Finding The Y-Intercept Of G(x) = 3x A Step-by-Step Guide

Hey guys! Let's dive into a fundamental concept in mathematics: the y-intercept. Specifically, we're going to explore how to find the y-intercept of a linear function. Linear functions, those neat lines we see on graphs, are described by equations, and understanding their components helps us grasp the line's behavior. Our focus today is on the function g(x) = 3x, a straightforward example that perfectly illustrates this concept. You might be thinking, "Okay, y-intercept... what's the big deal?" Well, the y-intercept is a crucial point on the graph of a line. It's the spot where the line crosses the y-axis, providing us with a valuable reference point. Think of it as the line's starting point on the vertical axis. Knowing the y-intercept, along with other information like the slope, allows us to accurately visualize and analyze the line's trajectory. In practical terms, the y-intercept can represent an initial value in a real-world scenario. For instance, if we're modeling the cost of a service based on the number of hours, the y-intercept might represent the fixed cost or the base fee charged regardless of the hours used. So, understanding how to find it is not just about math class; it has real-world implications. Now, before we jump directly into g(x) = 3x, let's establish some groundwork. Remember the standard form of a linear equation: y = mx + b. Here, m represents the slope of the line, which tells us how steep the line is and in what direction it's going. The slope is essentially the rate of change of y with respect to x. In simpler terms, it's how much y changes for every one unit change in x. The b in the equation is what we're most interested in today – the y-intercept. It's the value of y when x is equal to zero. This makes sense because any point on the y-axis has an x-coordinate of 0. So, to find the y-intercept, we essentially need to figure out what y is when x is zero. This is a fundamental principle that will guide us through solving our problem. We can find the y-intercept by simply substituting x = 0 into the equation and solving for y. This is a straightforward process, but it's built on a solid understanding of what the y-intercept represents. So, let's keep this foundational knowledge in mind as we proceed.

Analyzing the Function g(x) = 3x

Now that we've refreshed our understanding of the y-intercept and its significance, let's turn our attention to the specific function we're tackling today: g(x) = 3x. This function might look simple, but it holds all the keys to understanding the concept. First, let's recognize that g(x) is just another way of representing y. So, we can rewrite the function as y = 3x. This form makes it easier to see the function in the context of a standard linear equation. Comparing y = 3x to the standard form y = mx + b, we can immediately identify some key characteristics. The coefficient of x, which is 3 in this case, represents the slope (m) of the line. So, the line has a slope of 3, meaning it rises 3 units for every 1 unit increase in x. This indicates a fairly steep line that goes upwards as you move from left to right on the graph. Now, what about the y-intercept (b)? This is where things get interesting. Notice that there's no constant term added or subtracted in the equation y = 3x. In other words, we can think of it as y = 3x + 0. This is a crucial observation! By explicitly writing it as y = 3x + 0, we can clearly see that the y-intercept (b) is 0. This means the line crosses the y-axis at the point (0, 0), which is also the origin of the coordinate plane. Geometrically, this makes the line pass through the center of our graph, dividing it into quadrants. But, let's not just rely on visual inspection. We can mathematically confirm this by using the principle we discussed earlier: to find the y-intercept, we set x = 0 and solve for y. So, let's substitute x = 0 into the equation y = 3x: y = 3 * (0) y = 0 This calculation definitively shows that when x is 0, y is also 0. Thus, the y-intercept is indeed 0, which corresponds to the point (0, 0) on the graph. This process of substitution is a powerful tool, and it's a foolproof method for finding the y-intercept of any linear function. So, we've now both visually and mathematically determined the y-intercept of g(x) = 3x. It's a straightforward example, but it lays the foundation for tackling more complex linear functions. We can see how the absence of a constant term directly translates to a y-intercept of 0. This understanding is key to quickly analyzing linear equations and their corresponding graphs. We can confidently say that the line represented by g(x) = 3x passes through the origin, making the y-intercept a fundamental characteristic of this function.

