Finding The Linear Function From Point-Slope Form Y-8=1/2(x-4)
In the realm of linear equations, understanding different forms and how to convert between them is a fundamental skill. This article delves into the process of transforming a point-slope equation into its linear function representation. We will specifically address the equation y - 8 = (1/2)(x - 4), meticulously demonstrating the steps to arrive at the correct linear function. This comprehensive guide aims to equip you with the knowledge and techniques to confidently tackle similar problems, enhancing your understanding of linear functions and their various forms.
Understanding Point-Slope Form
The point-slope form is a powerful way to represent a linear equation, especially when you know a specific point on the line and the line's slope. The general form of the point-slope equation is:
y - y1 = m(x - x1)
Where:
- (x1, y1) represents a specific point on the line.
- m represents the slope of the line.
In our given equation, y - 8 = (1/2)(x - 4), we can immediately identify the point and the slope:
- (x1, y1) = (4, 8)
- m = 1/2
This tells us that the line passes through the point (4, 8) and has a slope of 1/2. The slope indicates that for every 2 units you move horizontally (run), the line rises 1 unit vertically (rise). This understanding is crucial for visualizing the line and its behavior on a coordinate plane. Furthermore, the point-slope form provides a direct link between the geometric properties of the line (a point and its slope) and its algebraic representation. This connection is invaluable for solving various problems in coordinate geometry and linear algebra. The ability to quickly identify the slope and a point from the point-slope form is a key skill in analyzing and manipulating linear equations.
Converting to Slope-Intercept Form
The slope-intercept form is another common way to represent a linear equation, and it's particularly useful for identifying the slope and y-intercept of the line. The general form of the slope-intercept equation is:
y = mx + b
Where:
- m represents the slope of the line (as in point-slope form).
- b represents the y-intercept, which is the point where the line crosses the y-axis.
Our goal is to transform the given point-slope equation, y - 8 = (1/2)(x - 4), into this slope-intercept form. This involves a few algebraic steps, primarily focusing on isolating y on one side of the equation. The process begins with distributing the slope (1/2) across the terms inside the parentheses. This eliminates the parentheses and sets the stage for further simplification. After distributing, we will add 8 to both sides of the equation to isolate y. This step maintains the equality of the equation while moving the constant term to the right-hand side. The final result will be an equation in the form y = mx + b, allowing us to easily identify the slope and y-intercept. This conversion process is a fundamental technique in linear algebra and is crucial for comparing and analyzing different linear equations.
Step-by-Step Conversion
Let's perform the conversion step-by-step:
-
Distribute the 1/2:
y - 8 = (1/2)(x - 4)
y - 8 = (1/2)x - (1/2)*4
y - 8 = (1/2)x - 2
This step involves multiplying the slope (1/2) by both terms inside the parentheses: x and -4. This operation removes the parentheses and expands the equation, making it easier to isolate y. The result, y - 8 = (1/2)x - 2, shows the equation with the slope distributed, setting the stage for the next step in isolating y. This distribution step is a direct application of the distributive property of multiplication over subtraction, a fundamental algebraic principle. Ensuring accuracy in this step is crucial, as any errors here will propagate through the rest of the solution.
-
Add 8 to both sides:
y - 8 + 8 = (1/2)x - 2 + 8
y = (1/2)x + 6
This step involves adding 8 to both sides of the equation. Adding the same value to both sides maintains the equality, a core principle of algebraic manipulation. This operation isolates y on the left-hand side, bringing us closer to the slope-intercept form. The result, y = (1/2)x + 6, is now in the desired slope-intercept form, where the slope and y-intercept are readily apparent. This step completes the transformation from point-slope form to slope-intercept form, making it easy to identify the line's characteristics. The addition operation effectively cancels out the -8 on the left-hand side, leaving y by itself.
Identifying the Linear Function
Now that we have the equation in slope-intercept form, y = (1/2)x + 6, we can easily identify the corresponding linear function. A linear function is simply a function that represents a straight line when graphed. It's typically written in the form:
f(x) = mx + b
Where:
- f(x) represents the y-value of the function for a given x-value.
- m is the slope.
- b is the y-intercept.
Comparing our slope-intercept equation, y = (1/2)x + 6, with the general form of a linear function, f(x) = mx + b, we can directly map the components. The slope (m) is 1/2, and the y-intercept (b) is 6. Therefore, the linear function that represents the given line is: f(x) = (1/2)x + 6. This function provides a concise and formal representation of the line, allowing us to calculate the y-value for any given x-value. The linear function notation f(x) is a powerful tool for expressing relationships between variables and is widely used in mathematics and various scientific fields. The ability to translate between slope-intercept form and linear function notation is a fundamental skill for understanding and applying linear equations.
The Correct Answer
Based on our step-by-step conversion and analysis, the linear function that represents the line given by the point-slope equation y - 8 = (1/2)(x - 4) is:
B. f(x) = (1/2)x + 6
This answer choice accurately reflects the slope and y-intercept derived from the conversion process. The other options present different slopes or y-intercepts, making them incorrect representations of the original equation. Understanding why this is the correct answer involves not just the algebraic manipulation but also the conceptual understanding of linear equations and their representations. The slope of 1/2 indicates the rate of change of the line, while the y-intercept of 6 indicates where the line crosses the y-axis. These two parameters uniquely define the line, and the function f(x) = (1/2)x + 6 captures these characteristics perfectly. Choosing the correct answer also demonstrates an understanding of the equivalence between different forms of linear equations and their graphical representations.
Importance of Understanding Linear Functions
Linear functions are fundamental in mathematics and have numerous applications in various fields, including physics, engineering, economics, and computer science. Understanding linear functions is crucial for modeling real-world phenomena that exhibit a constant rate of change. For example, the relationship between distance and time for an object moving at a constant speed can be represented by a linear function. Similarly, the relationship between the cost of a product and the number of units purchased can often be modeled linearly. Mastering linear functions provides a foundation for more advanced mathematical concepts, such as calculus and linear algebra. These concepts rely heavily on the properties of linear functions and their generalizations. Furthermore, the ability to manipulate and analyze linear equations is essential for solving practical problems in various disciplines. From predicting future trends to optimizing resource allocation, linear functions provide a powerful tool for quantitative analysis.
Real-World Applications
- Physics: Describing motion at a constant velocity.
- Economics: Modeling supply and demand curves.
- Engineering: Designing linear control systems.
- Computer Science: Developing linear algorithms.
These are just a few examples of the many applications of linear functions in the real world. Their simplicity and versatility make them an indispensable tool for problem-solving and decision-making. The ability to recognize, analyze, and apply linear functions is a valuable skill for anyone pursuing a STEM field or any profession that requires quantitative reasoning.
Conclusion
In conclusion, we've successfully converted the point-slope equation y - 8 = (1/2)(x - 4) into its linear function representation, which is f(x) = (1/2)x + 6. This process involved distributing the slope, isolating y, and then expressing the equation in function notation. Understanding the different forms of linear equations and how to convert between them is essential for solving a wide range of mathematical problems. The ability to translate between point-slope form, slope-intercept form, and linear function notation provides a flexible and powerful toolkit for analyzing and manipulating linear relationships. Furthermore, the practical applications of linear functions in various fields highlight their importance in real-world problem-solving. By mastering these fundamental concepts, you'll be well-equipped to tackle more complex mathematical challenges and apply your knowledge to diverse areas of study and work. The key takeaways from this article include the definition and interpretation of point-slope form, the conversion process to slope-intercept form, the representation of linear equations as functions, and the significance of linear functions in various applications. By understanding these concepts, you can confidently navigate the world of linear equations and their numerous applications.
