brick. Each step is important, and the final result is a testament to your hard work and attention to detail. Now, take a moment to step back and look at the completed table. Do the values make sense in relation to each other? Can you see any patterns? This is an important step in the problem-solving process – checking your work and making sure that your answer is reasonable. It's like proofreading a paper before you submit it. You want to make sure that everything is correct and that your message is clear. Also, now that we have four points, we could even plot them on a graph and see the line that our function rule represents. This visual representation can provide another layer of understanding and confirmation that our calculations are correct. Alright, we've reached the finish line! Let's take a look at our completed table and summarize our findings.

The Completed Table

Okay, let's take a look at the masterpiece we've created! Our completed table, filled with the 'y' values that correspond to each 'x' value according to the function rule y = -2x + 5, looks like this:

Isn't it satisfying to see all those values neatly arranged? This table represents a set of solutions to our function rule. Each row is a pair of 'x' and 'y' values that make the equation y = -2x + 5 true. It's like a little treasure map, showing us the coordinates of points that lie on the line represented by this equation. Take a moment to appreciate the pattern in the table. Notice how as 'x' increases, 'y' decreases? This is due to the negative coefficient (-2) in front of the 'x' in our function rule. For every increase of 1 in 'x', 'y' decreases by 2. This is a key characteristic of linear functions, and it's important to be able to recognize these patterns. This pattern recognition is what turns problem-solving from a rote exercise into a deeper understanding of the relationships between numbers. It's like learning to read the weather – you start to notice the signs and predict what's going to happen. Also, notice how the value of 'y' when x = 0 is 5? This is the y-intercept of the line, the point where the line crosses the y-axis on a graph. The y-intercept is always the constant term in our function rule, which in this case is +5. Understanding this connection can be a valuable shortcut when working with linear functions. Think of the completed table as a snapshot of our function. It gives us a glimpse into the behavior of the equation and the relationship between 'x' and 'y'. But it's not the whole story – there are infinitely many other 'x' and 'y' values that satisfy this function rule. Our table is just a small sample, but it's enough to give us a good understanding of what's going on. So, let's take this understanding and move on to the next step: summarizing what we've learned and reinforcing the key concepts.

Key Takeaways and Further Exploration

Alright, guys, we've reached the end of our journey! We've successfully filled in the table using the function rule y = -2x + 5, and along the way, we've learned some valuable lessons about functions and how they work. Let's recap the key takeaways and think about how we can take this knowledge even further. Firstly, we learned that a function rule is like a set of instructions that tells us how to transform an input ('x' value) into an output ('y' value). It's a consistent, methodical process that can be applied to any function, not just the one we worked with today. This understanding is the foundation for working with functions of all kinds, from simple linear equations to more complex mathematical models. Thinking of the function rule as a set of instructions or a recipe is a powerful analogy that can help to demystify the process. Secondly, we saw the importance of following the order of operations (PEMDAS/BODMAS) when simplifying expressions. This is a fundamental skill in mathematics that's crucial for getting the correct answer. Remember to always perform operations in the correct order: parentheses/brackets first, then exponents/orders, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). This consistent application of the rules is what ensures accuracy and prevents errors. Thirdly, we observed patterns in our table and connected them to the function rule. We saw how the negative coefficient in front of the 'x' caused 'y' to decrease as 'x' increased, and we identified the y-intercept as the constant term in the equation. These kinds of observations are what turn you from a problem-solver into a mathematical thinker. It's about seeing the connections and understanding the underlying principles. So, what's next? Well, the fun doesn't have to stop here! You can explore this topic further by: Trying different function rules: Experiment with different coefficients and constant terms and see how they affect the table of values. Graphing the function: Plot the points from your table on a graph and see the line that the function rule represents. This visual representation can provide a deeper understanding of the relationship between 'x' and 'y'. Solving for 'x': Try working backwards! Given a 'y' value, can you use the function rule to solve for 'x'? This will challenge your understanding of the process and help you to think about functions in a new way. The world of functions is vast and fascinating, and there's always more to learn. The key is to keep practicing, keep exploring, and keep asking questions. So, go forth and conquer those functions! You've got this!