Key Components of Graphing a Line

Oct 13, 2025 by magicalceylontours.com 62 views

Understanding the Problem

Graphing a line with a given slope and point is a fundamental concept in mathematics, particularly in algebra and geometry. In this article, we will focus on graphing a line with a slope of 1/4 that contains the point (6,3). To begin, let's understand the key components involved in graphing a line.

Key Components of Graphing a Line

When graphing a line, there are two main components to consider: the slope and the y-intercept. The slope, denoted by the letter 'm', represents the rate of change of the line. It can be calculated as the ratio of the vertical change (rise) to the horizontal change (run). In this case, the slope is given as 1/4, which means that for every 4 units moved horizontally, the line will rise by 1 unit.

The y-intercept, denoted by the letter 'b', represents the point where the line intersects the y-axis. It is the value of 'y' when 'x' is equal to 0. In this case, we are given a point (6,3), which means that the line passes through this point.

Graphing the Line

To graph the line, we can use the point-slope form of a linear equation, which is given by:

y - y1 = m(x - x1)

where (x1, y1) is the given point and m is the slope. In this case, the point is (6,3) and the slope is 1/4. Plugging in these values, we get:

y - 3 = (1/4)(x - 6)

To simplify this equation, we can multiply both sides by 4 to eliminate the fraction:

4(y - 3) = x - 6

Expanding the left-hand side, we get:

4y - 12 = x - 6

Adding 12 to both sides, we get:

4y = x + 6

Dividing both sides by 4, we get:

y = (1/4)x + 1.5

This is the equation of the line in slope-intercept form, where the slope is 1/4 and the y-intercept is 1.5.

Graphing the Line

To graph the line, we can use the equation y = (1/4)x + 1.5. We can start by plotting the y-intercept, which is the point (0,1.5). From this point, we can draw a line with a slope of 1/4, which means that for every 4 units moved horizontally, the line will rise by 1 unit.

We can also use the given point (6,3) to help us graph the line. Since the point lies on the line, we can draw a line through this point with a slope of 1/4.

Conclusion

Graphing a line with a given slope and point is a straightforward process that involves using the point-slope form of a linear equation. By plugging in the given values, we can simplify the equation and graph the line using the slope-intercept form. In this article, we graphed a line with a slope of 1/4 that contains the point (6,3). We hope that this article has provided a clear understanding of the key components involved in graphing a line and has demonstrated how to graph a line using the point-slope form.

Real-World Applications

Graphing a line with a given slope and point has numerous real-world applications in fields such as engineering, economics, and physics. For example, in engineering, graphing a line can be used to model the behavior of a system, such as the motion of a particle or the flow of a fluid. In economics, graphing a line can be used to model the relationship between two variables, such as the demand for a product and its price.

Example 1: Modeling the Motion of a Particle

Suppose we want to model the motion of a particle that is moving with a constant acceleration of 2 m/s^2. We can use the equation of motion to graph the position of the particle as a function of time. The equation of motion is given by:

s(t) = s0 + v0t + (1/2)at^2

where s(t) is the position of the particle at time t, s0 is the initial position, v0 is the initial velocity, and a is the acceleration. In this case, the acceleration is 2 m/s^2, and the initial position and velocity are 0. Plugging in these values, we get:

s(t) = 0 + 0t + (1/2)(2)t^2

Simplifying the equation, we get:

s(t) = t^2

This is the equation of the line in slope-intercept form, where the slope is 2 and the y-intercept is 0. We can graph this line using the equation s(t) = t^2.

Example 2: Modeling the Demand for a Product

Suppose we want to model the demand for a product as a function of its price. We can use the equation of demand to graph the demand as a function of the price. The equation of demand is given by:

Q = a - bp

where Q is the quantity demanded, a is the intercept, and b is the slope. In this case, the intercept is 100 and the slope is 2. Plugging in these values, we get:

Q = 100 - 2p

This is the equation of the line in slope-intercept form, where the slope is -2 and the y-intercept is 100. We can graph this line using the equation Q = 100 - 2p.

Conclusion

Graphing a line with a given slope and point is a fundamental concept in mathematics that has numerous real-world applications. By using the point-slope form of a linear equation, we can simplify the equation and graph the line using the slope-intercept form. In this article, we graphed a line with a slope of 1/4 that contains the point (6,3). We hope that this article has provided a clear understanding of the key components involved in graphing a line and has demonstrated how to graph a line using the point-slope form.

References

"Graphing Lines" by Math Open Reference
"Point-Slope Form" by Math Is Fun
"Slope-Intercept Form" by Mathway

Understanding the Problem

Key Components of Graphing a Line

Graphing the Line

To graph the line, we can use the point-slope form of a linear equation, which is given by:

y - y1 = m(x - x1)

where (x1, y1) is the given point and m is the slope. In this case, the point is (6,3) and the slope is 1/4. Plugging in these values, we get:

y - 3 = (1/4)(x - 6)

To simplify this equation, we can multiply both sides by 4 to eliminate the fraction:

4(y - 3) = x - 6

Expanding the left-hand side, we get:

4y - 12 = x - 6

Adding 12 to both sides, we get:

4y = x + 6

Dividing both sides by 4, we get:

y = (1/4)x + 1.5

This is the equation of the line in slope-intercept form, where the slope is 1/4 and the y-intercept is 1.5.

Graphing the Line

We can also use the given point (6,3) to help us graph the line. Since the point lies on the line, we can draw a line through this point with a slope of 1/4.

Conclusion

Q&A

Q: What is the slope of the line?

A: The slope of the line is 1/4.

Q: What is the y-intercept of the line?

A: The y-intercept of the line is 1.5.

Q: How do I graph the line?

A: To graph the line, you can use the equation y = (1/4)x + 1.5. Start by plotting the y-intercept, which is the point (0,1.5). From this point, draw a line with a slope of 1/4.

Q: What is the point-slope form of a linear equation?

A: The point-slope form of a linear equation is given by:

y - y1 = m(x - x1)

where (x1, y1) is the given point and m is the slope.

Q: How do I simplify the equation?

A: To simplify the equation, you can multiply both sides by 4 to eliminate the fraction.

Q: What is the equation of the line in slope-intercept form?

A: The equation of the line in slope-intercept form is y = (1/4)x + 1.5.

Q: How do I use the given point to help me graph the line?

A: Since the point lies on the line, you can draw a line through this point with a slope of 1/4.

Q: What are the key components involved in graphing a line?

A: The key components involved in graphing a line are the slope and the y-intercept.

Q: How do I calculate the slope?

A: The slope can be calculated as the ratio of the vertical change (rise) to the horizontal change (run).

Q: What is the y-intercept?

A: The y-intercept is the point where the line intersects the y-axis.

Q: How do I graph the line using the point-slope form?

A: To graph the line using the point-slope form, you can plug in the given values and simplify the equation.

Q: What are the real-world applications of graphing a line?

A: Graphing a line has numerous real-world applications in fields such as engineering, economics, and physics.

Q: How do I model the motion of a particle?

A: To model the motion of a particle, you can use the equation of motion, which is given by:

s(t) = s0 + v0t + (1/2)at^2

Q: How do I model the demand for a product?

A: To model the demand for a product, you can use the equation of demand, which is given by:

Q = a - bp

Conclusion

References

"Graphing Lines" by Math Open Reference
"Point-Slope Form" by Math Is Fun
"Slope-Intercept Form" by Mathway