Choosing The Right Linear Function For A Point-Slope Equation
Hey guys! Today, we're diving into the world of linear functions and tackling a common problem: how to choose the correct linear function when given a point-slope equation. It might seem a bit tricky at first, but trust me, once you understand the fundamentals, it's a piece of cake. So, let's break down the point-slope equation, explore how it relates to the slope-intercept form, and ultimately, figure out which linear function matches our given equation. This is a crucial skill in algebra, and it pops up everywhere, from graphing lines to solving real-world problems. We'll take a step-by-step approach, making sure everyone's on board. So, buckle up and let's get started!
Understanding the Point-Slope Form
Let's kick things off by understanding the point-slope form. This form is a fantastic way to represent a linear equation because it directly incorporates a point on the line and the line's slope. The general form looks like this: y - y₁ = m(x - x₁). Now, what does each part mean? Well, m represents the slope of the line, which tells us how steep the line is and its direction (whether it's going upwards or downwards). The point (x₁, y₁) is a specific point that the line passes through. Think of it as an anchor, fixing the line in a certain spot on the graph. The x and y are variables representing any point on the line. Essentially, this equation says: “The change in y from a specific point is equal to the slope times the change in x from that same point.” This form is super useful because if you know a point and the slope, you can immediately write the equation of the line. It avoids the need to calculate the y-intercept first, which is necessary in the slope-intercept form. The point-slope form is especially handy when you're given a slope and a point, or when you need to construct an equation quickly without plotting points and calculating rises and runs. It's all about capturing the essence of a line – its direction and a fixed location – in a neat algebraic package. Once you grasp this concept, converting to other forms or analyzing line behavior becomes a whole lot easier.
Converting Point-Slope to Slope-Intercept Form
Now, let's talk about converting from point-slope form to slope-intercept form. Why do we need to do this? Well, the slope-intercept form, y = mx + b, is a super common and easy-to-interpret way to represent a linear equation. It directly tells us the slope (m) and the y-intercept (b), which is the point where the line crosses the y-axis. Converting to this form often makes it easier to compare different linear functions and to quickly sketch the graph of the line. The process involves a bit of algebraic manipulation, but it's quite straightforward. The first step is to distribute the slope (m) across the terms inside the parentheses in the point-slope equation. So, if we have y - y₁ = m(x - x₁), we multiply m by both x and x₁. This gives us y - y₁ = mx - mx₁. The next step is to isolate y on the left side of the equation. To do this, we add y₁ to both sides of the equation. This results in y = mx - mx₁ + y₁. Now, notice that -mx₁ + y₁ is a constant term – it's just a number. This constant term represents the y-intercept (b) in the slope-intercept form. So, we can rewrite the equation as y = mx + b, where b = -mx₁ + y₁. Voila! We've successfully converted from point-slope to slope-intercept form. This conversion is a fundamental skill, allowing us to seamlessly move between different representations of linear equations and gain a deeper understanding of their properties. By mastering this conversion, you'll be able to tackle a wider range of linear equation problems with confidence.
Applying the Conversion to the Given Equation
Okay, let's get practical and apply what we've learned to the equation given in the problem: y - 5 = 3(x - 2). Our mission is to convert this from point-slope form into slope-intercept form, which will help us identify the correct linear function. Remember, the goal is to get the equation into the form y = mx + b. The first step, as we discussed, is to distribute the slope, which in this case is 3, across the terms inside the parentheses. So, we multiply 3 by both x and -2. This gives us y - 5 = 3x - 6. Notice how we've carefully handled the signs – it's crucial to get this right! Next, we need to isolate y on the left side of the equation. To do this, we add 5 to both sides. This gives us y = 3x - 6 + 5. Now, we simplify the constant terms on the right side. -6 plus 5 equals -1, so our equation becomes y = 3x - 1. Bingo! We've successfully converted the point-slope equation into slope-intercept form. Looking at this, we can immediately see that the slope (m) is 3 and the y-intercept (b) is -1. This form makes it super clear how the line behaves – it rises 3 units for every 1 unit it moves to the right, and it crosses the y-axis at the point (0, -1). This conversion process is a cornerstone of linear equation manipulation, and mastering it will significantly boost your confidence in solving related problems. Now that we have the equation in slope-intercept form, we're ready to compare it to the answer choices and find the correct linear function.
Identifying the Correct Linear Function
Now comes the fun part: identifying the correct linear function from the given options. We've successfully converted our point-slope equation, y - 5 = 3(x - 2), into the slope-intercept form, which is y = 3x - 1. Remember, linear functions are often written in function notation, where y is replaced with f(x). So, our equation in function notation is f(x) = 3x - 1. Now, let's take a look at the answer choices:
A. f(x) = 3x + 1 B. f(x) = 3x - 1 C. f(x) = 8x + 10 D. f(x) = 8x - 10
Comparing our derived function, f(x) = 3x - 1, to the options, we can clearly see that option B, f(x) = 3x - 1, matches perfectly. The slope (3) and the y-intercept (-1) are exactly the same. Options A, C, and D have different slopes and/or y-intercepts, meaning they represent different lines altogether. This process highlights the power of converting to slope-intercept form. It allows for a direct comparison and eliminates any ambiguity. By understanding the relationship between the point-slope form, slope-intercept form, and function notation, you can confidently tackle these types of problems. This skill is not just about finding the right answer; it's about developing a deeper understanding of linear functions and their various representations. This understanding will be invaluable as you progress in your mathematical journey.
Conclusion: Mastering Linear Functions
Alright, guys, we've reached the end of our journey through point-slope equations and linear functions! We've successfully navigated the conversion process, identified the correct linear function, and hopefully, you've gained a solid understanding of the concepts involved. To recap, we started by understanding the point-slope form, y - y₁ = m(x - x₁), and how it represents a line using a point and the slope. We then learned how to convert this to the slope-intercept form, y = mx + b, which makes it easy to identify the slope and y-intercept. By applying this conversion to the given equation, y - 5 = 3(x - 2), we arrived at y = 3x - 1, or f(x) = 3x - 1 in function notation. Finally, we compared this to the answer choices and confidently selected the correct linear function, which was option B, f(x) = 3x - 1. Mastering these skills is crucial for success in algebra and beyond. Linear functions are the building blocks for more advanced mathematical concepts, and a strong foundation here will pay dividends in the future. Keep practicing, keep exploring, and remember that math is a journey of discovery. So, until next time, keep those equations balanced and those slopes rising!
How to choose a linear function that represents the line given by the point-slope equation ?
Choosing the Right Linear Function for a Point-Slope Equation
