What Is The Point-slope Form Of The Line With Slope \[$-\frac{1}{4}\$\] That Passes Through The Point \[$(-2, 9)\$\]?A. \[$y - 9 = -\frac{1}{4}(x + 2)\$\]B. \[$y - 2 = -\frac{1}{4}(x + 9)\$\]C. \[$y + 9 =
The point-slope form of a line is a mathematical representation that describes the relationship between the slope of a line and a point on the line. It is a fundamental concept in mathematics, particularly in algebra and geometry. In this article, we will explore the point-slope form of a line and how to use it to find the equation of a line given its slope and a point on the line.
Understanding the Point-Slope Form
The point-slope form of a line is given by the equation:
y - y1 = m(x - x1)
where:
- y is the dependent variable (the variable being measured)
- x is the independent variable (the variable being manipulated)
- m is the slope of the line
- (x1, y1) is a point on the line
Finding the Point-Slope Form of a Line
To find the point-slope form of a line, we need to know the slope of the line and a point on the line. The slope of a line can be found using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
Example: Finding the Point-Slope Form of a Line
Let's say we want to find the point-slope form of a line with a slope of -1/4 that passes through the point (-2, 9). We can use the formula above to find the equation of the line.
First, we need to find the slope of the line. We can do this by using the formula:
m = (y2 - y1) / (x2 - x1)
We know that the slope is -1/4, so we can plug in the values:
m = (-1/4)
Next, we need to find a point on the line. We are given the point (-2, 9), so we can use this point as (x1, y1).
Now, we can plug in the values into the point-slope form equation:
y - 9 = -1/4(x - (-2))
Simplifying the equation, we get:
y - 9 = -1/4(x + 2)
Answer:
The point-slope form of the line with a slope of -1/4 that passes through the point (-2, 9) is:
y - 9 = -1/4(x + 2)
This is option A.
Conclusion
In conclusion, the point-slope form of a line is a mathematical representation that describes the relationship between the slope of a line and a point on the line. It is a fundamental concept in mathematics, particularly in algebra and geometry. By using the point-slope form equation, we can find the equation of a line given its slope and a point on the line.
Common Mistakes to Avoid
When finding the point-slope form of a line, there are several common mistakes to avoid:
- Incorrect slope: Make sure to use the correct slope value in the point-slope form equation.
- Incorrect point: Make sure to use the correct point on the line in the point-slope form equation.
- Incorrect equation: Make sure to simplify the equation correctly and avoid any algebraic errors.
Real-World Applications
The point-slope form of a line has several real-world applications, including:
- Physics: The point-slope form of a line is used to describe the motion of objects under constant acceleration.
- Engineering: The point-slope form of a line is used to design and optimize systems, such as electrical circuits and mechanical systems.
- Computer Science: The point-slope form of a line is used in computer graphics and game development to create realistic motion and animation.
Final Thoughts
Frequently Asked Questions
Q: What is the point-slope form of a line? A: The point-slope form of a line is a mathematical representation that describes the relationship between the slope of a line and a point on the line. It is given by the equation:
y - y1 = m(x - x1)
Q: How do I find the point-slope form of a line? A: To find the point-slope form of a line, you need to know the slope of the line and a point on the line. You can use the formula:
m = (y2 - y1) / (x2 - x1)
to find the slope, and then plug in the values into the point-slope form equation.
Q: What is the slope of a line? A: The slope of a line is a measure of how steep the line is. It is calculated by dividing the change in y by the change in x.
Q: How do I find the slope of a line? A: You can find the slope of a line by using the formula:
m = (y2 - y1) / (x2 - x1)
Q: What is the point-slope form equation? A: The point-slope form equation is:
y - y1 = m(x - x1)
Q: How do I use the point-slope form equation? A: To use the point-slope form equation, you need to plug in the values of the slope and the point on the line. Then, simplify the equation to get the final answer.
Q: What are some common mistakes to avoid when finding the point-slope form of a line? A: Some common mistakes to avoid when finding the point-slope form of a line include:
- Incorrect slope: Make sure to use the correct slope value in the point-slope form equation.
- Incorrect point: Make sure to use the correct point on the line in the point-slope form equation.
- Incorrect equation: Make sure to simplify the equation correctly and avoid any algebraic errors.
Q: What are some real-world applications of the point-slope form of a line? A: The point-slope form of a line has several real-world applications, including:
- Physics: The point-slope form of a line is used to describe the motion of objects under constant acceleration.
- Engineering: The point-slope form of a line is used to design and optimize systems, such as electrical circuits and mechanical systems.
- Computer Science: The point-slope form of a line is used in computer graphics and game development to create realistic motion and animation.
Q: How do I graph a line using the point-slope form equation? A: To graph a line using the point-slope form equation, you need to first find the slope and the point on the line. Then, use the point-slope form equation to find the equation of the line. Finally, plot the line on a coordinate plane.
Q: What is the difference between the point-slope form and the slope-intercept form of a line? A: The point-slope form and the slope-intercept form are two different ways to represent the equation of a line. The point-slope form is given by the equation:
y - y1 = m(x - x1)
while the slope-intercept form is given by the equation:
y = mx + b
Q: When to use the point-slope form and when to use the slope-intercept form? A: The point-slope form is used when you know the slope and a point on the line, while the slope-intercept form is used when you know the slope and the y-intercept.
Q: Can I use the point-slope form to find the equation of a line that is not a straight line? A: No, the point-slope form is only used to find the equation of a straight line. If the line is not a straight line, you need to use a different method to find its equation.
Q: Can I use the point-slope form to find the equation of a line that is not in the standard form? A: Yes, the point-slope form can be used to find the equation of a line that is not in the standard form. However, you need to first convert the line to the standard form before using the point-slope form.