Graphing Y = -2/3x + 1 A Step-by-Step Guide

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In mathematics, graphing linear equations is a fundamental skill that provides a visual representation of the relationship between two variables. A linear equation, when graphed, produces a straight line, hence the name. Understanding how to graph these equations is crucial for solving various mathematical problems and interpreting real-world scenarios. This article will delve into the specifics of graphing the equation $y = -\frac{2}{3}x + 1$, providing a step-by-step guide and exploring the underlying concepts.

Understanding the Slope-Intercept Form

The equation $y = -\frac{2}{3}x + 1$ is presented in the slope-intercept form, which is a standard way of writing linear equations. The slope-intercept form is expressed as $y = mx + b$, where:

• $y$ represents the vertical coordinate on the graph.
• $x$ represents the horizontal coordinate on the graph.
• $m$ represents the slope of the line, indicating its steepness and direction.
• $b$ represents the y-intercept, the point where the line crosses the y-axis.

In our equation, $y = -\frac{2}{3}x + 1$, we can identify the slope ($m$) and the y-intercept ($b$) directly:

• Slope ($m$) = -\frac{2}{3}
• Y-intercept ($b$) = 1

Interpreting the Slope

The slope of a line is a crucial concept in understanding its behavior. It quantifies the rate at which the line rises or falls as we move from left to right along the x-axis. A positive slope indicates that the line is increasing, while a negative slope indicates that the line is decreasing. The magnitude of the slope reflects the steepness of the line; a larger magnitude signifies a steeper line, while a smaller magnitude indicates a flatter line.

In our case, the slope is $-\frac{2}{3}$. This negative value tells us that the line slopes downward from left to right. The fraction $\frac{2}{3}$ signifies the change in $y$ for every unit change in $x$. Specifically, for every 3 units we move to the right along the x-axis, the line decreases by 2 units along the y-axis. This relationship is often described as "rise over run," where the rise is the vertical change and the run is the horizontal change. Here, the rise is -2 (indicating a decrease), and the run is 3.

Identifying the Y-intercept

The y-intercept is the point where the line intersects the y-axis. It is the value of $y$ when $x$ is equal to 0. In the slope-intercept form ($y = mx + b$), the y-intercept is represented by the constant term $b$.

In our equation, $y = -\frac{2}{3}x + 1$, the y-intercept is 1. This means that the line crosses the y-axis at the point (0, 1). The y-intercept serves as a crucial starting point for graphing the line, as it provides a fixed point on the coordinate plane.

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

Now that we have a clear understanding of the slope and y-intercept, we can proceed with graphing the equation $y = -\frac{2}{3}x + 1$. Here's a step-by-step guide to help you through the process:

Step 1: Plot the Y-intercept

Begin by plotting the y-intercept on the coordinate plane. As we identified earlier, the y-intercept for the equation $y = -\frac{2}{3}x + 1$ is 1. This means the line crosses the y-axis at the point (0, 1). Locate this point on the graph and mark it clearly. This point will serve as our starting point for drawing the line.

Step 2: Use the Slope to Find Another Point

The slope provides us with the information needed to find another point on the line. Recall that the slope is $-\frac{2}{3}$, which means that for every 3 units we move to the right along the x-axis, the line decreases by 2 units along the y-axis. Starting from the y-intercept (0, 1), we can use this information to find our next point.

Move 3 units to the right along the x-axis (the "run"). This brings us to an x-coordinate of 3. Then, move 2 units down along the y-axis (the "rise"). This brings us to a y-coordinate of -1. The new point we have found is (3, -1). Plot this point on the graph.

Step 3: Draw a Straight Line Through the Points

Now that we have two points on the line – the y-intercept (0, 1) and the point (3, -1) – we can draw a straight line that passes through both of them. Use a ruler or straightedge to ensure the line is accurate. Extend the line beyond the two points to show that it continues infinitely in both directions. This line represents the graph of the equation $y = -\frac{2}{3}x + 1$.

Step 4: Verify the Graph

To ensure the accuracy of our graph, it's always a good idea to verify it by choosing another point on the line and plugging its coordinates into the original equation. If the equation holds true, then our graph is likely correct.

For instance, let's choose the point (6, -3), which appears to lie on the line. Substitute $x = 6$ and $y = -3$ into the equation $y = -\frac{2}{3}x + 1$:

$-3 = -\frac{2}{3}(6) + 1$

$-3 = -4 + 1$

$-3 = -3$

Since the equation holds true, we can be confident that our graph accurately represents the equation $y = -\frac{2}{3}x + 1$.

Alternative Method: Using Two Points

While the slope-intercept method is highly efficient, another way to graph a linear equation is by finding any two points that satisfy the equation. This method is particularly useful when the equation is not in slope-intercept form or when you prefer to work with specific values of $x$.

