If \[$ F(x) \$\] Is Parallel To The Line \[$ Y = \frac{3}{4}x + \frac{2}{5} \$\], Which Function Could Be \[$ F(x) \$\]?A. \[$ F(x) = \frac{3}{4}x + 9 \$\]B. \[$ F(x) = -\frac{4}{3}x + 9 \$\]C. \[$ F(x) = 9x

Feb 27, 2025 by ADMIN 208 views

If ${$ f(x) $}$ is parallel to the line ${$ y = \frac{3}{4}x + \frac{2}{5} $}$, which function could be ${$ f(x) $}$?

Understanding Parallel Lines in Mathematics

In mathematics, two lines are considered parallel if they have the same slope but different y-intercepts. The slope of a line is a measure of how steep it is, and it can be calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In the case of the given line, ${$ y = \frac{3}{4}x + \frac{2}{5} $}$, the slope is ${$ \frac{3}{4} $}$.

The Concept of Slope in Parallel Lines

To determine if a function ${$ f(x) $}$ is parallel to the given line, we need to check if it has the same slope. The slope of a linear function ${$ f(x) = mx + b $}$ is given by the coefficient of x, which is m. If the slope of ${$ f(x) $}$ is the same as the slope of the given line, then the two lines are parallel.

Analyzing the Given Options

Let's analyze the given options to determine which one has the same slope as the given line.

Option A: ${$ f(x) = \frac{3}{4}x + 9 $}$

$Option A: ${$ f(x) = \frac{3}{4}x + 9 \$}$$

The slope of this function is ${$ \frac{3}{4} $}$, which is the same as the slope of the given line. Therefore, this function is parallel to the given line.

Option B: ${$ f(x) = -\frac{4}{3}x + 9 $}$

The slope of this function is ${$ -\frac{4}{3} $}$, which is not the same as the slope of the given line. Therefore, this function is not parallel to the given line.

Option C: ${$ f(x) = 9x + 9 $}$

The slope of this function is ${$ 9 $}$, which is not the same as the slope of the given line. Therefore, this function is not parallel to the given line.

Conclusion

Based on the analysis of the given options, the function that could be ${$ f(x) $}$ and is parallel to the line ${$ y = \frac3}{4}x + \frac{2}{5} $}$ is Option A ${$ f(x) = \frac{3{4}x + 9 $}$.

Key Takeaways

Two lines are parallel if they have the same slope but different y-intercepts.
The slope of a line is a measure of how steep it is, and it can be calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.
To determine if a function ${$ f(x) $}$ is parallel to a given line, we need to check if it has the same slope.

Additional Resources

For more information on parallel lines and linear functions, you can refer to the following resources:

Khan Academy: Linear Equations and Functions
Mathway: Parallel Lines and Slope
Wolfram MathWorld: Parallel Lines and Slope
Q&A: Understanding Parallel Lines and Linear Functions

Frequently Asked Questions

In the previous article, we discussed the concept of parallel lines and linear functions. Here are some frequently asked questions and answers to help you better understand the topic.

Q: What is the definition of parallel lines in mathematics?

A: In mathematics, two lines are considered parallel if they have the same slope but different y-intercepts.

Q: How do I calculate the slope of a line?

A: The slope of a line can be calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. It can also be calculated using the formula: ${$ m = \frac{y_2 - y_1}{x_2 - x_1} $}$, where ${$ (x_1, y_1) $}$ and ${$ (x_2, y_2) $}$ are two points on the line.

Q: What is the difference between a slope and a y-intercept?

A: The slope of a line is a measure of how steep it is, while the y-intercept is the point where the line intersects the y-axis. The slope is a measure of the rate of change of the line, while the y-intercept is a measure of the starting point of the line.

Q: How do I determine if a function is parallel to a given line?

A: To determine if a function is parallel to a given line, you need to check if it has the same slope as the given line. If the slope of the function is the same as the slope of the given line, then the two lines are parallel.

Q: What is the significance of parallel lines in real-world applications?

A: Parallel lines have many real-world applications, including architecture, engineering, and physics. For example, in architecture, parallel lines are used to design buildings and structures that are aesthetically pleasing and functional. In engineering, parallel lines are used to design systems and mechanisms that are efficient and effective. In physics, parallel lines are used to describe the motion of objects and the behavior of forces.

Q: Can parallel lines intersect?

A: No, parallel lines cannot intersect. By definition, parallel lines are lines that lie in the same plane and never intersect.

Q: Can parallel lines be perpendicular?

A: No, parallel lines cannot be perpendicular. Perpendicular lines are lines that intersect at a right angle, while parallel lines are lines that lie in the same plane and never intersect.

Q: Can parallel lines have the same y-intercept?

A: Yes, parallel lines can have the same y-intercept. However, they must also have the same slope in order to be parallel.

Q: Can parallel lines have different slopes?

A: No, parallel lines cannot have different slopes. By definition, parallel lines are lines that have the same slope but different y-intercepts.