Graphing A Line Through (4, 3) With Slope 1/2 A Step-by-Step Guide

Jul 8, 2025 by Admin 67 views

Graphing a Line Passing Through (4, 3) with Slope 1/2: A Comprehensive Guide

To graph a line that passes through the point $(4, 3)$ and has a slope of $\frac{1}{2}$ , we will delve into the fundamental concepts of linear equations and graphing techniques. This comprehensive guide will not only walk you through the step-by-step process but also enhance your understanding of slope-intercept form, point-slope form, and the significance of slope and y-intercept in defining a line. Let's embark on this mathematical journey to master the art of graphing lines.

Understanding the Basics

Before we dive into the graphing process, it is crucial to grasp the fundamental concepts that underpin linear equations and their graphical representation. The slope of a line, often denoted by 'm', quantifies the steepness and direction of the line. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. A positive slope indicates an upward slant, while a negative slope indicates a downward slant. A slope of zero represents a horizontal line, and an undefined slope corresponds to a vertical line. The y-intercept, denoted by 'b', is the point where the line intersects the y-axis. It represents the value of y when x is zero. Understanding these concepts forms the bedrock for accurately graphing linear equations.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as $y = mx + b$ , where 'm' is the slope and 'b' is the y-intercept. This form provides a direct and intuitive way to identify the slope and y-intercept of a line. For instance, in the equation $y = 2x + 3$ , the slope is 2, and the y-intercept is 3. This means that the line rises 2 units for every 1 unit it moves to the right and intersects the y-axis at the point (0, 3). The slope-intercept form is particularly useful for graphing lines because it readily provides the two key parameters needed to define a line: its steepness and its starting point on the y-axis.

Point-Slope Form

The point-slope form of a linear equation is expressed as $y - y_1 = m(x - x_1)$ , where 'm' is the slope and $(x_1, y_1)$ is a point on the line. This form is especially useful when you know a point on the line and its slope. It allows you to write the equation of the line directly without having to first find the y-intercept. For example, if a line passes through the point (1, 2) and has a slope of -1, the point-slope form of the equation would be $y - 2 = -1(x - 1)$ . This form can then be rearranged into slope-intercept form or standard form if desired. The point-slope form provides a flexible approach to defining a line when a specific point and slope are known.

Step-by-Step Graphing Process

Now, let's apply these concepts to graph a line that passes through the point $(4, 3)$ and has a slope of $\frac{1}{2}$ .

Step 1: Plot the Given Point

The first step is to plot the given point $(4, 3)$ on the Cartesian plane. This point serves as a reference point for drawing the line. Locate the point on the coordinate plane where the x-coordinate is 4 and the y-coordinate is 3. Mark this point clearly, as it will be one of the points through which your line will pass. This initial point is crucial as it anchors the line in the correct location on the graph.

Step 2: Use the Slope to Find Another Point

The slope of $\frac{1}{2}$ tells us that for every 2 units we move to the right (run), we move 1 unit up (rise). Starting from the point $(4, 3)$ , move 2 units to the right along the x-axis and then 1 unit up along the y-axis. This will lead you to a new point. To illustrate, moving 2 units to the right from x = 4 takes us to x = 6, and moving 1 unit up from y = 3 takes us to y = 4. Therefore, the new point is $(6, 4)$ . This process utilizes the fundamental definition of slope to find a second point on the line, which is essential for accurately drawing the line.

Step 3: Draw the Line

Now that we have two points, $(4, 3)$ and $(6, 4)$ , we can draw a straight line through these points. Use a ruler or straightedge to ensure the line is drawn accurately. Extend the line beyond the two points to cover a reasonable portion of the graph. The line should pass precisely through the two plotted points, and its straightness is critical for a correct graphical representation. This step visually connects the two points, creating the line that represents the given conditions.

Alternative Method: Using the Slope-Intercept Form

Let's explore an alternative method using the slope-intercept form ( $y = mx + b$ ) to graph the line.

