Which Linear Function Represents The Line Given By The Point-slope Equation $y - 2 = 4(x - 3$\]?A. $f(x) = 6x - 1$ B. $f(x) = 8x - 6$ C. $f(x) = 4x - 14$ D. $f(x) = 4x - 10$

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 $y - 2 = 4(x - 3)$. 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 $m = 4$ and the term inside the parentheses is $(x - 3)$. We can distribute the slope as follows:

$y - 2 = 4(x - 3)$

$y - 2 = 4x - 12$

Step 2: Add $y_1$ to Both Sides

The next step is to add $y_1$ to both sides of the equation. In this case, we have $y_1 = 2$. We can add $y_1$ to both sides as follows:

$y - 2 + 2 = 4x - 12 + 2$

$y = 4x - 10$

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 $m = 4$ and $b = -10$. We can write the linear function as follows:

$f(x) = 4x - 10$

Conclusion

In this article, we explored how to convert a point-slope equation to a linear function. We used the point-slope equation $y - 2 = 4(x - 3)$ as an example and converted it to a linear function in the form $f(x) = mx + b$. We also discussed the importance of understanding the point-slope equation and how to use it to represent lines in mathematics.

Which Linear Function Represents the Line?

Now that we have converted the point-slope equation to a linear function, we can compare it to the given options. The linear function we obtained is $f(x) = 4x - 10$. Let's compare it to the given options:

A. $f(x) = 6x - 1$ B. $f(x) = 8x - 6$ C. $f(x) = 4x - 14$ D. $f(x) = 4x - 10$

The only option that matches our linear function is option D. Therefore, the correct answer is:

D. $f(x) = 4x - 10$

Final Answer

Introduction

In our previous article, we explored how to convert a point-slope equation to a linear function. We also discussed the importance of understanding the point-slope equation and how to use it to represent lines in mathematics. In this article, we will answer some frequently asked questions (FAQs) about point-slope equations and linear functions.

Q: What is a point-slope equation?

A: A point-slope equation is a mathematical equation that represents 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: 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 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 difference between a point-slope equation and a linear function?

A: A point-slope equation is a mathematical equation that represents a line in the Cartesian plane, while a linear function is a mathematical function that represents a line in the Cartesian plane. A point-slope equation is typically used to find the equation of a line, while a linear function is used to find the value of a function at a given point.

Q: How do I find the slope of a line given a point-slope equation?

A: To find the slope of a line given a point-slope equation, you need to look at the coefficient of the $x$ term. The coefficient of the $x$ term is the slope of the line.

Q: How do I find the y-intercept of a line given a point-slope equation?

A: To find the y-intercept of a line given a point-slope equation, you need to look at the constant term. The constant term is the y-intercept of the line.

Q: Can I use a point-slope equation to find the equation of a line that is not a straight line?

A: No, a point-slope equation can only be used to find the equation of a straight line. If you have a line that is not a straight line, you will need to use a different type of equation, such as a quadratic equation.

Q: Can I use a point-slope equation to find the equation of a line that is not in the Cartesian plane?

A: No, a point-slope equation can only be used to find the equation of a line that is in the Cartesian plane. If you have a line that is not in the Cartesian plane, you will need to use a different type of equation.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about point-slope equations and linear functions. We hope that this article has been helpful in clarifying any confusion you may have had about these topics. If you have any further questions, please don't hesitate to ask.

Additional Resources

If you are interested in learning more about point-slope equations and linear functions, we recommend checking out the following resources:

• Khan Academy: Point-Slope Form
• Mathway: Point-Slope Form
• Wolfram Alpha: Point-Slope Form

We hope that this article has been helpful in your understanding of point-slope equations and linear functions.