Which Linear Function Represents The Line Given By The Point-slope Equation $y + 1 = -3(x - 5$\]?A. $f(x) = -3x - 6$B. $f(x) = -3x - 4$C. $f(x) = -3x + 16$D. $f(x) = -3x + 14$

Feb 26, 2025 by ADMIN 176 views

Introduction

In mathematics, the point-slope equation is a fundamental concept used to represent a line in the Cartesian plane. It is given by the equation $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is a point on the line and $m$ is the slope of the line. In this article, we will explore how to convert a point-slope equation to a linear function in the form $f(x) = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

Understanding the Point-Slope Equation

The point-slope equation is a powerful tool for representing lines in mathematics. It is given by the equation $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is a point on the line and $m$ is the slope of the line. The slope-intercept form of a linear equation is given by $f(x) = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

Converting Point-Slope Equation to Linear Function

To convert a point-slope equation to a linear function, we need to isolate the variable $y$ on one side of the equation. We can do this by adding $y_1$ to both sides of the equation and then simplifying the resulting expression.

Let's consider the point-slope equation $y + 1 = -3(x - 5)$ . To convert this equation to a linear function, we need to isolate the variable $y$ on one side of the equation.

Step 1: Distribute the Slope

The first step is to distribute the slope $m$ to the terms inside the parentheses. In this case, we have $-3(x - 5)$ , which can be distributed as $-3x + 15$ .

y + 1 = -3(x - 5)
y + 1 = -3x + 15

Step 2: Isolate the Variable y

The next step is to isolate the variable $y$ on one side of the equation. We can do this by subtracting $1$ from both sides of the equation.

y + 1 - 1 = -3x + 15 - 1
y = -3x + 14

Step 3: Write the Linear Function

The final step is to write the linear function in the form $f(x) = mx + b$ . In this case, we have $f(x) = -3x + 14$ .

f(x) = -3x + 14

Conclusion

In this article, we explored how to convert a point-slope equation to a linear function. We used the point-slope equation $y + 1 = -3(x - 5)$ as an example and showed how to distribute the slope, isolate the variable $y$ , and write the linear function in the form $f(x) = mx + b$ . The resulting linear function is $f(x) = -3x + 14$ .

Answer

The correct answer is D. $f(x) = -3x + 14$ .

Discussion

This problem is a great example of how to convert a point-slope equation to a linear function. It requires the student to understand the concept of the point-slope equation and how to isolate the variable $y$ on one side of the equation. The student must also be able to distribute the slope and write the linear function in the form $f(x) = mx + b$ .

Tips and Tricks

Make sure to distribute the slope correctly.
Isolate the variable $y$ on one side of the equation.
Write the linear function in the form $f(x) = mx + b$ .

Practice Problems

Convert the point-slope equation $y - 2 = 4(x - 3)$ to a linear function.
Convert the point-slope equation $y + 3 = -2(x - 1)$ to a linear function.
Convert the point-slope equation $y - 1 = 3(x - 2)$ to a linear function.

Answer Key

$f(x) = 4x - 10$
$f(x) = -2x + 1$
$f(x) = 3x - 5$
Q&A: Converting Point-Slope Equations to Linear Functions ===========================================================

Introduction

In our previous article, we explored how to convert a point-slope equation to a linear function. We used the point-slope equation $y + 1 = -3(x - 5)$ as an example and showed how to distribute the slope, isolate the variable $y$ , and write the linear function in the form $f(x) = mx + b$ . In this article, we will answer some frequently asked questions about converting point-slope equations to linear functions.

Q: What is the point-slope equation?

A: The point-slope equation is a fundamental concept used to represent a line in the Cartesian plane. It is given by the equation $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is a point on the line and $m$ is the slope of the line.

Q: How do I convert a point-slope equation to a linear function?

A: To convert a point-slope equation to a linear function, you need to isolate the variable $y$ on one side of the equation. You can do this by adding $y_1$ to both sides of the equation and then simplifying the resulting expression.

Q: What is the slope-intercept form of a linear equation?

A: The slope-intercept form of a linear equation is given by $f(x) = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

Q: How do I distribute the slope in a point-slope equation?

A: To distribute the slope, you need to multiply the slope by each term inside the parentheses. For example, if you have the point-slope equation $y + 1 = -3(x - 5)$ , you would distribute the slope as follows:

y + 1 = -3(x - 5)
y + 1 = -3x + 15

Q: How do I isolate the variable y in a point-slope equation?

A: To isolate the variable $y$ , you need to add or subtract the same value from both sides of the equation. For example, if you have the point-slope equation $y + 1 = -3(x - 5)$ , you would isolate the variable $y$ as follows:

y + 1 = -3x + 15
y = -3x + 14

Q: What is the y-intercept of a linear function?

A: The y-intercept of a linear function is the value of $y$ when $x = 0$ . It is denoted by the letter $b$ in the slope-intercept form of a linear equation.

Q: How do I write a linear function in the form f(x) = mx + b?

A: To write a linear function in the form $f(x) = mx + b$ , you need to identify the slope and the y-intercept of the function. The slope is the coefficient of $x$ , and the y-intercept is the constant term.

Conclusion

In this article, we answered some frequently asked questions about converting point-slope equations to linear functions. We covered topics such as the point-slope equation, distributing the slope, isolating the variable $y$ , and writing a linear function in the form $f(x) = mx + b$ . We hope that this article has been helpful in clarifying any confusion you may have had about converting point-slope equations to linear functions.

Practice Problems

Convert the point-slope equation $y - 2 = 4(x - 3)$ to a linear function.
Convert the point-slope equation $y + 3 = -2(x - 1)$ to a linear function.
Convert the point-slope equation $y - 1 = 3(x - 2)$ to a linear function.

Answer Key

$f(x) = 4x - 10$
$f(x) = -2x + 1$
$f(x) = 3x - 5$