Graphing A Line With Slope 1/2 Through Point (4, -3)

In this comprehensive guide, we will explore the fundamental concepts of graphing linear equations, with a specific focus on how to graph a line given its slope and a point it passes through. This is a crucial skill in mathematics, with applications across various fields, from basic algebra to advanced calculus and beyond. We will delve into the slope-intercept form of a linear equation, which provides a clear and concise way to understand the relationship between a line's slope, its y-intercept, and the coordinates of points on the line. By understanding these concepts, you'll be able to visualize linear relationships, solve equations, and make predictions based on data.

Our specific task is to graph a line with a slope of 1/2 that passes through the point (4, -3). We will break down this process into manageable steps, starting with a review of the basic principles of graphing lines. This involves understanding the coordinate plane, which is the foundation for plotting points and visualizing relationships between variables. We'll also discuss the concept of slope, which represents the steepness and direction of a line. The slope is a critical parameter that determines how much the line rises or falls for each unit of horizontal change. A positive slope indicates an increasing line, while a negative slope indicates a decreasing line. A slope of zero represents a horizontal line, and an undefined slope represents a vertical line. We will explore how to interpret the slope as a ratio of the change in y to the change in x, and how this ratio can be used to find additional points on the line.

We will then apply the point-slope form of a linear equation, which allows us to directly construct the equation of a line given a point and the slope. This form is particularly useful when we have specific information about a line but not necessarily the y-intercept. The point-slope form is expressed as y - y₁ = m(x - x₁), where (x₁, y₁) is a known point on the line and m is the slope. By substituting the given point (4, -3) and the slope 1/2 into this equation, we can derive the equation of the line in point-slope form. From there, we can transform the equation into slope-intercept form (y = mx + b), which provides the y-intercept (b) of the line, allowing us to easily graph it. The y-intercept is the point where the line crosses the y-axis, and it is a crucial reference point for graphing the line. Understanding how to convert between point-slope form and slope-intercept form is essential for working with linear equations and their graphs.

Before we begin graphing, let's first define the key concepts of slope and points on a coordinate plane. A coordinate plane is a two-dimensional space formed by two perpendicular lines, the x-axis (horizontal) and the y-axis (vertical). Points on the plane are identified by ordered pairs (x, y), where x represents the horizontal distance from the origin (0, 0) and y represents the vertical distance. The origin is the point where the x-axis and y-axis intersect, and it serves as the reference point for all other points on the plane. Understanding the coordinate plane is essential for visualizing linear relationships and graphing equations.

The slope of a line, often denoted by the letter 'm', is a measure of its steepness and direction. It is defined as the change in y (vertical change) divided by the change in x (horizontal change) between any two points on the line. Mathematically, the slope can be calculated using the formula: m = (y₂ - y₁) / (x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are two distinct points on the line. A positive slope indicates that the line rises as we move from left to right, while a negative slope indicates that the line falls. A slope of zero corresponds to a horizontal line, and an undefined slope corresponds to a vertical line.

In our case, we are given a slope of 1/2. This means that for every 2 units we move to the right along the x-axis, the line rises 1 unit along the y-axis. This information will be crucial in finding additional points on the line once we have our initial point. We are also given the point (4, -3), which is a specific location on the coordinate plane where the line passes through. This point provides a fixed reference for drawing the line. The x-coordinate, 4, tells us to move 4 units to the right from the origin, and the y-coordinate, -3, tells us to move 3 units down. Understanding how to plot points on the coordinate plane is essential for visualizing and graphing linear equations.

By combining the information about the slope and the point, we can determine the unique line that satisfies these conditions. The slope provides the direction and steepness of the line, while the point anchors the line to a specific location on the coordinate plane. This combination allows us to graph the line accurately and understand its behavior. The relationship between slope and points is fundamental to understanding linear equations and their graphical representations.

To graph the line, we can use the point-slope form of a linear equation. The point-slope form is a powerful tool for finding the equation of a line when we know a point on the line and its slope. It is expressed as: y - y₁ = m(x - x₁), where (x₁, y₁) is a known point on the line, and m is the slope. This form is derived from the definition of slope and provides a direct way to construct the equation of a line given the necessary information.

In our case, we have the slope m = 1/2 and the point (4, -3). We can substitute these values into the point-slope form to obtain the equation of our line. Substituting x₁ = 4, y₁ = -3, and m = 1/2, we get: y - (-3) = (1/2)(x - 4). Simplifying this equation, we get: y + 3 = (1/2)(x - 4). This is the equation of the line in point-slope form. While this form is useful for finding the equation, it is not the most convenient form for graphing the line directly. To graph the line, it is often easier to convert the equation to slope-intercept form.

The slope-intercept form of a linear equation is given by: y = mx + b, where m is the slope and b is the y-intercept. The y-intercept is the point where the line crosses the y-axis, and it is a crucial reference point for graphing the line. To convert our equation from point-slope form to slope-intercept form, we need to isolate y on one side of the equation. Starting with y + 3 = (1/2)(x - 4), we can distribute the 1/2 on the right side: y + 3 = (1/2)x - 2. Then, we subtract 3 from both sides to isolate y: y = (1/2)x - 2 - 3, which simplifies to: y = (1/2)x - 5. This is the equation of the line in slope-intercept form.

