Graphing The Equation Y=(2/5)x+3 By Plotting Points

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To effectively graph an equation by plotting points, especially one like y = \frac{2}{5}x + 3, a systematic approach is crucial. This guide provides a detailed, step-by-step method to ensure accurate and insightful graphical representation. Understanding how to graph linear equations is a fundamental skill in algebra, serving as the building block for more complex mathematical concepts. By the end of this guide, you'll not only be able to graph this specific equation but also grasp the underlying principles applicable to various linear equations.

Step 1: Understanding the Equation

The given equation, y = \frac{2}{5}x + 3, is in slope-intercept form, which is a particularly useful format for graphing linear equations. The slope-intercept form is generally written as y = mx + b, where m represents the slope of the line and b represents the y-intercept. In our equation:

  • Slope (m): The slope is \frac{2}{5}. This means that for every 5 units you move to the right along the x-axis, you move 2 units up along the y-axis. The slope indicates the steepness and direction of the line. A positive slope, like ours, indicates that the line rises from left to right.
  • Y-intercept (b): The y-intercept is 3. This is the point where the line crosses the y-axis. In coordinate terms, this point is (0, 3). The y-intercept serves as our starting point when graphing the line.

Recognizing these components is the first step towards accurately graphing the equation. The slope and y-intercept provide crucial information about the line's position and orientation on the coordinate plane. With this understanding, we can proceed to selecting appropriate points for plotting.

Step 2: Choosing Points for Plotting

To graph a linear equation, you need at least two points. However, plotting three points is recommended to ensure accuracy and avoid errors. The third point acts as a check to verify that all points lie on the same line. When choosing points, it's strategic to select values for x that will result in integer values for y, as these are easier to plot accurately. Given the equation y = \frac{2}{5}x + 3, selecting multiples of 5 for x will eliminate the fraction, making the calculation simpler.

Here are three points we can choose:

  1. x = 0: This is a straightforward choice as it directly gives us the y-intercept. Substituting x = 0 into the equation:

    • y = \frac{2}{5}(0) + 3 = 3
    • So, the first point is (0, 3).

  2. x = 5: Choosing x = 5 eliminates the fraction in the equation:

    • y = \frac{2}{5}(5) + 3 = 2 + 3 = 5
    • Thus, the second point is (5, 5).

  3. x = -5: Selecting a negative value helps to spread the points across the graph, providing a more comprehensive view of the line:

    • y = \frac{2}{5}(-5) + 3 = -2 + 3 = 1
    • Hence, the third point is (-5, 1).

By strategically selecting these points, we ensure that the resulting y values are integers, making them easier to plot on the graph. Now that we have our points, we can move on to plotting them on the coordinate plane.

Step 3: Plotting the Points on the Coordinate Plane

Now that we have three coordinate pairs: (0, 3), (5, 5), and (-5, 1), we can plot them on the coordinate plane. The coordinate plane consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). The point where these axes intersect is called the origin, which is represented by the coordinates (0, 0).

Here's how to plot each point:

  1. (0, 3):

    • Start at the origin (0, 0).
    • Since the x-coordinate is 0, you do not move left or right along the x-axis.
    • Move 3 units up along the y-axis.
    • Mark the point. This is the y-intercept, as expected.

  2. (5, 5):

    • Start at the origin (0, 0).
    • Move 5 units to the right along the x-axis.
    • Move 5 units up along the y-axis.
    • Mark the point.

  3. (-5, 1):

    • Start at the origin (0, 0).
    • Move 5 units to the left along the x-axis (since the x-coordinate is negative).
    • Move 1 unit up along the y-axis.
    • Mark the point.

After plotting these points, you should see them forming a linear pattern. If one of the points seems out of line, it's a good idea to recheck your calculations and plotting. Accurate plotting is crucial for obtaining the correct graph of the equation. With the points plotted, the next step is to draw the line that connects them.

Step 4: Drawing the Line

Once the points are plotted accurately on the coordinate plane, the next step is to draw a straight line that passes through all of them. This line represents all the possible solutions to the equation y = \frac{2}{5}x + 3. Use a ruler or straightedge to ensure the line is drawn accurately.

Here’s how to draw the line:

  1. Align the Ruler: Place the ruler so that it aligns with at least two of the plotted points. Ideally, it should align with all three points. If it doesn't, double-check your point plotting, as this could indicate an error in your calculations or plotting.
  2. Draw the Line: Draw a straight line through the points, extending it beyond the outermost points on both ends. This indicates that the line continues infinitely in both directions, representing all possible solutions to the equation.
  3. Arrows on the Ends: Add arrows on both ends of the line to emphasize that it extends indefinitely. These arrows are a standard convention in graphing linear equations.

After drawing the line, take a moment to visually inspect it. Does it match the slope and y-intercept you identified in Step 1? The line should cross the y-axis at y = 3 (the y-intercept) and should rise 2 units for every 5 units it moves to the right (the slope of \frac{2}{5}). If the line appears to deviate from these characteristics, it's wise to review your work for any potential errors.

Step 5: Verifying the Graph

Verifying the graph is an essential step to ensure accuracy. There are several methods to verify that the graphed line correctly represents the equation y = \frac{2}{5}x + 3. This step helps catch any errors in calculation or plotting.

Here are a few methods to verify the graph:

  1. Check the Y-intercept: Ensure the line crosses the y-axis at the correct y-intercept, which is (0, 3) in this case. Visually inspect the graph to confirm that the line indeed intersects the y-axis at this point.

  2. Verify the Slope: Use the plotted points to calculate the slope of the line. The slope can be calculated using the formula:

    • m = \frac{y_2 - y_1}{x_2 - x_1}

    Using the points (0, 3) and (5, 5):

    • m = \frac{5 - 3}{5 - 0} = \frac{2}{5}

    This confirms that the slope of the graphed line matches the slope in the equation.

  3. Substitute Another Point: Choose another point on the line (other than the ones you used for plotting) and substitute its coordinates into the equation. For instance, the point (10, 7) appears to lie on the line. Let’s substitute these values into the equation:

    • y = \frac{2}{5}x + 3
    • 7 = \frac{2}{5}(10) + 3
    • 7 = 4 + 3
    • 7 = 7

    Since the equation holds true, this further verifies the accuracy of the graph.

By using these verification methods, you can be confident that the graphed line accurately represents the equation y = \frac{2}{5}x + 3. This step is a crucial part of the graphing process and should not be overlooked.

Conclusion

Graphing the equation y = \frac{2}{5}x + 3 by plotting points involves a systematic process of understanding the equation, selecting appropriate points, plotting them on the coordinate plane, drawing the line, and verifying the graph. Each step is crucial for ensuring accuracy and a solid understanding of linear equations. Mastering this method provides a strong foundation for more advanced topics in mathematics.

By following this guide, you have not only learned how to graph this specific equation but also gained a valuable skill applicable to graphing various linear equations. Remember, practice is key to mastering this skill. The more you graph equations by plotting points, the more proficient you will become. Happy graphing!