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 − 10 F(x) = \frac{1}{2}x - 10 F ( X ) = 2 1 X − 10 B. F ( X ) = 1 2 X − 12 F(x) = \frac{1}{2}x - 12 F ( X ) = 2 1 X − 12 C. F ( X ) = 1 2 X + 4 F(x) = \frac{1}{2}x + 4 F ( X ) = 2 1 X + 4 D. $f(x) =
Understanding the Point-Slope Equation
The point-slope equation is a fundamental concept in mathematics, particularly in algebra and geometry. It is used to describe a line that passes through a given point and has a specific slope. The general form of the point-slope equation is:
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope of the line.
Given Point-Slope Equation
In this problem, we are given the point-slope equation:
y - 8 = \frac{1}{2}(x - 4)
Our goal is to determine which linear function represents the line given by this equation.
Converting Point-Slope Equation to Slope-Intercept Form
To convert the point-slope equation to slope-intercept form, we need to isolate y. We can do this by adding 8 to both sides of the equation:
y = \frac{1}{2}(x - 4) + 8
Next, we can simplify the equation by distributing the \frac{1}{2} to the terms inside the parentheses:
y = \frac{1}{2}x - 2 + 8
Now, we can combine the constant terms:
y = \frac{1}{2}x + 6
Comparing with the Given Options
Now that we have the slope-intercept form of the equation, we can compare it with the given options:
A. B. C. D.
We can see that option D matches our derived equation.
Conclusion
In conclusion, the linear function that represents the line given by the point-slope equation is .
Why is this Important?
Understanding the point-slope equation and converting it to slope-intercept form is crucial in mathematics, particularly in algebra and geometry. It allows us to describe lines in a more concise and intuitive way, making it easier to solve problems and visualize geometric shapes.
Real-World Applications
The point-slope equation has numerous real-world applications, including:
- Physics: The point-slope equation is used to describe the motion of objects under constant acceleration.
- Engineering: The point-slope equation is used to design and optimize systems, such as electrical circuits and mechanical systems.
- Computer Science: The point-slope equation is used in computer graphics and game development to create realistic and interactive 3D models.
Tips and Tricks
Here are some tips and tricks to help you master the point-slope equation:
- Practice, practice, practice: The more you practice, the more comfortable you will become with the point-slope equation.
- Use visual aids: Visual aids, such as graphs and charts, can help you understand the point-slope equation and its applications.
- Break down complex problems: Break down complex problems into smaller, more manageable parts to make them easier to solve.
Common Mistakes to Avoid
Here are some common mistakes to avoid when working with the point-slope equation:
- Not distributing the slope: Failing to distribute the slope to the terms inside the parentheses can lead to incorrect results.
- Not combining constant terms: Failing to combine constant terms can lead to incorrect results.
- Not checking units: Failing to check units can lead to incorrect results.
Conclusion
Q: What is the point-slope equation?
A: The point-slope equation is a fundamental concept in mathematics, particularly in algebra and geometry. It is used to describe a line that passes through a given point and has a specific slope. The general form of the point-slope equation is:
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope of the line.
Q: How do I convert the point-slope equation to slope-intercept form?
A: To convert the point-slope equation to slope-intercept form, you need to isolate y. You can do this by adding y1 to both sides of the equation and then subtracting m(x - x1) from both sides. This will give you the slope-intercept form of the equation, which is:
y = mx + b
where m is the slope and b is the y-intercept.
Q: What is the slope-intercept form of the equation?
A: The slope-intercept form of the equation is:
y = mx + b
where m is the slope and b is the y-intercept.
Q: How do I find the slope of the line?
A: To find the slope of the line, you need to look at the coefficient of x in the point-slope equation. The coefficient of x is the slope of the line.
Q: How do I find the y-intercept of the line?
A: To find the y-intercept of the line, you need to look at the constant term in the point-slope equation. The constant term is the y-intercept of the line.
Q: What is the difference between the point-slope equation and the slope-intercept form?
A: The point-slope equation is used to describe a line that passes through a given point and has a specific slope. The slope-intercept form is used to describe a line that has a specific slope and y-intercept.
Q: When should I use the point-slope equation?
A: You should use the point-slope equation when you are given a point and the slope of the line, and you want to find the equation of the line.
Q: When should I use the slope-intercept form?
A: You should use the slope-intercept form when you are given the slope and y-intercept of the line, and you want to find the equation of the line.
Q: Can I use the point-slope equation to find the equation of a line that is not a straight line?
A: No, the point-slope equation is only used to describe straight lines. If you want to find the equation of a line that is not a straight line, you will need to use a different equation, such as the parametric equation or the polar equation.
Q: Can I use the slope-intercept form to find the equation of a line that is not a straight line?
A: No, the slope-intercept form is only used to describe straight lines. If you want to find the equation of a line that is not a straight line, you will need to use a different equation, such as the parametric equation or the polar equation.
Q: How do I graph a line using the point-slope equation?
A: To graph a line using the point-slope equation, you need to plot the given point and then draw a line that passes through the point and has the given slope.
Q: How do I graph a line using the slope-intercept form?
A: To graph a line using the slope-intercept form, you need to plot the y-intercept and then draw a line that passes through the y-intercept and has the given slope.
Q: Can I use the point-slope equation to find the equation of a line that is parallel to another line?
A: Yes, you can use the point-slope equation to find the equation of a line that is parallel to another line. To do this, you need to find the slope of the parallel line and then use the point-slope equation to find the equation of the line.
Q: Can I use the slope-intercept form to find the equation of a line that is parallel to another line?
A: Yes, you can use the slope-intercept form to find the equation of a line that is parallel to another line. To do this, you need to find the slope of the parallel line and then use the slope-intercept form to find the equation of the line.
Q: Can I use the point-slope equation to find the equation of a line that is perpendicular to another line?
A: Yes, you can use the point-slope equation to find the equation of a line that is perpendicular to another line. To do this, you need to find the slope of the perpendicular line and then use the point-slope equation to find the equation of the line.
Q: Can I use the slope-intercept form to find the equation of a line that is perpendicular to another line?
A: Yes, you can use the slope-intercept form to find the equation of a line that is perpendicular to another line. To do this, you need to find the slope of the perpendicular line and then use the slope-intercept form to find the equation of the line.