Finding The Zero Of A Linear Function: A Step-by-Step Guide

Hey guys! Today, we're diving into a fundamental concept in algebra: finding the zero of a linear function. Don't worry, it's not as scary as it sounds! It's all about finding the x-value where the function's output (y-value) equals zero. This is super important because it helps us understand where the function crosses the x-axis, which is a crucial point for graphing and solving equations. We'll be working through the specific problem of determining the zero of the function f(x) = 4x - 36, and I'll walk you through each step. Get ready to flex those math muscles and understand why this concept is so essential. Let's make this journey fun and easy to grasp. Ready to get started?

Understanding the Zero of a Function

Alright, before we get to the solution, let's break down what the 'zero of a function' actually means. When we talk about the zero of a function, we're essentially looking for the x-value that makes the function equal to zero. In other words, it's the point where the graph of the function intersects the x-axis. Think of it like this: the x-axis is your horizontal line, and the zero is where your function line 'touches down' on it. This point is also frequently referred to as the root or x-intercept of the function. Understanding this is key because it helps us visualize the function and how it behaves. Knowing the zero allows us to pinpoint where the function changes its sign, moving from negative to positive values, or vice versa. This is particularly useful in various real-world applications, such as physics, engineering, and economics, where we need to analyze how different variables relate to each other and find critical points in a system. By knowing the zero, we gain valuable insights into the function's properties and behavior.

So, finding the zero involves setting f(x) equal to 0 and solving for x. This simple concept underpins a lot of more complex mathematical ideas, so it’s super important to nail it down. For f(x) = 4x - 36, our goal is to find the value of x that makes the entire expression equal to zero. This is a fundamental skill in algebra and is essential for more advanced concepts. Think of the zero as a crucial landmark that helps us to understand the behavior of the function. It is important to remember that the zero of the function can be found by substituting 0 for the function f(x), or by putting the function equal to 0, which makes it easier to solve the equation. The process itself is not complicated, but understanding the concept is key. Are you with me so far? Let's move on to the actual solution!

Solving for the Zero: Step-by-Step

Alright, let’s get down to business and solve for the zero of our function, f(x) = 4x - 36. The approach is straightforward: We're going to set the function equal to zero and then isolate x. Here's the step-by-step breakdown. Remember, this is about finding the x-value that makes the function equal to zero. First, we replace f(x) with 0, which gives us the equation 0 = 4x - 36. Then, our goal is to get x alone on one side of the equation. To do this, we need to get rid of that -36. We do this by adding 36 to both sides of the equation, which keeps everything balanced. This leaves us with 36 = 4x.

Now, we need to get x completely by itself. It is multiplied by 4, so we need to do the opposite and divide both sides of the equation by 4. This gives us 36 / 4 = x. Finally, we simplify the left side of the equation: 36 / 4 = 9. Therefore, x = 9. Congratulations, you've found the zero of the function! This means that when x is equal to 9, the function f(x) = 4x - 36 equals zero. It's like finding a treasure; now you know the magic number that makes the function vanish! So, the zero of the function is the x-value where the graph intersects the x-axis. This process is applicable to other linear functions. To clarify, in this case, when x is 9, the value of the function is zero. It indicates the point where the line crosses the x-axis. Knowing this, we can now confidently select the correct answer from the multiple-choice options. You did a great job following along! You are ready to apply these steps to other linear functions. Understanding this is crucial because it helps us grasp the function's overall behavior. So, let’s choose the correct answer.

Selecting the Correct Answer

Okay, now that we've found the zero of the function f(x) = 4x - 36 to be x = 9, we can easily select the correct answer from the options provided. The question gives us several choices: (A) 9, (B) 36, (C) 4, (D) 12, and (E) 0. We've done the math, and we know that the x-value where the function equals zero is 9. This means that when we substitute x = 9 into our function, we should get zero. Let's do a quick check to confirm: f(9) = 4(9) - 36 = 36 - 36 = 0. Yep, it checks out! So, the correct answer is (A) 9. It’s that simple. By going through the steps, you've not only solved the problem but also deepened your understanding of the concept. Finding the zero of a linear function is a fundamental skill in algebra, and now you have a strong grasp of it. Remembering that the zero represents the x-intercept makes it easier to visualize and understand the function's behavior. We correctly identified 9 as the x-value that makes the function equal zero. You can now confidently solve similar problems. Feel proud of your success in understanding and solving for the zero of the function. Now that you've successfully found the zero, you can move on to other related concepts, such as graphing linear functions and understanding their properties. Well done! It's a great feeling, isn't it?

Conclusion: Zeroing in on Success

Awesome work, everyone! You've successfully found the zero of the linear function f(x) = 4x - 36. We started by understanding what the zero of a function means – the x-value where the function equals zero – and then went through a simple, step-by-step process to solve for it. By setting the function equal to zero and isolating x, we found our answer: x = 9. You should now feel confident in identifying the zero of any linear function! Remember that this concept forms the basis for more advanced mathematical ideas. It has real-world applications in many fields. You’ve unlocked a key piece of the puzzle. Being able to solve for the zero empowers you to analyze and understand how functions behave. This is an essential skill in algebra, and you've conquered it! Now, the next time you encounter a similar problem, you'll know exactly what to do. Keep practicing, and you'll become even more proficient. Remember that with practice, you'll become a pro at these problems. Well done, everyone. Keep up the excellent work, and I'll see you in the next lesson! You’re on the right track; keep exploring and practicing. See you next time! You can now apply this to other similar functions. Your skills are improving with each step!