From the slope-intercept form, we can clearly identify the slope as 1/2 and the y-intercept as -5. This means that the line crosses the y-axis at the point (0, -5). Having the slope and the y-intercept makes it straightforward to graph the line. We can plot the y-intercept on the coordinate plane and then use the slope to find additional points on the line. For example, since the slope is 1/2, we can move 2 units to the right from the y-intercept and 1 unit up to find another point on the line. By connecting these points, we can accurately graph the line.

Now that we have the equation of the line in slope-intercept form, y = (1/2)x - 5, we can proceed with graphing it. The slope-intercept form provides us with two key pieces of information: the slope (m) and the y-intercept (b). In this case, the slope m is 1/2, and the y-intercept b is -5. The y-intercept is the point where the line crosses the y-axis, so we can plot the point (0, -5) as our first point on the graph.

To find additional points on the line, we can use the slope. Remember that the slope is the ratio of the change in y (rise) to the change in x (run). A slope of 1/2 means that for every 2 units we move to the right along the x-axis, the line rises 1 unit along the y-axis. Starting from the y-intercept (0, -5), we can move 2 units to the right and 1 unit up to find another point on the line. This brings us to the point (2, -4). We can repeat this process to find more points: move 2 units to the right and 1 unit up from (2, -4) to reach the point (4, -3), which is the point given in the problem statement. This confirms that our line passes through the given point.

We can also use the slope to find points to the left of the y-intercept. Since the slope is 1/2, we can move 2 units to the left and 1 unit down from the y-intercept (0, -5) to find another point on the line. This brings us to the point (-2, -6). We can repeat this process to find more points on the left side of the graph.

Once we have a few points plotted, we can draw a straight line through them. This line represents the graph of the equation y = (1/2)x - 5. It is important to use a ruler or straightedge to ensure that the line is straight and accurate. The line should extend beyond the points we have plotted, as it represents all possible solutions to the equation. The graph of the line provides a visual representation of the relationship between x and y, and it allows us to quickly determine the value of y for any given value of x, or vice versa.

Another way to graph the line is to start from the given point (4, -3) and use the slope to find other points. This method avoids the need to convert to slope-intercept form and can be more direct in some cases. We are given the point (4, -3) and the slope 1/2. Starting from the point (4, -3), we can use the slope to find other points on the line. The slope of 1/2 tells us that for every 2 units we move to the right along the x-axis, the line rises 1 unit along the y-axis.

So, from the point (4, -3), we can move 2 units to the right to x = 6 and 1 unit up to y = -2. This gives us the point (6, -2), which is another point on the line. We can repeat this process to find more points: move 2 units to the right from (6, -2) to x = 8 and 1 unit up to y = -1, giving us the point (8, -1).

Similarly, we can move in the opposite direction. From the point (4, -3), we can move 2 units to the left to x = 2 and 1 unit down to y = -4. This gives us the point (2, -4), which is also on the line. We can repeat this process to find more points on the left side of the given point. This method is particularly useful when we are only interested in graphing the line and do not need to find the equation of the line in slope-intercept form.

Once we have a few points plotted using this method, we can draw a straight line through them to graph the line. This method reinforces the understanding of slope as the ratio of rise over run and provides a visual way to construct the line. By starting from the given point and using the slope, we can accurately graph the line without explicitly using the slope-intercept form. This alternative approach highlights the flexibility and interconnectedness of different methods in solving mathematical problems.

In conclusion, we have successfully graphed the line with a slope of 1/2 passing through the point (4, -3). We explored two main methods for graphing the line: using the slope-intercept form and using the slope directly from the given point. Both methods provide accurate ways to visualize the linear relationship defined by the slope and the point. Understanding how to graph lines given their slope and a point is a fundamental skill in mathematics, with applications in various fields.

We started by reviewing the basic concepts of the coordinate plane and the definition of slope. The slope is a measure of the steepness and direction of a line, and it is defined as the change in y divided by the change in x. The point provides a fixed location on the coordinate plane where the line passes through. By combining the information about the slope and the point, we can uniquely determine the line.

We then applied the point-slope form of a linear equation, which allowed us to construct the equation of the line directly from the given point and slope. The point-slope form is expressed as y - y₁ = m(x - x₁), where (x₁, y₁) is a known point on the line, and m is the slope. We substituted the given values into this equation and then converted it to slope-intercept form, y = mx + b, which provides the y-intercept of the line. The slope-intercept form is particularly useful for graphing lines because it clearly shows the slope and the y-intercept.

Finally, we used the slope and the y-intercept to graph the line. We plotted the y-intercept on the coordinate plane and then used the slope to find additional points on the line. By connecting these points, we created an accurate graph of the line. We also explored an alternative method of graphing the line by starting from the given point and using the slope to find other points. This method reinforces the understanding of slope as the ratio of rise over run and provides a visual way to construct the line.

By mastering these techniques, you can confidently graph linear equations and understand the relationships between variables represented by linear functions. The ability to visualize linear relationships is crucial for solving mathematical problems and applying mathematical concepts in real-world scenarios. This skill will serve as a foundation for more advanced topics in mathematics and other disciplines.