Completing Function Tables: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into the world of linear equations and completing a function table. We'll be working with the equation $y = 4 - 3x$, and our goal is to find the corresponding $y$ values for the given $x$ values. It's like a fun puzzle, and I'll walk you through it step-by-step. So, grab your pencils, and let's get started!

Understanding the Function Rule and Table

Alright, first things first, let's understand what we're dealing with. The function rule, $y = 4 - 3x$, is a linear equation. This means that when you plot it on a graph, you'll get a straight line. The equation tells us how to calculate the value of $y$ for any given value of $x$. Think of $x$ as the input and $y$ as the output. The function rule is the magical formula that transforms the input into the output. Now, let's talk about the table. The table is a neat way to organize the input ($x$) and output ($y$) values. It helps us visualize the relationship between the $x$ and $y$ values. In our table, we already have some $x$ values and their corresponding $y$ values. Our job is to fill in the missing $y$ values. This is a crucial step in understanding how the function works and how the values of $x$ influence the values of $y$. We will use the equation $y = 4 - 3x$ to find the missing $y$ values. Remember, this equation tells us how to transform an $x$ value into its corresponding $y$ value.

To kick things off, let's consider the function rule $y = 4 - 3x$. This equation acts as a set of instructions. It tells us how to transform an input value ($x$) into an output value ($y$). For every $x$ value we plug in, we'll perform the operations described in the equation: multiply $x$ by $-3$ and then add $4$. This process will give us the corresponding $y$ value. So, when we say 'complete the table,' we are essentially finding the missing outputs ($y$ values) for the given inputs ($x$ values). It’s like following a recipe; the equation is the recipe, $x$ is an ingredient, and $y$ is the final dish! Understanding this relationship is super important because it forms the foundation of many mathematical concepts. Now, it is worth emphasizing that we're dealing with a linear equation. This means the relationship between $x$ and $y$ is constant, and the changes in $y$ will be proportional to the changes in $x$. As we progress through the table, notice how the values of $y$ change as the values of $x$ change. This consistent pattern is characteristic of all linear equations. The function table is a useful tool to explore the function rule and to understand how the values of $x$ and $y$ are related. Let's move on to the next section, where we will start calculating the missing $y$ values.

Filling in the Blanks: Calculating the Missing $y$ Values

Alright, let's get down to business and fill in those missing values! We'll tackle this one $x$ value at a time, using the equation $y = 4 - 3x$. It's like a treasure hunt, and we're looking for the hidden $y$ values!

Step 1: When $x = 0$

When $x = 0$, the equation becomes $y = 4 - 3(0)$. Following the order of operations (PEMDAS/BODMAS), we first multiply: $3 imes 0 = 0$. Then, we subtract: $4 - 0 = 4$. Therefore, when $x = 0$, $y = 4$. So, the first missing value in our table is 4. Remember that, we substitute the value of x into the given equation and simplify to find the corresponding y value. It's like plugging in the value of x into a machine, and the output is the value of y. Let us take a closer look at the steps: We start with the given equation $y = 4 - 3x$. We substitute $x$ with $0$ to get $y = 4 - 3(0)$. This means we multiply $-3$ by $0$, which equals $0$. Thus, the equation now reads $y = 4 - 0$, which simplifies to $y = 4$. That gives us our first missing $y$ value. Each time we substitute the given $x$ value into the equation, it’s like we're activating a calculation process. The function rule dictates what mathematical operation we need to perform with $x$ in order to find $y$. It's a straightforward approach, yet a really powerful tool for understanding how the $x$ and $y$ values are connected to each other.

Step 2: When $x = 1$

Now, let's find the $y$ value when $x = 1$. The equation becomes $y = 4 - 3(1)$. Again, following the order of operations, we first multiply: $3 imes 1 = 3$. Then, we subtract: $4 - 3 = 1$. So, when $x = 1$, $y = 1$. Let us break it down once again: We'll start with $y = 4 - 3x$. We substitute $x$ with $1$ to get $y = 4 - 3(1)$. We perform the multiplication first: $-3$ times $1$ is $-3$. The equation then reads $y = 4 - 3$, and we get $y = 1$. This demonstrates how a small change in $x$ leads to a change in $y$ values. Keep in mind that the relationship between the values of $x$ and $y$ is described by the function rule, meaning that any change in $x$ directly influences the final outcome in the function, which is represented by the value of $y$. We continue this process for each $x$ value to get the corresponding $y$ value. We have seen in the last examples that the function rule, $y = 4 - 3x$ is the key to find the corresponding values of $y$, by substituting the value of $x$ given. Now, we have another $y$ value to fill into the table.

Step 3: When $x = 2$

Finally, let's find the $y$ value when $x = 2$. The equation becomes $y = 4 - 3(2)$. Following the order of operations, we first multiply: $3 imes 2 = 6$. Then, we subtract: $4 - 6 = -2$. So, when $x = 2$, $y = -2$. Let's go through it one more time to ensure we have fully grasped the concept: We begin with $y = 4 - 3x$. We substitute $x$ with $2$ to get $y = 4 - 3(2)$. We multiply $-3$ by $2$, which is equal to $-6$. This leads to the equation $y = 4 - 6$. Consequently, we will get $y = -2$. As we fill in the table, it gives us a visual representation of the function's behavior, showing how the y-values change in response to the x-values. The pattern created helps us understand the rate of change of the function. These calculations showcase the impact of each x-value in generating the corresponding y-value. The final missing $y$ value in our table is $-2$. Great job, everyone! We have successfully found all the missing $y$ values.

Completed Table

Here's the completed table:

See, wasn't that fun? We've now completed the table for the function rule $y = 4 - 3x$. You've successfully found all the missing $y$ values by using the function rule. You did great, guys!

Conclusion: Mastering Function Tables

Congratulations! You've now successfully completed a function table. This is a fundamental skill in mathematics, and it's a great way to understand the relationship between $x$ and $y$ values. Remember that, function tables are more than just a way to organize numbers; they're a visual tool that helps us understand the behavior of functions. As we have seen, in the example, $x$ values affect the $y$ values in a predictable pattern. It's a crucial foundation for more advanced topics in algebra and calculus. You can use this approach to complete other tables for different functions. Keep practicing, and you'll become a pro in no time! Well done, guys, and keep exploring the wonderful world of math!