Select The Correct Answer.What Is The Slope Of The Line That Goes Through The Points { (-4, 2)$}$ And { (8, 5)$}$?A. -4 B. { -\frac{1}{4}$}$ C. { \frac{1}{4}$}$ D. 4
Introduction
In mathematics, the slope of a line is a fundamental concept used to describe the steepness or incline of a line. It is a crucial concept in geometry, algebra, and calculus. The slope of a line can be calculated using the coordinates of two points on the line. In this article, we will discuss how to calculate the slope of a line that passes through two given points.
What is Slope?
The slope of a line is a measure of how much the line rises (or falls) vertically over a given horizontal distance. It is denoted by the letter 'm' and is calculated as the ratio of the vertical change (rise) to the horizontal change (run). The slope of a line can be positive, negative, or zero, depending on the direction and orientation of the line.
Calculating Slope
To calculate the slope of a line that passes through two points, we can use the following formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points.
Example Problem
Let's consider the problem of finding the slope of the line that passes through the points (-4, 2) and (8, 5). We can use the formula above to calculate the slope.
Step 1: Identify the coordinates of the two points
The coordinates of the two points are (-4, 2) and (8, 5).
Step 2: Plug the coordinates into the formula
m = (5 - 2) / (8 - (-4)) m = (3) / (12) m = 1/4
Step 3: Simplify the fraction
The slope of the line is 1/4.
Answer
The correct answer is C. 1/4.
Why is this important?
Understanding the concept of slope is essential in mathematics, as it is used to describe the behavior of lines and curves. It is also used in real-world applications, such as calculating the steepness of a roof, the angle of a ramp, or the incline of a road.
Real-World Applications
The concept of slope has numerous real-world applications, including:
- Architecture: Calculating the slope of a roof or a ramp to ensure safety and stability.
- Engineering: Designing roads, bridges, and buildings that take into account the slope of the land.
- Physics: Calculating the motion of objects, such as the trajectory of a projectile or the motion of a pendulum.
- Navigation: Using slope to determine the direction and distance of a journey.
Conclusion
In conclusion, the slope of a line is a fundamental concept in mathematics that describes the steepness or incline of a line. It can be calculated using the coordinates of two points on the line. Understanding the concept of slope is essential in mathematics and has numerous real-world applications. By mastering the concept of slope, you will be able to solve problems and make informed decisions in a variety of fields.
Additional Resources
For further learning, we recommend the following resources:
- Math textbooks: "Algebra and Trigonometry" by Michael Sullivan and "Calculus" by James Stewart.
- Online resources: Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.
- Practice problems: Try solving problems on websites like Mathway, Symbolab, and IXL.
Final Thoughts
Introduction
In our previous article, we discussed the concept of slope and how to calculate it using the coordinates of two points. In this article, we will answer some frequently asked questions about slope to help you better understand this important concept in mathematics.
Q: What is the difference between slope and rate of change?
A: 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 of a function changes when the input changes.
Q: How do I determine if a line is increasing or decreasing?
A: To determine if a line is increasing or decreasing, you can look at the slope of the line. If the slope is positive, the line is increasing. If the slope is negative, the line is decreasing. If the slope is zero, the line is horizontal.
Q: Can a line have a slope of zero?
A: Yes, a line can have a slope of zero. This occurs when the line is horizontal, meaning it does not rise or fall at all.
Q: Can a line have an undefined slope?
A: Yes, a line can have an undefined slope. This occurs when the line is vertical, meaning it does not have a horizontal change.
Q: How do I calculate the slope of a line that passes through a point and has a given slope?
A: To calculate the slope of a line that passes through a point and has a given slope, you can use the point-slope form of a linear equation:
y - y1 = m(x - x1)
where (x1, y1) is the point and m is the slope.
Q: Can I use the slope formula to find the equation of a line?
A: Yes, you can use the slope formula to find the equation of a line. Once you have the slope and a point on the line, you can use the point-slope form of a linear equation to find the equation of the line.
Q: How do I graph a line using its slope and a point?
A: To graph a line using its slope and a point, you can use the following steps:
- Plot the point on the coordinate plane.
- Draw a line through the point with the given slope.
- Continue drawing the line until you reach the desired endpoint.
Q: Can I use the slope formula to find the equation of a line that passes through two points?
A: Yes, you can use the slope formula to find the equation of a line that passes through two points. Once you have the slope and the two points, you can use the point-slope form of a linear equation to find the equation of the line.
Q: How do I determine if two lines are parallel or perpendicular?
A: To determine if two lines are parallel or perpendicular, you can look at their slopes. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals of each other, the lines are perpendicular.
Conclusion
In conclusion, the concept of slope is a fundamental concept in mathematics that has numerous real-world applications. By understanding the concept of slope, you will be able to solve problems and make informed decisions in a variety of fields. We hope this Q&A article has helped you better understand the concept of slope and how to apply it in different situations.
Additional Resources
For further learning, we recommend the following resources:
- Math textbooks: "Algebra and Trigonometry" by Michael Sullivan and "Calculus" by James Stewart.
- Online resources: Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.
- Practice problems: Try solving problems on websites like Mathway, Symbolab, and IXL.
Final Thoughts
The concept of slope is a powerful tool in mathematics that has numerous real-world applications. By understanding the concept of slope, you will be able to solve problems and make informed decisions in a variety of fields. Remember to practice regularly and seek help when needed to master the concept of slope.