Step 1: Write the Equation

We know the slope ( $m = \frac{1}{2}$ ) and a point $(4, 3)$ . We can use the point-slope form to find the equation of the line:

$y - y_1 = m(x - x_1)$

Substitute the values:

$y - 3 = \frac{1}{2}(x - 4)$

Step 2: Convert to Slope-Intercept Form

Simplify the equation to slope-intercept form ( $y = mx + b$ ):

$y - 3 = \frac{1}{2}x - 2$

$y = \frac{1}{2}x + 1$

Step 3: Identify the Y-Intercept

From the slope-intercept form, we can see that the y-intercept ( $b$ ) is 1. This means the line intersects the y-axis at the point $(0, 1)$ . The y-intercept is a crucial piece of information for graphing, as it provides a fixed point on the line.

Step 4: Plot the Y-Intercept

Plot the y-intercept $(0, 1)$ on the Cartesian plane. This is the point where the line crosses the vertical y-axis. Marking this point accurately is essential for correctly positioning the line on the graph.

Step 5: Use the Slope to Find Another Point

Similar to the previous method, use the slope of $\frac{1}{2}$ to find another point. Starting from the y-intercept $(0, 1)$ , move 2 units to the right and 1 unit up. This brings us to the point $(2, 2)$ . This step applies the definition of slope to locate a second point on the line, providing another reference for drawing the line.

Step 6: Draw the Line

Draw a straight line through the y-intercept $(0, 1)$ and the point $(2, 2)$ . Use a ruler or straightedge for accuracy. Extend the line beyond the two points to cover a reasonable portion of the graph. Ensuring the line passes precisely through these two points and is drawn straight is crucial for a correct graphical representation.

Key Takeaways

Graphing a line requires a solid understanding of the concepts of slope and y-intercept. The slope determines the steepness and direction of the line, while the y-intercept indicates where the line crosses the y-axis. Both the point-slope form and slope-intercept form of a linear equation provide valuable tools for graphing lines. The point-slope form is particularly useful when you know a point on the line and its slope, while the slope-intercept form directly reveals the slope and y-intercept. By mastering these techniques, you can confidently graph any linear equation.

Importance of Accurate Graphing

Accurate graphing is essential in various fields, including mathematics, physics, engineering, and economics. Graphs provide a visual representation of relationships between variables, making it easier to analyze and interpret data. In mathematics, accurate graphs are crucial for solving equations, understanding functions, and visualizing geometric concepts. In physics, graphs are used to represent motion, forces, and energy. In engineering, graphs are used to design structures, analyze circuits, and model systems. In economics, graphs are used to represent supply and demand, track economic indicators, and forecast trends. The ability to accurately graph lines and other functions is a fundamental skill that is applicable across a wide range of disciplines.

Practice Problems

To solidify your understanding, try graphing the following lines:

Passes through $(1, 2)$ with a slope of $3$ .
Passes through $(-2, 1)$ with a slope of $-\frac{1}{2}$ .
Passes through $(0, -3)$ with a slope of $2$ .
Passes through $(5, 0)$ with a slope of $-\frac{2}{3}$ .

By working through these practice problems, you will reinforce your skills in graphing lines and gain confidence in applying the concepts of slope and y-intercept. Consistent practice is key to mastering this fundamental mathematical skill.

Conclusion

In conclusion, graphing a line that passes through the point $(4, 3)$ and has a slope of $\frac{1}{2}$ involves understanding the fundamental concepts of slope, y-intercept, point-slope form, and slope-intercept form. By following the step-by-step process outlined in this guide, you can accurately graph the line and enhance your understanding of linear equations. Whether you choose to use the point-slope method or the slope-intercept method, the key is to accurately plot points and draw a straight line through them. Accurate graphing is a valuable skill that has applications in various fields, making it an essential tool for problem-solving and analysis. Remember to practice regularly to solidify your understanding and build confidence in your graphing abilities. With a solid grasp of these concepts, you'll be well-equipped to tackle more advanced mathematical challenges.