Which Linear Function Represents The Line Given By The Point-slope Equation Y − 8 = 1 2 ( X − 4 Y - 8 = \frac{1}{2}(x - 4 Y − 8 = 2 1 ​ ( X − 4 ]?A. F ( X ) = 1 2 X + 4 F(x) = \frac{1}{2} X + 4 F ( X ) = 2 1 ​ X + 4 B. F ( X ) = 1 2 X + 6 F(x) = \frac{1}{2} X + 6 F ( X ) = 2 1 ​ X + 6 C. F ( X ) = 1 2 X − 10 F(x) = \frac{1}{2} X - 10 F ( X ) = 2 1 ​ X − 10 D.

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 $y - 8 = \frac{1}{2}(x - 4)$. To convert this equation to a linear function, we need to isolate the variable $y$ on one side of the equation.

Step 1: Add $y_1$ to Both Sides of the Equation

We can add $y_1$ to both sides of the equation to get:

$$y - 8 + 8 = \frac{1}{2}(x - 4) + 8$$

This simplifies to:

$$y = \frac{1}{2}(x - 4) + 8$$

Step 2: Simplify the Expression

We can simplify the expression by distributing the $\frac{1}{2}$ to the terms inside the parentheses:

$$y = \frac{1}{2}x - 2 + 8$$

This simplifies to:

$$y = \frac{1}{2}x + 6$$

Conclusion

In this article, we have explored how to convert a point-slope equation to a linear function. We have seen that the point-slope equation $y - 8 = \frac{1}{2}(x - 4)$ can be converted to the linear function $f(x) = \frac{1}{2}x + 6$. This is a powerful tool for representing lines in mathematics and is used extensively in algebra and calculus.

Answer

The correct answer is B. $f(x) = \frac{1}{2} x + 6$.

Discussion

The point-slope equation is a fundamental concept in mathematics and is used extensively in algebra and calculus. It is a powerful tool for representing lines in the Cartesian plane and is used to solve a wide range of problems in mathematics.

Example Problems

1. Convert the point-slope equation $y - 3 = 2(x - 2)$ to a linear function.
2. Convert the point-slope equation $y - 5 = \frac{1}{3}(x - 3)$ to a linear function.
3. Convert the point-slope equation $y - 2 = -4(x - 1)$ to a linear function.

Solutions

1. $f(x) = 2x - 1$
2. $f(x) = \frac{1}{3}x + 2$
3. $f(x) = -4x + 10$

Conclusion

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Q: What is the point-slope equation?

A: The point-slope equation is a fundamental concept in mathematics that is 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 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 identify the slope $m$ in the equation $y - y_1 = m(x - x_1)$. The slope is the coefficient of the $x$ term.

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 isolate the constant term on one side of the equation. The constant term is the y-intercept.

Q: What is the difference between a point-slope equation and a slope-intercept equation?

A: A point-slope equation is given by $y - y_1 = m(x - x_1)$, while a slope-intercept equation is given by $f(x) = mx + b$. The point-slope equation is used to represent a line in the Cartesian plane, while the slope-intercept equation is used to find the equation of a line given the slope and y-intercept.

Q: Can I use a point-slope equation to find the equation of a line given two points?

A: Yes, you can use a point-slope equation to find the equation of a line given two points. You need to identify the two points and then use the point-slope equation to find the equation of the line.

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

A: To use a point-slope equation to find the equation of a line given a point and the slope, you need to identify the point and the slope, and then use the point-slope equation to find the equation of the line.

Q: Can I use a point-slope equation to find the equation of a line given the slope and the y-intercept?

A: No, you cannot use a point-slope equation to find the equation of a line given the slope and the y-intercept. You need to use the slope-intercept equation to find the equation of the line given the slope and the y-intercept.

Q: What are some common mistakes to avoid when working with point-slope equations?

A: Some common mistakes to avoid when working with point-slope equations include:

• Not identifying the slope and the point correctly
• Not isolating the variable $y$ on one side of the equation
• Not simplifying the resulting expression
• Not using the correct form of the point-slope equation

Q: How do I check my work when working with point-slope equations?

A: To check your work when working with point-slope equations, you need to:

• Verify that the equation is in the correct form
• Check that the slope and the point are correctly identified
• Check that the variable $y$ is isolated on one side of the equation
• Check that the resulting expression is simplified

Conclusion

In this article, we have answered some common questions about point-slope equations. We have seen that the point-slope equation is a fundamental concept in mathematics that is used to represent a line in the Cartesian plane. We have also seen that the point-slope equation can be converted to a linear function, and that the slope-intercept form of a linear equation is given by $f(x) = mx + b$.