Which Linear Function Represents The Line Given By The Point-slope Equation $y + 1 = -3(x - 5$\]?A. $f(x) = -3x - 6$ B. $f(x) = -3x - 4$ C. $f(x) = -3x + 16$ D. $f(x) = -3x + 14$
Introduction
In mathematics, a point-slope equation is a type of linear equation that represents a line in the Cartesian plane. It is defined by a point on the line and the slope of the line. The point-slope equation is given by the formula:
y - y1 = m(x - x1)
where (x1, y1) 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.
Converting Point-Slope Equation to Linear Function
To convert a point-slope equation to a linear function, we need to isolate y on one side of the equation. We can do this by adding y1 to both sides of the equation and then subtracting m(x - x1) from both sides.
Let's consider the point-slope equation given in the problem:
y + 1 = -3(x - 5)
We can start by isolating y on one side of the equation:
y = -3(x - 5) - 1
Now, we can simplify the equation by distributing the negative sign to the terms inside the parentheses:
y = -3x + 15 - 1
y = -3x + 14
Comparing with the Given Options
Now that we have the linear function in the form f(x) = mx + b, we can compare it with the given options:
A. f(x) = -3x - 6 B. f(x) = -3x - 4 C. f(x) = -3x + 16 D. f(x) = -3x + 14
We can see that option D matches our linear function.
Conclusion
In this article, we have explored how to convert a point-slope equation to a linear function in the form f(x) = mx + b. We have used the given point-slope equation y + 1 = -3(x - 5) as an example and have shown that the linear function f(x) = -3x + 14 represents the line given by the point-slope equation.
Key Takeaways
- A point-slope equation is a type of linear equation that represents a line in the Cartesian plane.
- The point-slope equation is given by the formula y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line.
- To convert a point-slope equation to a linear function, we need to isolate y on one side of the equation.
- The linear function in the form f(x) = mx + b represents the line given by the point-slope equation.
Practice Problems
- Convert the point-slope equation y - 2 = 2(x - 3) to a linear function in the form f(x) = mx + b.
- Convert the point-slope equation y + 3 = -2(x - 1) to a linear function in the form f(x) = mx + b.
- Convert the point-slope equation y - 4 = 3(x - 2) to a linear function in the form f(x) = mx + b.
Answer Key
- f(x) = 2x - 4
- f(x) = -2x + 1
- f(x) = 3x - 2
Frequently Asked Questions (FAQs) About Point-Slope Equations and Linear Functions =====================================================================================
Q: What is a point-slope equation?
A: A point-slope equation is a type of linear equation that represents a line in the Cartesian plane. It is defined by a point on the line and the slope of the line.
Q: How is a point-slope equation written?
A: A point-slope equation is written in the form y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line.
Q: What is the slope of a line in a point-slope equation?
A: The slope of a line in a point-slope equation is the value of m in the equation y - y1 = m(x - x1).
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 y on one side of the equation. You can do this by adding y1 to both sides of the equation and then subtracting m(x - x1) from both sides.
Q: What is the y-intercept of a line in a linear function?
A: The y-intercept of a line in a linear function is the value of b in the equation f(x) = mx + b.
Q: How do I find the y-intercept of a line in a point-slope equation?
A: To find the y-intercept of a line in a point-slope equation, you need to substitute x = 0 into the equation and solve for y.
Q: What is the difference between a point-slope equation and a linear function?
A: A point-slope equation is a type of linear equation that represents a line in the Cartesian plane, while a linear function is a mathematical expression that represents a line in the form f(x) = mx + b.
Q: Can a point-slope equation have a slope of zero?
A: Yes, a point-slope equation can have a slope of zero. In this case, the equation will be of the form y = c, where c is a constant.
Q: Can a point-slope equation have a slope of infinity?
A: No, a point-slope equation cannot have a slope of infinity. The slope of a line in a point-slope equation is always a finite value.
Q: How do I graph a line represented by a point-slope equation?
A: To graph a line represented by a point-slope equation, you need to find the slope of the line and the coordinates of a point on the line. You can then use this information to draw the line on a coordinate plane.
Q: Can a point-slope equation have multiple solutions?
A: No, a point-slope equation has only one solution. The equation represents a single line in the Cartesian plane.
Q: Can a point-slope equation be used to represent a non-linear relationship?
A: No, a point-slope equation can only be used to represent a linear relationship. If you need to represent a non-linear relationship, you will need to use a different type of equation, such as a quadratic equation or a polynomial equation.