Determining the Y-Intercept: Step-by-Step Solution

Alright guys, let's formalize the step-by-step process of finding the y-intercept for the function g(x) = 3x. This will not only solidify our understanding but also provide a clear method that can be applied to any linear function. We've already intuitively and visually grasped the concept, but let's break it down into concrete steps. Step 1: Rewrite the function in the form y = mx + b. As we've discussed, g(x) is simply another way of writing y, so we can rewrite the function as y = 3x. This step emphasizes that we're dealing with a linear equation, and it sets us up to easily identify the slope and y-intercept. Step 2: Identify the slope (m) and the y-intercept (b). Comparing y = 3x to the standard form y = mx + b, we can see that the slope (m) is 3. This is the coefficient of x, and it tells us the steepness and direction of the line. Now, for the y-intercept (b), we notice that there's no constant term added or subtracted in the equation y = 3x. This is the same as saying y = 3x + 0. Therefore, the y-intercept (b) is 0. This means the line crosses the y-axis at the point (0, 0). Step 3: Verify the y-intercept by substituting x = 0 into the equation. This is a crucial step to confirm our findings and ensure accuracy. Substitute x = 0 into the equation y = 3x: y = 3 * (0) y = 0 This calculation confirms that when x is 0, y is also 0. This reinforces our conclusion that the y-intercept is 0. Step 4: State the y-intercept. Based on our analysis and calculations, we can confidently state that the y-intercept of the function g(x) = 3x is 0. This corresponds to the point (0, 0) on the graph. So, there you have it! A clear, step-by-step method for determining the y-intercept. These steps are not just applicable to this specific function; they can be used for any linear equation. By following this process, you can quickly and accurately identify the y-intercept, which is a fundamental characteristic of the line. Remember, the key is to rewrite the function in the y = mx + b form, identify the constant term (b), and verify your answer by substitution. This method will become second nature with practice, and you'll be able to find y-intercepts in a snap! We've now successfully navigated through the process, ensuring a solid understanding of how to pinpoint the y-intercept of a linear function. This foundation will be invaluable as we tackle more complex mathematical concepts.

Choosing the Correct Answer

Having thoroughly analyzed the function g(x) = 3x and determined its y-intercept, we can now confidently choose the correct answer from the given options. Let's quickly recap what we've established. We started by understanding the significance of the y-intercept as the point where the line crosses the y-axis. We then rewrote g(x) = 3x as y = 3x and compared it to the standard linear equation form, y = mx + b. By recognizing that the constant term (b) is 0 in this case, we identified the y-intercept as 0. To verify this, we substituted x = 0 into the equation and confirmed that y is indeed 0. Therefore, the y-intercept of g(x) = 3x is 0, corresponding to the point (0, 0) on the graph. Now, let's look at the answer choices provided:

A. 0 B. 1 C. 2 D. 3

Clearly, option A, which is 0, matches our calculated y-intercept. The other options (1, 2, and 3) are incorrect as they do not align with our analysis. Therefore, the correct answer is A. 0. Choosing the correct answer is the final step in the problem-solving process, and it demonstrates our comprehensive understanding of the concept. We didn't just guess; we systematically worked through the problem, applying our knowledge of linear functions and y-intercepts. This methodical approach is crucial for success in mathematics. We not only arrived at the correct answer but also gained a deeper understanding of the underlying principles. This understanding will serve us well as we encounter more complex problems in the future. By taking the time to analyze each step and verify our results, we ensure accuracy and build confidence in our problem-solving abilities. So, we can confidently select option A and move forward, knowing that we have a solid grasp of the y-intercept concept. This process exemplifies how a strong foundation in mathematical concepts, combined with a systematic approach, leads to correct solutions and a deeper appreciation for the subject.

Conclusion: Mastering the Y-Intercept

In conclusion, guys, we've successfully navigated the process of finding the y-intercept of the function g(x) = 3x. We started with a fundamental understanding of what the y-intercept represents – the point where a line crosses the y-axis. We then applied this understanding to the specific function g(x) = 3x, rewriting it as y = 3x to clearly see its linear form. By comparing it to the standard form y = mx + b, we identified the y-intercept (b) as 0. We further validated our result by substituting x = 0 into the equation, confirming that y is indeed 0 when x is 0. This step-by-step approach not only led us to the correct answer (A. 0) but also reinforced our understanding of the underlying mathematical principles. Mastering the concept of the y-intercept is crucial in mathematics as it forms the basis for understanding linear functions and their graphs. It's not just about memorizing a formula; it's about grasping the significance of this point and how it relates to the overall behavior of the line. The y-intercept often represents a starting point or an initial value in real-world scenarios, making it a practical concept beyond the classroom. Our journey through this problem highlights the importance of a systematic approach to problem-solving. We didn't just jump to the answer; we broke down the problem into manageable steps, analyzed each step carefully, and verified our results. This methodical approach is applicable to a wide range of mathematical problems and is a key skill for success. Furthermore, we've seen how seemingly simple functions like g(x) = 3x can reveal fundamental concepts. Understanding these basic building blocks is essential for tackling more complex mathematical challenges. The ability to analyze linear equations, identify their components, and interpret their graphs is a valuable asset in various fields, from science and engineering to economics and finance. So, by mastering the y-intercept, we've not only solved a specific problem but also strengthened our mathematical foundation and equipped ourselves with valuable problem-solving skills. Keep practicing, keep exploring, and remember that every mathematical concept, no matter how simple it may seem, is a stepping stone to greater understanding and achievement.