Select The Correct Answer.Two Points Located On $\overleftrightarrow{JK}$ Are $J(1, -4$\] And $K(-2, 8$\]. What Is The Slope Of $\overleftrightarrow{JK}$?A. -4 B. -2 C. $-\frac{1}{4}$ D.

Mar 1, 2025 by ADMIN 190 views

**Slope of a Line: A Comprehensive Guide**

=====================================================

Introduction

In mathematics, the slope of a line is a fundamental concept that helps us understand the steepness or incline of a line. It is a measure of how much the line rises (or falls) vertically over a given horizontal distance. In this article, we will delve into the concept of slope, its importance, and how to calculate it using real-world examples.

What is Slope?

The slope of a line is denoted by the letter 'm' and is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. It can be represented mathematically as:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are two points on the line.

Types of Slope

There are two types of slope: positive and negative.

Positive Slope: A line with a positive slope rises from left to right. This means that as the x-coordinate increases, the y-coordinate also increases.
Negative Slope: A line with a negative slope falls from left to right. This means that as the x-coordinate increases, the y-coordinate decreases.

Calculating Slope

To calculate the slope of a line, we need to know the coordinates of two points on the line. Let's consider an example to illustrate this.

Example 1

Suppose we have two points on the line: (1, -4) and (-2, 8). We can use the formula for slope to calculate the slope of the line:

m = (y2 - y1) / (x2 - x1) = (8 - (-4)) / (-2 - 1) = (8 + 4) / (-2 - 1) = 12 / -3 = -4

Therefore, the slope of the line is -4.

Example 2

Suppose we have two points on the line: (2, 3) and (4, 6). We can use the formula for slope to calculate the slope of the line:

m = (y2 - y1) / (x2 - x1) = (6 - 3) / (4 - 2) = 3 / 2 = 1.5

Therefore, the slope of the line is 1.5.

Real-World Applications

Slope has numerous real-world applications, including:

Physics: Slope is used to calculate the acceleration of an object moving along a curved path.
Engineering: Slope is used to design roads, bridges, and buildings.
Economics: Slope is used to analyze the relationship between two variables, such as supply and demand.

Conclusion

In conclusion, slope is a fundamental concept in mathematics that helps us understand the steepness or incline of a line. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. We have discussed the types of slope, how to calculate slope, and its real-world applications. By understanding slope, we can better analyze and solve problems in various fields.

Final Answer

To answer the original question, the slope of the line passing through points J(1, -4) and K(-2, 8) is -4.

Answer Key

A. -4 B. -2 C. $-\frac{1}{4}$ D. -4

The correct answer is A. -4.

=====================================================

Introduction

In our previous article, we discussed the concept of slope, its importance, and how to calculate it using real-world examples. In this article, we will address some frequently asked questions related to slope.

Q&A

Q1: What is the difference between slope and rate of change?

A1: The slope and rate of change are related but distinct concepts. The slope of a line is a measure of how much the line rises (or falls) vertically over a given horizontal distance, while the rate of change is a measure of how much the output changes when the input changes.

Q2: How do I determine if a line is parallel or perpendicular to another line?

A2: To determine if two lines are parallel or perpendicular, you can compare their slopes. If the slopes are equal, the lines are parallel. If the product of the slopes is -1, the lines are perpendicular.

Q3: Can a line have a slope of zero?

A3: Yes, a line can have a slope of zero. This occurs when the line is horizontal, meaning it does not rise or fall vertically over a given horizontal distance.

Q4: How do I calculate the slope of a line given two points and a third point?

A4: To calculate the slope of a line given two points and a third point, you can use the formula for slope:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are the two given points, and (x3, y3) is the third point.

Q5: Can a line have a negative slope?

A5: Yes, a line can have a negative slope. This occurs when the line falls from left to right, meaning that as the x-coordinate increases, the y-coordinate decreases.

Q6: How do I determine if a line is increasing or decreasing?

A6: To determine if a line is increasing or decreasing, you can look at its slope. If the slope is positive, the line is increasing. If the slope is negative, the line is decreasing.

Q7: Can a line have a slope of infinity?

A7: No, a line cannot have a slope of infinity. However, a line can have a vertical slope, which is represented by the symbol ∞.

Q8: How do I calculate the slope of a line given a point and a line equation?

A8: To calculate the slope of a line given a point and a line equation, you can substitute the point into the equation and solve for the slope.

Q9: Can a line have a slope of zero and still be increasing?

A9: No, a line cannot have a slope of zero and still be increasing. If a line has a slope of zero, it is horizontal and does not rise or fall vertically over a given horizontal distance.

Q10: How do I determine if a line is tangent to a curve?

A10: To determine if a line is tangent to a curve, you can look at the slope of the line and the slope of the curve at the point of tangency. If the slopes are equal, the line is tangent to the curve.