Determining Linear Function From Point-Slope Form Y-5=3(x-2)

Jun 24, 2025 by ADMIN 61 views

Determine the linear function for the line given by the point-slope equation y-5=3(x-2). A. f(x)=3x+1 B. f(x)=3x-1 C. f(x)=8x+10 D. f(x)=8x-10

Choosing the Correct Linear Function from Point-Slope Form

In this comprehensive guide, we will walk through the process of selecting the correct linear function that corresponds to a given point-slope equation. Specifically, we will focus on the equation y - 5 = 3(x - 2) and identify the equivalent slope-intercept form from a set of options. Understanding how to convert between point-slope and slope-intercept forms is crucial for mastering linear functions and their applications in mathematics. We will break down each step, explain the underlying concepts, and ensure that you can confidently tackle similar problems in the future.

Understanding Point-Slope Form

The point-slope form of a linear equation is a powerful way to represent a line, especially when you know a point on the line and its slope. The general form of the point-slope equation is:

y - y₁ = m(x - x₁)

Where:

m represents the slope of the line.
(x₁, y₁) is a specific point on the line.

The given equation y - 5 = 3(x - 2) is already in point-slope form. By comparing it to the general form, we can identify the slope and a point on the line:

The slope, m, is 3.
The point (x₁, y₁) is (2, 5).

The point-slope form is particularly useful because it directly incorporates the slope and a point, making it easy to visualize and understand the line's characteristics. However, to compare it with the provided options, which are in slope-intercept form, we need to convert the equation.

Converting to Slope-Intercept Form

The slope-intercept form of a linear equation is written as:

y = mx + b

Where:

m represents the slope of the line.
b represents the y-intercept (the point where the line crosses the y-axis).

To convert the given point-slope equation y - 5 = 3(x - 2) to slope-intercept form, we need to isolate y on one side of the equation. This involves a few algebraic steps:

Distribute the slope:

Multiply the slope (3) by each term inside the parentheses:
```
y - 5 = 3x - 6
```
Isolate y:

Add 5 to both sides of the equation to isolate y:
```
y - 5 + 5 = 3x - 6 + 5
y = 3x - 1
```

Now, the equation is in slope-intercept form, y = 3x - 1. This form tells us that the line has a slope of 3 and a y-intercept of -1. Converting to slope-intercept form allows us to easily compare our equation with the provided options and identify the correct linear function.

Identifying the Correct Linear Function

Now that we have the equation in slope-intercept form, y = 3x - 1, we can compare it with the given options:

A. f(x) = 3x + 1 B. f(x) = 3x - 1 C. f(x) = 8x + 10 D. f(x) = 8x - 10

By comparing our derived equation with the options, we can see that option B, f(x) = 3x - 1, matches exactly. This confirms that the correct linear function representing the line given by the point-slope equation y - 5 = 3(x - 2) is f(x) = 3x - 1.

Detailed Explanation of Each Option

To further clarify why option B is correct and the others are not, let's examine each option in detail. This will reinforce your understanding of slope-intercept form and how to accurately identify linear functions.

Option A: f(x) = 3x + 1

This linear function has a slope of 3, which matches the slope we found in our conversion. However, the y-intercept is +1, while our converted equation has a y-intercept of -1. Therefore, this option is incorrect because it does not match the y-intercept of the line represented by the given point-slope equation. The y-intercept is a crucial part of the linear function, and any discrepancy means the function does not accurately represent the same line.

Option B: f(x) = 3x - 1

This linear function has a slope of 3 and a y-intercept of -1. Both values match the slope and y-intercept we found when converting the point-slope equation to slope-intercept form. Therefore, this option is the correct answer. The slope and y-intercept are the defining characteristics of a linear function in slope-intercept form, and this option aligns perfectly with the line described by the original equation. Understanding these key components is vital for correctly identifying linear functions.

Option C: f(x) = 8x + 10

This linear function has a slope of 8 and a y-intercept of +10. Neither the slope nor the y-intercept matches the values we derived from the point-slope equation. The slope of 8 is significantly different from the slope of 3 in the original equation, and the y-intercept of +10 is far from the -1 we found. Thus, this option is incorrect. The discrepancy in both slope and y-intercept indicates that this function represents a completely different line.

Option D: f(x) = 8x - 10

This linear function has a slope of 8 and a y-intercept of -10. Similar to option C, neither the slope nor the y-intercept matches our derived values. The slope of 8 again differs from the original slope of 3, and the y-intercept of -10 is not the same as -1. Therefore, this option is also incorrect. Identifying the correct slope and y-intercept is fundamental in recognizing linear functions, and this option fails to align with the equation in question.

Alternative Method: Verifying with the Point

Another way to verify the correct linear function is by plugging the point (2, 5) from the point-slope form into each option. If the point lies on the line, the equation should hold true.

Let's test each option:

Option A: f(x) = 3x + 1

f(2) = 3(2) + 1 = 6 + 1 = 7

Since f(2) = 7 and not 5, this option is incorrect.

Option B: f(x) = 3x - 1

f(2) = 3(2) - 1 = 6 - 1 = 5

Since f(2) = 5, this option is correct, as the point (2, 5) lies on the line.

Option C: f(x) = 8x + 10

f(2) = 8(2) + 10 = 16 + 10 = 26

Since f(2) = 26 and not 5, this option is incorrect.

Option D: f(x) = 8x - 10

f(2) = 8(2) - 10 = 16 - 10 = 6

Since f(2) = 6 and not 5, this option is incorrect.

This method provides a direct way to confirm whether a given point satisfies the linear function, further validating our conclusion that option B is the correct choice. Substituting the x-coordinate into the function and checking if the result matches the y-coordinate of the point is a powerful verification technique.

Importance of Understanding Linear Functions

Linear functions are fundamental in mathematics and have numerous applications in real-world scenarios. They are used in various fields, including physics, engineering, economics, and computer science. Mastering the concepts of slope-intercept form, point-slope form, and how to convert between them is crucial for solving problems related to linear relationships. Understanding linear functions allows you to model and predict outcomes in diverse situations, making it a valuable skill in both academic and professional settings.

Practical Applications of Linear Functions

Linear functions are not just theoretical concepts; they are used extensively in practical applications. For example:

Economics: Linear functions can model the relationship between supply and demand, cost and revenue, and other economic variables. Understanding these relationships helps economists make predictions and informed decisions.
Physics: Linear functions describe motion with constant velocity, the relationship between force and displacement in a spring, and other physical phenomena. Engineers and physicists rely on these models to design and analyze systems.
Engineering: Linear functions are used in circuit analysis, structural design, and many other engineering applications. Engineers use these functions to ensure the stability and efficiency of their designs.
Computer Science: Linear functions are used in computer graphics, data analysis, and machine learning. Understanding linear relationships helps in creating efficient algorithms and models.

Conclusion

In conclusion, the correct linear function for the line represented by the point-slope equation y - 5 = 3(x - 2) is B. f(x) = 3x - 1. We arrived at this answer by converting the point-slope equation to slope-intercept form and verifying the result through substitution. Understanding the relationship between different forms of linear equations and being able to convert between them is a critical skill in mathematics. By mastering these concepts, you will be well-equipped to tackle a wide range of problems involving linear functions. The ability to accurately identify and manipulate linear functions is essential for success in various fields, making this a valuable skill to develop and refine. Consistent practice and a solid understanding of the underlying principles are key to mastering linear functions and their applications. This article has provided a comprehensive guide to help you achieve that mastery.