Step 1: Choose Two Values for $x$

Select any two values for $x$ that are easy to work with. For simplicity, you might choose $x = 0$ and $x = 3$. These values will give us two distinct points on the line.

Step 2: Calculate the Corresponding Values for $y$

Substitute each chosen value of $x$ into the equation $y = -\frac{2}{3}x + 1$ and solve for $y$.

• When $x = 0$:

  $y = -\frac{2}{3}(0) + 1$

  $y = 1$

  This gives us the point (0, 1).

• When $x = 3$:

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

  $y = -2 + 1$

  $y = -1$

  This gives us the point (3, -1).

Step 3: Plot the Points and Draw the Line

Plot the two points (0, 1) and (3, -1) on the coordinate plane. Then, use a ruler or straightedge to draw a straight line that passes through both points. This line represents the graph of the equation $y = -\frac{2}{3}x + 1$.

This method provides an alternative approach to graphing linear equations, ensuring you have a solid understanding of the relationship between the equation and its graphical representation.

Common Mistakes to Avoid

When graphing linear equations, several common mistakes can lead to inaccurate graphs. Being aware of these pitfalls can help you avoid them and ensure your graphs are correct.

Misinterpreting the Slope

One of the most common errors is misinterpreting the slope. Remember that the slope is the ratio of the change in $y$ (rise) to the change in $x$ (run). A negative slope indicates that the line slopes downward from left to right, while a positive slope indicates an upward slope. Confusing the sign or the order of the rise and run can result in an incorrectly graphed line.

For example, if you misinterpret the slope $-\frac{2}{3}$ as $\frac{3}{2}$, you will draw a line that slopes upward instead of downward, and the steepness will also be incorrect.

Incorrectly Plotting the Y-intercept

The y-intercept is the point where the line crosses the y-axis, and it is crucial to plot this point accurately. An error in plotting the y-intercept will shift the entire line, resulting in an incorrect graph. Make sure to identify the y-intercept correctly from the equation and plot it precisely on the coordinate plane.

Not Using a Straightedge

A linear equation, by definition, produces a straight line when graphed. Drawing a line freehand can introduce inaccuracies and make it difficult to read the graph correctly. Always use a ruler or straightedge to draw a precise line through the points you have plotted.

Not Extending the Line

The line representing a linear equation extends infinitely in both directions. It's important to extend the line beyond the points you have plotted to indicate this infinite extent. A line segment that stops at the plotted points does not fully represent the equation.

Not Verifying the Graph

To ensure the accuracy of your graph, it's always a good practice to verify it by choosing an additional point on the line and substituting its coordinates into the original equation. If the equation holds true, your graph is likely correct. If not, you should re-examine your steps to identify and correct any errors.

Real-World Applications of Graphing Linear Equations

Graphing linear equations is not just a theoretical exercise; it has numerous practical applications in various fields. Understanding how to represent linear relationships graphically can help in problem-solving and decision-making in real-world scenarios.

Business and Economics

In business and economics, linear equations are used to model various relationships, such as cost, revenue, and profit. For example, a company's total cost can be represented as a linear function of the number of units produced, where the slope represents the variable cost per unit and the y-intercept represents the fixed costs. Graphing these equations can help businesses visualize their cost structure and determine break-even points.

Science and Engineering

In science and engineering, linear equations are used to describe relationships between physical quantities. For instance, the relationship between distance, speed, and time can be represented by a linear equation. Similarly, in physics, Ohm's Law (V = IR) is a linear equation that relates voltage, current, and resistance. Graphing these relationships allows scientists and engineers to analyze data, make predictions, and design systems efficiently.

Everyday Life

Linear equations also find applications in everyday life. For example, calculating the cost of a taxi ride, where there is a fixed initial charge plus a charge per mile, can be modeled using a linear equation. Similarly, converting temperatures between Celsius and Fahrenheit involves a linear relationship. Understanding these relationships and how to graph them can help in making informed decisions and solving practical problems.

Conclusion

Graphing linear equations is a fundamental skill in mathematics with wide-ranging applications. By understanding the slope-intercept form, identifying the slope and y-intercept, and following a step-by-step graphing process, you can accurately represent linear equations on a coordinate plane. Whether using the slope-intercept method or plotting two points, the key is to grasp the underlying concepts and avoid common mistakes. Mastering this skill not only enhances your mathematical proficiency but also equips you to tackle real-world problems that involve linear relationships. In the case of $y = -\frac{2}{3}x + 1$, the negative slope and the y-intercept of 1 provide a clear guide to graphing a line that slopes downward from left to right, crossing the y-axis at the point (0, 1).