Graphing $y=-\frac{2}{3}x+1$ A Step-by-Step Guide

Introduction: Understanding Linear Equations

In the realm of mathematics, linear equations play a fundamental role. These equations, characterized by their straight-line graphs, are essential for modeling real-world phenomena and solving various problems across different disciplines. Graphing linear equations is a core skill in algebra, and it provides a visual representation of the relationship between two variables. In this article, we will delve into the process of graphing the linear equation $y=-\frac{2}{3}x+1$, providing a step-by-step guide and exploring the underlying concepts. This guide aims to provide a detailed explanation, ensuring that readers understand not just the 'how' but also the 'why' behind each step. Mastering the art of graphing linear equations opens doors to more advanced mathematical concepts and their applications in the real world. Let's embark on this journey to understand and graph the equation $y=-\frac{2}{3}x+1$ effectively.

Decoding the Equation: Slope-Intercept Form

The equation $y=-\frac{2}{3}x+1$ is presented in slope-intercept form, a format that provides immediate insights into the line's characteristics. The slope-intercept form is generally expressed as $y=mx+b$, where 'm' represents the slope and 'b' represents the y-intercept. In our equation, $y=-\frac{2}{3}x+1$, we can identify that the slope (m) is $-\frac{2}{3}$ and the y-intercept (b) is 1. Understanding the slope and y-intercept is crucial for graphing the line accurately. The slope, $-\frac{2}{3}$, tells us how steeply the line rises or falls. A negative slope indicates that the line decreases as we move from left to right. The fraction $-\frac{2}{3}$ means that for every 3 units we move to the right on the x-axis, the line goes down 2 units on the y-axis. The y-intercept, 1, is the point where the line crosses the y-axis, which is the point (0, 1). This form provides a straightforward method for visualizing and plotting linear equations, making it an indispensable tool in algebra. By recognizing and interpreting the slope and y-intercept, we can quickly sketch the line and understand its behavior.

Plotting the First Point: The Y-Intercept

The first step in graphing the equation $y=-\frac{2}{3}x+1$ is to plot the y-intercept. As we identified earlier, the y-intercept is the point where the line intersects the y-axis. In this case, the y-intercept is 1, which corresponds to the point (0, 1) on the coordinate plane. To plot this point, locate the y-axis and find the point where y equals 1. Mark this point clearly on the graph. This point serves as our starting point for drawing the line. The y-intercept is a crucial reference point because it anchors the line on the graph. Without correctly plotting the y-intercept, the entire line will be shifted, leading to an inaccurate representation of the equation. This initial step is simple yet fundamental, setting the stage for using the slope to find additional points and complete the graph. Accuracy in plotting the y-intercept ensures the correctness of the final graph. Therefore, take a moment to carefully locate and mark the point (0, 1) on your graph.

Using the Slope: Finding Additional Points

Once we've plotted the y-intercept, the next step is to use the slope to find additional points on the line. The slope, which we identified as $-\frac{2}{3}$, tells us the rate of change of the line. Remember, the slope is the ratio of the “rise” (vertical change) to the “run” (horizontal change). In this case, a slope of $-\frac{2}{3}$ means that for every 3 units we move to the right (run) along the x-axis, the line goes down 2 units (rise) along the y-axis. Starting from the y-intercept (0, 1), we can apply this slope to find another point. Move 3 units to the right from x=0 to x=3, and then move 2 units down from y=1 to y=-1. This gives us the point (3, -1). Plot this point on the graph. We can repeat this process to find additional points if needed. For instance, starting from (3, -1), move 3 more units to the right (to x=6) and 2 units down (to y=-3), giving us the point (6, -3). Plotting these additional points helps ensure the accuracy of the line and makes it easier to draw a straight line through the points. The slope acts as a guide, allowing us to extend the line and visualize its path across the coordinate plane.

Drawing the Line: Connecting the Points

With at least two points plotted on the graph, we can now draw the line. Using a ruler or a straight edge, carefully connect the points we've plotted: the y-intercept (0, 1) and the point (3, -1), which we found using the slope. Extend the line beyond these points to cover the entire graph. This line represents all the solutions to the equation $y=-\frac{2}{3}x+1$. The line should be straight and pass precisely through the plotted points. If the line doesn't align perfectly with the points, it may indicate an error in plotting the points or applying the slope. Double-check your calculations and the placement of the points. A well-drawn line accurately represents the linear equation, providing a visual depiction of the relationship between x and y. Remember, the line extends infinitely in both directions, so ensure that your drawn line covers a significant portion of the graph. This step is crucial as it visually solidifies the equation, making it easier to understand the behavior and solutions of the linear function.

Verifying the Graph: Checking Additional Points

After drawing the line, it's essential to verify the graph to ensure its accuracy. One way to do this is by choosing an additional point on the line and checking if its coordinates satisfy the equation $y=-\frac{2}{3}x+1$. For example, let's take the point (6, -3), which we identified earlier using the slope. Substitute x=6 into the equation: $y=-\frac{2}{3}(6)+1$. Simplify the equation: $y=-4+1$, which gives us $y=-3$. Since the calculated y-value matches the y-coordinate of the point (6, -3), we can confirm that this point lies on the line. You can also choose another point, such as (-3, 3), and verify it similarly. Substitute x=-3 into the equation: $y=-\frac{2}{3}(-3)+1$. Simplify the equation: $y=2+1$, which gives us $y=3$. This confirms that the point (-3, 3) also lies on the line. By verifying multiple points, we can be confident that the line is drawn correctly and accurately represents the equation. This step is a crucial quality check, ensuring that the graph is a reliable visual representation of the linear equation.

Conclusion: The Visual Representation of $y=-\frac{2}{3}x+1$

In conclusion, we have successfully graphed the linear equation $y=-\frac{2}{3}x+1$ by following a step-by-step process. We began by understanding the slope-intercept form of the equation, identifying the slope as $-\frac{2}{3}$ and the y-intercept as 1. We then plotted the y-intercept (0, 1) on the graph and used the slope to find additional points, such as (3, -1) and (6, -3). By connecting these points with a straight line, we created a visual representation of the equation. To ensure accuracy, we verified the graph by checking if the coordinates of additional points on the line satisfied the equation. This entire process demonstrates how a linear equation can be transformed into a visual form, making it easier to understand and analyze. Graphing linear equations is a fundamental skill in mathematics, and mastering it opens the door to more advanced concepts and applications. The graph of $y=-\frac{2}{3}x+1$ is a straight line that slopes downwards from left to right, crossing the y-axis at the point (0, 1). This visual representation allows us to quickly understand the relationship between x and y as defined by the equation.