Finding The Linear Function From Point-Slope Form

Which linear function represents the line given by the point-slope equation $y+1=-3(x-5)$? Options: A. $f(x)=-3 x-6$ B. $f(x)=-3 x-4$ C. $f(x)=-3 x+16$ D. $f(x)=-3 x+14$

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In the realm of linear equations, understanding the various forms and their interrelationships is paramount. Among these forms, the point-slope equation holds a unique position, providing a direct link between a line's slope and a specific point it traverses. This article delves into the process of converting a point-slope equation into the more familiar slope-intercept form, thereby revealing the underlying linear function. Specifically, we will dissect the equation $y + 1 = -3(x - 5)$, meticulously transforming it to identify the equivalent linear function from the options provided. This exploration not only reinforces algebraic manipulation skills but also deepens the comprehension of how different equation forms represent the same linear relationship. So, let's embark on this mathematical journey, unraveling the intricacies of linear functions and their diverse representations.

Understanding the Point-Slope Form

Before we dive into the specific problem, let's first solidify our understanding of the point-slope form of a linear equation. This form, mathematically expressed as $y - y_1 = m(x - x_1)$, elegantly encapsulates the essence of a line's characteristics. Here, 'm' represents the slope, a measure of the line's steepness and direction, while $(x_1, y_1)$ denotes a specific point that lies on the line. The beauty of this form lies in its ability to directly incorporate a known point and slope, making it exceptionally useful when these two pieces of information are readily available. For instance, if we know a line passes through the point (2, 3) and has a slope of 4, we can immediately write its equation in point-slope form as $y - 3 = 4(x - 2)$. This form serves as a powerful bridge, connecting the geometric properties of a line—its slope and a point it passes through—with its algebraic representation. Mastering the point-slope form is not just about memorizing a formula; it's about grasping how the equation reflects the fundamental nature of a line's trajectory in the coordinate plane. It enables us to visualize and analyze linear relationships with greater clarity and precision.

Transforming Point-Slope to Slope-Intercept Form

The slope-intercept form, represented as $y = mx + b$, is another fundamental way to express a linear equation. In this form, 'm' retains its meaning as the slope of the line, while 'b' represents the y-intercept, the point where the line crosses the vertical axis. The slope-intercept form is particularly valuable because it readily reveals two key attributes of the line: its steepness (slope) and its intersection with the y-axis. Converting from point-slope form to slope-intercept form involves a series of algebraic manipulations aimed at isolating 'y' on one side of the equation. This process typically involves distributing the slope, combining like terms, and adding or subtracting constants to both sides of the equation. By transforming a point-slope equation into slope-intercept form, we gain a clearer picture of the line's behavior, making it easier to graph, compare with other lines, and analyze its characteristics. For example, if we have the equation $y - 2 = 3(x + 1)$ in point-slope form, we can distribute the 3 to get $y - 2 = 3x + 3$, and then add 2 to both sides to arrive at $y = 3x + 5$, which is the slope-intercept form. This transformation highlights that the line has a slope of 3 and intersects the y-axis at the point (0, 5). Understanding this conversion process is crucial for a comprehensive grasp of linear equations and their applications.

Solving the Problem: $y + 1 = -3(x - 5)$

Now, let's tackle the given equation: $y + 1 = -3(x - 5)$. Our mission is to transform this point-slope form into the slope-intercept form, which will allow us to identify the correct linear function from the provided options. The first step in this transformation involves distributing the -3 on the right side of the equation. This means multiplying -3 by both 'x' and -5 within the parentheses: $y + 1 = -3 * x + (-3) * (-5)$. This simplifies to $y + 1 = -3x + 15$. Our next goal is to isolate 'y' on the left side of the equation. To achieve this, we subtract 1 from both sides of the equation, maintaining the balance: $y + 1 - 1 = -3x + 15 - 1$. This simplifies to $y = -3x + 14$. Now, we have successfully converted the equation into slope-intercept form, $y = -3x + 14$. This form clearly reveals that the line has a slope of -3 and a y-intercept of 14. Comparing this result with the given options, we can confidently identify the correct linear function.

Identifying the Correct Linear Function

Having transformed the equation $y + 1 = -3(x - 5)$ into the slope-intercept form $y = -3x + 14$, we can now easily identify the correct linear function. Recall that the options provided are in the form $f(x) = mx + b$, which is essentially the same as the slope-intercept form, just using function notation. By comparing our transformed equation, $y = -3x + 14$, with the general form $f(x) = mx + b$, we can see that 'm', the slope, is -3, and 'b', the y-intercept, is 14. Now, let's examine the given options:

A. $f(x) = -3x - 6$ B. $f(x) = -3x - 4$ C. $f(x) = -3x + 16$ D. $f(x) = -3x + 14$

By direct comparison, we can see that option D, $f(x) = -3x + 14$, perfectly matches our transformed equation. The slope is -3, and the y-intercept is 14, exactly as we calculated. Therefore, option D is the correct linear function that represents the line given by the point-slope equation $y + 1 = -3(x - 5)$. This exercise highlights the power of algebraic manipulation in converting between different forms of linear equations and the importance of understanding the significance of slope and y-intercept in defining a line.

Conclusion

In conclusion, we've successfully navigated the process of converting a point-slope equation into its equivalent slope-intercept form and, in turn, identified the correct linear function. By meticulously applying algebraic principles—distributing, combining like terms, and isolating the variable 'y'—we transformed $y + 1 = -3(x - 5)$ into $y = -3x + 14$. This transformation allowed us to directly match the equation with the correct option, $f(x) = -3x + 14$. This exercise underscores the interconnectedness of different representations of linear equations and the importance of mastering algebraic manipulation techniques. Furthermore, it reinforces the understanding of slope and y-intercept as key characteristics that define a line. The ability to seamlessly transition between equation forms empowers us to analyze and interpret linear relationships with greater confidence and accuracy. As we continue our exploration of mathematics, this foundational understanding of linear functions will undoubtedly serve as a cornerstone for more advanced concepts and applications.