Finding The Linear Function From Point-Slope Form Y-2=4(x-3)

Hey everyone! Today, we're diving into a super common math problem: figuring out which linear function matches a line given in point-slope form. We'll break down the equation y - 2 = 4(x - 3) step-by-step, making sure everyone understands how to transform it into the standard slope-intercept form. By the end of this article, you'll be a pro at converting point-slope equations and identifying the correct linear function. Let's get started!

Understanding Point-Slope Form

Before we jump into solving our specific problem, let's quickly recap what point-slope form actually means. The point-slope form is a way to represent a linear equation, and it looks like this: y - y₁ = m(x - x₁). In this equation:

• m represents the slope of the line.
• (x₁, y₁) represents a specific point that the line passes through.

Point-slope form is super handy because it lets us write the equation of a line if we know just one point on the line and its slope. In our case, the given equation y - 2 = 4(x - 3) is already in point-slope form, which is awesome! We can immediately see that the slope m is 4, and the point (x₁, y₁) is (3, 2). This is our starting point for converting this into a more familiar form.

Understanding this form is crucial because it bridges the gap between geometric information (a point and a slope) and the algebraic representation of a line. Recognizing the components immediately—the slope and the point—sets us up for easily manipulating the equation. Think of it like having a map; point-slope form gives you a starting location and a direction (slope), which you can then use to navigate to other forms of the equation. Without grasping this foundational concept, transforming the equation can feel like trying to solve a puzzle without knowing the rules. So, let's keep this definition in mind as we move forward and unravel the mystery of converting y - 2 = 4(x - 3) into a recognizable linear function.

Converting to Slope-Intercept Form

Now, our main goal is to figure out which linear function from the options (A, B, C, or D) represents the same line as y - 2 = 4(x - 3). To do this, we need to convert the point-slope form into the slope-intercept form, which is y = mx + b. Here, m is still the slope, and b is the y-intercept (the point where the line crosses the y-axis).

So, how do we do this? It's actually pretty straightforward. We'll use the distributive property and some basic algebra to isolate y on one side of the equation. Let's break it down step-by-step:

1. Distribute the 4: Start by distributing the 4 on the right side of the equation: 4 * x and 4 * -3. This gives us y - 2 = 4x - 12. This step is crucial because it begins to unravel the parentheses, allowing us to separate the terms and eventually isolate y. Think of it as the first major move in our algebraic dance – we're expanding the equation to reveal its hidden structure.
2. Isolate y: Now, we want to get y all by itself on the left side. To do this, we need to get rid of the -2. The opposite of subtracting 2 is adding 2, so we'll add 2 to both sides of the equation. This ensures we maintain the balance of the equation. This step is all about bringing the equation into a cleaner, more recognizable format, similar to organizing your desk before starting a big project. This gives us y - 2 + 2 = 4x - 12 + 2.
3. Simplify: Now, let's simplify both sides. -2 + 2 cancels out on the left, leaving us with just y. On the right side, -12 + 2 simplifies to -10. So, we have y = 4x - 10. Voila! We've successfully converted the equation into slope-intercept form. This is our final transformation, and it presents the equation in a clear, easy-to-understand manner. We've arrived at our destination after navigating through the algebraic steps.

By following these steps, we've transformed the point-slope equation into the slope-intercept equation, making it much easier to compare with the options provided and identify the correct linear function. This process highlights the power of algebraic manipulation in revealing the underlying structure of equations and making them more accessible.

Identifying the Correct Linear Function

Okay, now that we've converted our equation into slope-intercept form (y = 4x - 10), we're ready to match it with one of the given options. Remember, the slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.

Let's quickly look at our equation again: y = 4x - 10. We can see that:

• The slope (m) is 4.
• The y-intercept (b) is -10.

Now, let's review the options:

A. f(x) = 6x - 1 B. f(x) = 8x - 6 C. f(x) = 4x - 14 D. f(x) = 4x - 10

Comparing our equation y = 4x - 10 with the options, we can see that option D. f(x) = 4x - 10 perfectly matches our converted equation. The slope is 4, and the y-intercept is -10, just like in our equation. It’s like finding the missing piece of a puzzle—everything lines up perfectly!

Options A, B, and C have different slopes and/or y-intercepts, meaning they represent different lines. They might look similar at first glance, but the key is to focus on the specific values of the slope and y-intercept. This careful comparison is what allows us to confidently identify the correct answer. So, by systematically converting the point-slope form to slope-intercept form, we were able to clearly see which linear function represents the same line, highlighting the importance of understanding these different forms and their components.

Why Other Options Are Incorrect

To really solidify our understanding, let's briefly discuss why the other options are incorrect. This is super helpful because it reinforces what we've learned about slope-intercept form and helps us avoid similar mistakes in the future. Sometimes, knowing why something is wrong is just as important as knowing why something is right!

• Option A: f(x) = 6x - 1
  • This equation has a slope of 6 and a y-intercept of -1. The slope is different from our equation's slope of 4, and the y-intercept is different from our equation's y-intercept of -10. So, this line is completely different from the one represented by y - 2 = 4(x - 3). It’s like comparing apples and oranges – they are both fruits, but they are not the same.
• Option B: f(x) = 8x - 6
  • This equation has a slope of 8 and a y-intercept of -6. Again, the slope and y-intercept do not match our equation. The slope is way off, and the y-intercept is also different. This further emphasizes the importance of getting both the slope and y-intercept correct. It’s not enough to just have one match; both must align for the lines to be the same.
• Option C: f(x) = 4x - 14
  • This equation has the correct slope of 4, but its y-intercept is -14, which is different from our equation's y-intercept of -10. This option is a tricky one because it shares the same slope, meaning the lines are parallel. However, since the y-intercepts differ, these parallel lines will never intersect at the same point on the y-axis. This highlights that even if the slopes are the same, the y-intercepts must also match for the equations to represent the same line.

By understanding why these options are incorrect, we gain a deeper appreciation for the nuances of linear equations. We see that both the slope and y-intercept play critical roles in defining a line, and any deviation in these values results in a different line. This level of understanding not only helps us solve the current problem but also prepares us for more complex mathematical challenges in the future.

Conclusion

So, there you have it! We've successfully converted the point-slope equation y - 2 = 4(x - 3) into slope-intercept form and identified the correct linear function: D. f(x) = 4x - 10. We've journeyed through understanding point-slope form, converting it to slope-intercept form, and carefully comparing our result with the given options.

This process is a fundamental skill in algebra, and mastering it opens doors to solving more complex problems. Remember, the key is to take it step-by-step, understand the underlying concepts, and double-check your work. Whether you're prepping for a test or just want to brush up on your algebra skills, knowing how to convert between different forms of linear equations is a valuable asset. Keep practicing, and you'll become a pro in no time!

If you found this article helpful, give it a thumbs up and share it with your friends who might be struggling with linear functions. And as always, keep exploring the fascinating world of mathematics!