Identifying The Linear Function For Y-8=1/2(x-4)

Introduction: Understanding Linear Functions and Point-Slope Form

In the realm of mathematics, linear functions form the bedrock of numerous concepts and applications. Understanding these functions is crucial for grasping more advanced mathematical ideas. A linear function is typically represented in the slope-intercept form, which is y = mx + b, where m denotes the slope and b represents the y-intercept. However, lines can also be expressed in point-slope form, offering a flexible way to define a line using a single point and the slope. The point-slope form of a linear equation is given by y - y₁ = m(x - x₁), where (x₁, y₁) is a specific point on the line and m is the slope. This form is particularly useful when you have a point and the slope, allowing you to easily write the equation of the line. Converting between different forms of linear equations, such as from point-slope to slope-intercept form, is a fundamental skill in algebra. This conversion involves manipulating the equation to isolate y on one side, revealing the slope and y-intercept explicitly. In this article, we'll explore how to convert a point-slope equation into a linear function in slope-intercept form. Understanding the relationship between these forms enhances our ability to analyze and solve linear equations effectively. By mastering these concepts, we can tackle a wide range of problems involving linear relationships, making this knowledge invaluable in both academic and practical contexts. We will focus on how to identify the correct linear function in slope-intercept form that corresponds to a given point-slope equation. This involves algebraic manipulation and a clear understanding of the properties of linear equations, ensuring we arrive at the accurate representation of the line.

Problem Statement: Converting Point-Slope Form to Slope-Intercept Form

To effectively address the problem, we begin by stating it clearly: Which linear function represents the line given by the point-slope equation y - 8 = (1/2)(x - 4)? This question requires us to transform the given equation, which is in point-slope form, into slope-intercept form. The slope-intercept form of a linear equation is expressed as f(x) = mx + b, where m represents the slope of the line and b represents the y-intercept. The point-slope form, on the other hand, is given by y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. Our task is to manipulate the given point-slope equation algebraically to match the slope-intercept form. This involves distributing the slope, adding constants, and isolating y to one side of the equation. The key to solving this problem lies in understanding the relationship between these two forms and applying algebraic principles correctly. By carefully performing each step, we can accurately convert the equation and identify the linear function that represents the given line. The options provided are in the form of f(x) = mx + b, which makes it easier to directly compare our result with the given choices. This methodical approach ensures that we arrive at the correct answer by systematically converting the point-slope equation into its equivalent slope-intercept form.

Step-by-Step Solution: Transforming the Equation

Let's embark on a detailed, step-by-step journey to convert the given point-slope equation, y - 8 = (1/2)(x - 4), into the slope-intercept form, which is f(x) = mx + b. This process involves a series of algebraic manipulations to isolate y and express the equation in the desired format.

1. Distribute the slope:

   The first step in converting the equation is to distribute the slope, which is 1/2, across the terms inside the parentheses. This means multiplying 1/2 by both x and -4. This gives us:

   y - 8 = (1/2)x - (1/2)(4)

   Simplifying the multiplication, we get:

   y - 8 = (1/2)x - 2

   This step is crucial because it begins the process of separating y from the rest of the equation, bringing us closer to the slope-intercept form.

2. Isolate y:

   Now, our goal is to isolate y on one side of the equation. To do this, we need to eliminate the -8 on the left side. We achieve this by adding 8 to both sides of the equation. This maintains the balance of the equation while moving us closer to the desired form:

   y - 8 + 8 = (1/2)x - 2 + 8

   Simplifying both sides, we obtain:

   y = (1/2)x + 6

   At this point, we have successfully isolated y, and the equation is now in slope-intercept form.

3. Express as a function f(x):

   The final step is to express the equation as a function f(x). In slope-intercept form, we replace y with f(x). So, our equation becomes:

   f(x) = (1/2)x + 6

   This completes the conversion, giving us the linear function in slope-intercept form that represents the given line.

Through these steps, we have systematically transformed the point-slope equation into the slope-intercept form, making it easy to identify the slope and y-intercept. This methodical approach ensures accuracy and a clear understanding of the underlying algebraic principles. With the equation now in the form f(x) = (1/2)x + 6, we can readily compare it with the given options to find the correct answer.

Identifying the Correct Option: Matching the Solution

Having converted the point-slope equation y - 8 = (1/2)(x - 4) into the slope-intercept form f(x) = (1/2)x + 6, our next step is to identify the correct option among the given choices. This involves a direct comparison of our derived equation with the provided options to find a perfect match.

Let's revisit the options:

A. f(x) = (1/2)x + 4 B. f(x) = (1/2)x + 6 C. f(x) = (1/2)x - 10 D. f(x) = (1/2)x - 12

By carefully comparing our solution, f(x) = (1/2)x + 6, with the options, we can see that option B, f(x) = (1/2)x + 6, exactly matches our result. The slope (1/2) and the y-intercept (6) are identical in both the derived equation and option B.

The other options can be easily ruled out:

• Option A, f(x) = (1/2)x + 4, has the correct slope but an incorrect y-intercept.
• Options C, f(x) = (1/2)x - 10, and D, f(x) = (1/2)x - 12, also have the correct slope but incorrect y-intercepts.

Thus, the correct option is B, f(x) = (1/2)x + 6. This process of matching our solution with the given options highlights the importance of accuracy in each step of the conversion. A single error in the algebraic manipulation could lead to an incorrect equation, and consequently, the wrong choice. By methodically working through the steps and carefully comparing the result, we ensure that we select the correct linear function that represents the given line.

Conclusion: The Linear Function in Slope-Intercept Form

In conclusion, the linear function that represents the line given by the point-slope equation y - 8 = (1/2)(x - 4) is f(x) = (1/2)x + 6. This solution was achieved through a methodical conversion process, which involved distributing the slope, isolating y, and expressing the equation in slope-intercept form. The key steps included distributing the 1/2 across the terms in the parentheses, adding 8 to both sides of the equation to isolate y, and finally, writing the equation in the functional notation f(x). The accuracy of each step was crucial in arriving at the correct result.

The process of converting from point-slope form to slope-intercept form is a fundamental skill in algebra, and it is essential for understanding and working with linear functions. The point-slope form, y - y₁ = m(x - x₁), is particularly useful when you have a point and the slope, while the slope-intercept form, f(x) = mx + b, provides a clear view of the slope and y-intercept of the line. By mastering this conversion, we can easily switch between these forms and use them to solve various problems involving linear equations.

The correct option, f(x) = (1/2)x + 6, matches our derived equation perfectly, confirming the accuracy of our step-by-step solution. The other options were ruled out because they did not have the correct y-intercept, highlighting the importance of careful comparison and attention to detail. This exercise demonstrates the significance of understanding the properties of linear equations and the ability to manipulate them algebraically to find the desired form. Through this detailed exploration, we have reinforced the understanding of linear functions and their representation in different forms.